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Journal of Inequalities and Applications

Erschienen in:

Open Access 01.12.2013 | Research

Inequalities involving Dresher variance mean

verfasst von: Jiajin Wen, Tianyong Han, Sui Sun Cheng

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2013

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Abstract

Let p be a real density function defined on a compact subset Ω of

R^{m}

, and let

E (f, p) = \int_{Ω} p f d ω

be the expectation of f with respect to the density function p. In this paper, we define a one-parameter extension

{Var}_{γ} (f, p)

of the usual variance

Var (f, p) = E (f^{2}, p) - E^{2} (f, p)

of a positive continuous function f defined on Ω. By means of this extension, a two-parameter mean

V_{r, s} (f, p)

, called the Dresher variance mean, is then defined. Their properties are then discussed. In particular, we establish a Dresher variance mean inequality

{min}_{t \in Ω} {f (t)} \leq V_{r, s} (f, p) \leq {max}_{t \in Ω} {f (t)}

, that is to say, the Dresher variance mean

V_{r, s} (f, p)

is a true mean of f. We also establish a Dresher-type inequality

V_{r, s} (f, p) \geq V_{r^{*}, s^{*}} (f, p)

under appropriate conditions on r, s,

r^{*}

s^{*}

; and finally, a V-E inequality

V_{r, s} (f, p) \geq {(\frac{s}{r})}^{1 / (r - s)} E (f, p)

that shows that

V_{r, s} (f, p)

can be compared with

E (f, p)

. We are also able to illustrate the uses of these results in space science.

MSC:26D15, 26E60, 62J10.

Authors’ original file for figure 1

Authors’ original file for figure 2

Authors’ original file for figure 3

Authors’ original file for figure 4

Electronic supplementary material

The online version of this article (doi:10.1186/1029-242X-2013-366) contains supplementary material, which is available to authorized users.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in this paper. All authors read and approved the final manuscript.

1 Introduction and main results

As indicated in the monograph [1], the concept of mean is basic in the theory of inequalities and its applications. Indeed, there are many inequalities involving different types of mean in [1‐18], and a great number of them have been used in mathematics and other natural sciences.

Dresher in [14], by means of moment space techniques, proved the following inequality: If

ρ \geq 1 \geq σ \geq 0

f, g \geq 0

, and ϕ is a distribution function, then

{[\frac{\int {(f + g)}^{ρ} d ϕ}{\int {(f + g)}^{σ} d ϕ}]}^{1 / (ρ - σ)} \leq {(\frac{\int f^{ρ} d ϕ}{\int f^{σ} d ϕ})}^{1 / (ρ - σ)} + {(\frac{\int g^{ρ} d ϕ}{\int g^{σ} d ϕ})}^{1 / (ρ - σ)} .

This result is referred to as Dresher’s inequality by Daskin [15], Beckenbach and Bellman [16] (§24 in Ch. 1) and Hu [18] (p.21). Note that if we define

\bar{E} (f, ϕ) = \int f d ϕ,

and

D_{r, s} (f, ϕ) = {(\frac{\bar{E} (f^{r}, ϕ)}{\bar{E} (f^{s}, ϕ)})}^{1 / (r - s)}, r, s \in R,

(1)

then the above inequality can be rewritten as

D_{ρ, σ} (f + g, ϕ) \leq D_{ρ, σ} (f, ϕ) + D_{ρ, σ} (g, ϕ) .

D_{r, s} (f, ϕ)

is the well-known Dresher mean of the function f (see [10, 11, 14‐18]), which involves two parameters r and s and has applications in the theory of probability.

However, variance is also a crucial quantity in probability and statistics theory. It is, therefore, of interest to establish inequalities for various variances as well. In this paper, we introduce generalized ‘variances’ and establish several inequalities involving them. Although we may start out under a more general setting, for the sake of simplicity, we choose to consider the variance of a continuous function f with respect to a weight function p (including probability densities) defined on a closed and bounded domain Ω in

R^{m}

instead of a distribution function.

More precisely, unless stated otherwise, in all later discussions, let Ω be a fixed, nonempty, closed and bounded domain in

R^{m}

and let

p : Ω \to (0, \infty)

be a fixed function which satisfies

\int_{Ω} p d ω = 1

. For any continuous function

f : Ω \to R

, we write

E (f, p) = \int_{Ω} p f d ω,

which may be regarded as the weighted mean of the function f with respect to the weight function p.

Recall that the standard variance (see [12] and [19]) of a random variable f with respect to a density function p is

Var (f, p) = E (f^{2}, p) - E^{2} (f, p) .

We may, however, generalize this to the γ-variance of the function

f : Ω \to (0, \infty)

defined by

{Var}_{γ} (f, p) = {\begin{matrix} \frac{2}{γ (γ - 1)} [E (f^{γ}, p) - E^{γ} (f, p)], & γ \neq 0, 1, \\ 2 [ln E (f, p) - E (ln f, p)], & γ = 0, \\ 2 [E (f ln f, p) - E (f, p) ln E (f, p)], & γ = 1 . \end{matrix}

According to this definition, we know that γ-variance

{Var}_{γ} (f, p)

is a functional of the function

f : Ω \to (0, \infty)

and

p : Ω \to (0, \infty)

, and such a definition is compatible with the generalized integral means studied elsewhere (see, e.g., [1‐6]). Indeed, according to the power mean inequality (see, e.g., [1‐9]), we may see that

{Var}_{γ} (f, p) \geq 0, \forall γ \in R .

Let

f : Ω \to (0, \infty)

and

g : Ω \to (0, \infty)

be two continuous functions. We define

{Cov}_{γ} (f, g) = E {[f^{γ} - E^{γ} (f, p)] \circ [g^{γ} - E^{γ} (g, p)], p}

is the γ-covariance of the function

f : Ω \to (0, \infty)

and the function

g : Ω \to (0, \infty)

, where

γ \in R

and the function

a \circ b : R^{2} \to R

is defined as follows:

a \circ b = {\begin{matrix} \sqrt{a b}, & a \neq b, a b \geq 0, \\ - \sqrt{- a b}, & a \neq b, a b < 0, \\ a, & a = b . \end{matrix}

According to this definition, we get

{Cov}_{0} (f, g) \equiv 0, {Cov}_{1} (f, f) \equiv 0

and

{Var}_{γ} (f, p) = {\begin{matrix} \frac{2}{γ (γ - 1)} {Cov}_{γ} (f, f), & γ \neq 0, 1, \\ {lim}_{γ \to 0} \frac{2}{γ (γ - 1)} {Cov}_{γ} (f, f), & γ = 0, \\ {lim}_{γ \to 1} \frac{2}{γ (γ - 1)} {Cov}_{γ} (f, f), & γ = 1 . \end{matrix}

If we define

{Abscov}_{γ} (f, g) = E {| [f^{γ} - E^{γ} (f, p)] \circ [g^{γ} - E^{γ} (g, p)] |, p}

is the γ-absolute covariance of the function

f : Ω \to (0, \infty)

and the function

g : Ω \to (0, \infty)

, then we have that

| \frac{{Cov}_{γ} (f, g)}{\sqrt{{Abscov}_{γ} (f, f)} \sqrt{{Abscov}_{γ} (g, g)}} | \leq 1

for

γ \neq 0

by the Cauchy inequality

| \int_{Ω} p f g d ω | \leq \sqrt{\int_{Ω} p f^{2} d ω} \sqrt{\int_{Ω} p g^{2} d ω} .

Therefore, we can define the γ-correlation coefficient of the function

f : Ω \to (0, \infty)

and the function

g : Ω \to (0, \infty)

as follows:

ρ_{γ} (f, g) = {\begin{matrix} \frac{{Cov}_{γ} (f, g)}{\sqrt{{Abscov}_{γ} (f, f)} \sqrt{{Abscov}_{γ} (g, g)}}, & γ \neq 0, \\ {lim}_{γ \to 0} \frac{{Cov}_{γ} (f, g)}{\sqrt{{Abscov}_{γ} (f, f)} \sqrt{{Abscov}_{γ} (g, g)}}, & γ = 0, \end{matrix}

where

ρ_{γ} (f, g) \in [- 1, 1]

By means of

{Var}_{γ} (f, p)

, we may then define another two-parameter mean. This new two-parameter mean

V_{r, s} (f, p)

will be called the Dresher variance mean of the function f. It is motivated by (1) and [10, 11, 14‐18] and is defined as follows. Given

(r, s) \in R^{2}

and the continuous function

f : Ω \to (0, \infty)

. If f is a constant function defined by

f (x) = c

for any

x \in Ω

, we define the functional

V_{r, s} (f, p) = c,

and if f is not a constant function, we define the functional

V_{r, s} (f, p) = {\begin{matrix} {[\frac{{Var}_{r} (f, p)}{{Var}_{s} (f, p)}]}^{1 / (r - s)}, & r \neq s, \\ exp [\frac{E (f^{r} ln f, p) - E^{r} (f, p) ln E (f, p)}{E (f^{r}, p) - E^{r} (f, p)} - (\frac{1}{r} + \frac{1}{r - 1})], & r = s \neq 0, 1, \\ exp {\frac{E ({ln}^{2} f, p) - {ln}^{2} E (f, p)}{2 [E (ln f, p) - ln E (f, p)]} + 1}, & r = s = 0, \\ exp [\frac{E (f {ln}^{2} f, p) - E (f, p) {ln}^{2} E (f, p)}{2 [E (f ln f, p) - E (f, p) ln E (f, p)]} - 1], & r = s = 1 . \end{matrix}

Since the function

f : Ω \to (0, \infty)

is continuous, we know that the functions

p f^{γ}

and

p f^{γ} ln f

are integrable for any

γ \in R

. Thus

{Var}_{γ} (f, p)

and

V_{r, s} (f, p)

are well defined. Since

\begin{matrix} lim_{γ \to γ^{*}} {Var}_{γ} (f, p) = {Var}_{γ^{*}} (f, p), \\ lim_{(r, s) \to (r^{*}, s^{*})} V_{r, s} (f, p) = V_{r^{*}, s^{*}} (f, p), \end{matrix}

{Var}_{γ} (f, p)

is continuous with respect to

γ \in R

and

V_{r, s} (f, p)

continuous with respect to

(r, s) \in R^{2}

We will explain why we are concerned with our one-parameter variance

{Var}_{γ} (f, p)

and the two-parameter mean

V_{r, s} (f, p)

by illustrating their uses in statistics and space science.

Before doing so, we first state three main theorems of our investigations.

Theorem 1 (Dresher variance mean inequality)

For any continuous function

f : Ω \to (0, \infty)

, we have

min_{t \in Ω} {f (t)} \leq V_{r, s} (f, p) \leq max_{t \in Ω} {f (t)} .

(2)

Theorem 2 (Dresher-type inequality)

Let the function

f : Ω \to (0, \infty)

be continuous. If

(r, s) \in R^{2}

(r^{*}, s^{*}) \in R^{2}

max {r, s} \geq max {r^{*}, s^{*}}

and

min {r, s} \geq min {r^{*}, s^{*}}

, then

V_{r, s} (f, p) \geq V_{r^{*}, s^{*}} (f, p) .

(3)

Theorem 3 (V-E inequality)

For any continuous function

f : Ω \to (0, \infty)

, we have

V_{r, s} (f, p) \geq {(\frac{s}{r})}^{\frac{1}{r - s}} E (f, p),

(4)

moreover, the coefficient

{(\frac{s}{r})}^{\frac{1}{r - s}}

in (4) is the best constant.

From Theorem 1, we know that

V_{r, s} (f, p)

is a certain mean value of f. Theorem 2 is similar to the well-known Dresher inequality stated in Lemma 3 below (see, e.g., [2], p.74 and [10, 11]). By Theorem 2, we see that

V_{r, s} (f, p)

and

V_{r^{*}, s^{*}} (f, p)

can be compared under appropriate conditions on r, s,

r^{*}

s^{*}

. Theorem 3 states a connection of

V_{r, s} (f, p)

with the weighted mean

E (f, p)

Let Ω be a fixed, nonempty, closed and bounded domain in

R^{m}

, and let

p = p (X)

be the density function with support in Ω for the random vector

X = (x_{1}, x_{2}, \dots, x_{m})

. For any function

f : Ω \to (0, \infty)

, the mean of the random variable

f (X)

E [f (X)] = \int_{Ω} p f d ω = E (f, p);

moreover, its variance is

Var [f (X)] = E [f^{2} (X)] - E^{2} [f (X)] = \int_{Ω} p {f - E [f (X)]}^{2} d ω = {Var}_{2} (f, p) .

Therefore,

{Var}_{γ} [f (X)] = {Var}_{γ} (f, p), γ \in R

may be regarded as a γ-variance and

V_{r, s} [f (X)] = V_{r, s} (f, p), (r, s) \in R^{2}

may be regarded as a Dresher variance mean of the random variable

f (X)

Note that by Theorem 1, we have

min_{X \in Ω} {f (X)} \leq V_{r, s} [f (X)] \leq max_{X \in Ω} {f (X)}, (r, s) \in R^{2};

(5)

and by Theorem 2, we see that for any r, s,

r^{*}

s^{*} \in R

such that

max {r, s} \geq max {r^{*}, s^{*}} and min {r, s} \geq min {r^{*}, s^{*}},

then

V_{r, s} [f (X)] \geq V_{r^{*}, s^{*}} [f (X)];

(6)

and by Theorem 3, if

r > s \geq 1

, then

\frac{{Var}_{r} [f (X)]}{{Var}_{s} [f (X)]} \geq \frac{s}{r} E^{r - s} [f (X)],

(7)

where the coefficient

s / r

is the best constant.

In the above results, Ω is a closed and bounded domain of

R^{m}

. However, we remark that our results still hold if Ω is an unbounded domain of

R^{m}

or some values of f are 0, as long as the integrals in Theorems 1-3 are convergent. Such extended results can be obtained by standard techniques in real analysis by applying continuity arguments and Lebesgue’s dominated convergence theorem, and hence we need not spell out all the details in this paper.

2 Proof of Theorem 1

For the sake of simplicity, we employ the following notations. Let n be an integer greater than or equal to 2, and let

N_{n} = {1, 2, \dots, n}

. For real n-vectors

x = (x_{1}, \dots, x_{n})

and

p = (p_{1}, \dots, p_{n})

, the dot product of p and x is denoted by

A (x, p) = p \cdot x = \sum_{i = 1}^{n} p_{i} x_{i}

, where

p \in S^{n}

and

S^{n} = {p \in {[0, \infty)}^{n} ∣ \sum_{i = 1}^{n} p_{i} = 1}

S^{n}

is an

(n - 1)

-dimensional simplex. If ϕ is a real function of a real variable, for the sake of convenience, we set the vector function

ϕ (x) = (ϕ (x_{1}), ϕ (x_{2}), \dots, ϕ (x_{n})) .

Suppose that

p \in S^{n}

and

γ, r, s \in R

. If

x \in {(0, \infty)}^{n}

, then

{Var}_{γ} (x, p) = {\begin{matrix} \frac{2}{γ (γ - 1)} [A (x^{γ}, p) - A^{γ} (x, p)], & γ \neq 0, 1, \\ 2 [ln A (x, p) - A (ln x, p)], & γ = 0, \\ 2 [A (x ln x, p) - A (x, p) ln A (x, p)], & γ = 1 \end{matrix}

is called the γ-variance of the vector x with respect to p. If

x \in {(0, \infty)}^{n}

is a constant n-vector, then we define

V_{r, s} (x, p) = x_{1},

while if x is not a constant vector (i.e., there exist

i, j \in N_{n}

such that

x_{i} \neq x_{j}

), then we define

V_{r, s} (x, p) = {\begin{matrix} {[\frac{{Var}_{r} (x, p)}{{Var}_{s} (x, p)}]}^{\frac{1}{r - s}}, & r \neq s, \\ exp [\frac{A (x^{r} ln x, p) - A^{r} (x, p) ln A (x, p)}{A (x^{r}, p) - A^{r} (x, p)} - (\frac{1}{r} + \frac{1}{r - 1})], & r = s \neq 0, 1, \\ exp {\frac{A ({ln}^{2} x, p) - {ln}^{2} A (x, p)}{2 [A (ln x, p) - ln A (x, p)]} + 1}, & r = s = 0, \\ exp {\frac{A (x {ln}^{2} x, p) - A (x, p) {ln}^{2} A (x, p)}{2 [A (x ln x, p) - A (x, p) ln A (x, p)]} - 1}, & r = s = 1 . \end{matrix}

V_{r, s} (x, p)

is called the Dresher variance mean of the vector x.

Clearly,

{Var}_{γ} (x, p)

is nonnegative and is continuous with respect to

γ \in R

V_{r, s} (x, p) = V_{s, r} (x, p)

and is also continuous with respect to

(r, s)

R^{2}

Lemma 1 Let I be a real interval. Suppose the function

ϕ : I \to R

C^{(2)}

, i.e., twice continuously differentiable. If

x \in I^{n}

and

p \in S^{n}

, then

A (ϕ (x), p) - ϕ (A (x, p)) = \sum_{1 \leq i < j \leq n} p_{i} p_{j} {\int \int_{Φ} ϕ^{''} [w_{i, j} (x, p, t_{1}, t_{2})] d t_{1} d t_{2}} {(x_{i} - x_{j})}^{2},

(8)

where Φ is the triangle

{(t_{1}, t_{2}) \in {[0, \infty)}^{2} ∣ t_{1} + t_{2} \leq 1}

and

w_{i, j} (x, p, t_{1}, t_{2}) = t_{1} x_{i} + t_{2} x_{j} + (1 - t_{1} - t_{2}) A (x, p) .

Proof Note that

\begin{aligned} \int \int_{Φ} ϕ^{''} [w_{i, j} (x, p, t_{1}, t_{2})] d t_{1} d t_{2} \\ = {\int_{0}}^{1} d t_{1} \int_{0}^{1 - t_{1}} ϕ^{''} [w_{i, j} (x, p, t_{1}, t_{2})] d t_{2} \\ = \frac{1}{x_{j} - A (x, p)} \int_{0}^{1} d t_{1} {\int_{0}}^{1 - t_{1}} ϕ^{''} [w_{i, j} (x, p, t_{1}, t_{2})] d [w_{i, j} (x, p, t_{1}, t_{2})] \\ = \frac{1}{x_{j} - A (x, p)} \int_{0}^{1} d t_{1} ϕ^{'} [t_{1} x_{i} + t_{2} x_{j} + (1 - t_{1} - t_{2}) A (x, p)] |_{0}^{1 - t_{1}} \\ = \frac{1}{x_{j} - A (x, p)} \int_{0}^{1} {ϕ^{'} [t_{1} x_{i} + (1 - t_{1}) x_{j}] - ϕ^{'} [t_{1} x_{i} + (1 - t_{1}) A (x, p)]} d t_{1} \\ = \frac{1}{x_{j} - A (x, p)} {\frac{ϕ [t_{1} x_{i} + (1 - t_{1}) x_{j}]}{x_{i} - x_{j}} - \frac{ϕ [t_{1} x_{i} + (1 - t_{1}) A (x, p)]}{x_{i} - A (x, p)}} |_{0}^{1} \\ = \frac{1}{x_{j} - A (x, p)} [\frac{ϕ (x_{i}) - f (x_{j})}{x_{i} - x_{j}} - \frac{ϕ (x_{i}) - ϕ (A (x, p))}{x_{i} - A (x, p)}] \\ = \frac{1}{(x_{i} - x_{j}) [x_{j} - A (x, p)] [x_{i} - A (x, p)]} | \begin{array}{c} ϕ (A (x, p)) & A (x, p) & 1 \\ ϕ (x_{i}) & x_{i} & 1 \\ ϕ (x_{j}) & x_{j} & 1 \end{array} | . \end{aligned}

Hence,

\begin{aligned} \sum_{1 \leq i < j \leq n} p_{i} p_{j} {\int \int_{Φ} ϕ^{''} [t_{1} x_{i} + t_{2} x_{j} + (1 - t_{1} - t_{2}) A (x, p)] d t_{1} d t_{2}} {(x_{i} - x_{j})}^{2} \\ = \sum_{1 \leq i < j \leq n} p_{i} p_{j} \frac{x_{i} - x_{j}}{[x_{j} - A (x, p)] [x_{i} - A (x, p)]} | \begin{array}{c} ϕ (A (x, p)) & A (x, p) & 1 \\ ϕ (x_{i}) & x_{i} & 1 \\ ϕ (x_{j}) & x_{j} & 1 \end{array} | \\ = \frac{1}{2} \sum_{1 \leq i, j \leq n} p_{i} p_{j} [\frac{1}{x_{j} - A (x, p)} - \frac{1}{x_{i} - A (x, p)}] | \begin{array}{c} ϕ (A (x, p)) & A (x, p) & 1 \\ ϕ (x_{i}) & x_{i} & 1 \\ ϕ (x_{j}) & x_{j} & 1 \end{array} | \\ = \frac{1}{2} [\sum_{1 \leq i, j \leq n} p_{i} p_{j} \frac{1}{x_{j} - A (x, p)} | \begin{array}{c} ϕ (A (x, p)) & A (x, p) & 1 \\ ϕ (x_{i}) & x_{i} & 1 \\ ϕ (x_{j}) & x_{j} & 1 \end{array} | \\ - \sum_{1 \leq i, j \leq n} p_{i} p_{j} \frac{1}{x_{i} - A (x, p)} | \begin{array}{c} ϕ (A (x, p)) & A (x, p) & 1 \\ ϕ (x_{i}) & x_{i} & 1 \\ ϕ (x_{j}) & x_{j} & 1 \end{array} |] \\ = \frac{1}{2} [\sum_{j = 1}^{n} \frac{p_{j}}{x_{j} - A (x, p)} \sum_{i = 1}^{n} | \begin{array}{c} ϕ (A (x, p)) & A (x, p) & 1 \\ p_{i} ϕ (x_{i}) & p_{i} x_{i} & p_{i} \\ ϕ (x_{j}) & x_{j} & 1 \end{array} | \\ - \sum_{i = 1}^{n} \frac{p_{i}}{x_{i} - A (x, p)} \sum_{j = 1}^{n} | \begin{array}{c} ϕ (A (x, p)) & A (x, p) & 1 \\ ϕ (x_{i}) & x_{i} & 1 \\ p_{j} ϕ (x_{j}) & p_{j} x_{j} & p_{j} \end{array} |] \\ = \frac{1}{2} [\sum_{j = 1}^{n} \frac{p_{j}}{x_{j} - A (x, p)} | \begin{array}{c} ϕ (A (x, p)) & A (x, p) & 1 \\ \sum_{i = 1}^{n} p_{i} ϕ (x_{i}) & \sum_{i = 1}^{n} p_{i} x_{i} & \sum_{i = 1}^{n} p_{i} \\ ϕ (x_{j}) & x_{j} & 1 \end{array} | \\ - \sum_{i = 1}^{n} \frac{p_{i}}{x_{i} - A (x, p)} | \begin{array}{c} ϕ (A (x, p)) & A (x, p) & 1 \\ ϕ (x_{i}) & x_{i} & 1 \\ \sum_{j = 1}^{n} p_{j} ϕ (x_{j}) & \sum_{j = 1}^{n} p_{j} x_{j} & \sum_{j = 1}^{n} p_{j} \end{array} |] \\ = \frac{1}{2} [\sum_{j = 1}^{n} \frac{p_{j}}{x_{j} - A (x, p)} | \begin{array}{c} ϕ (A (x, p)) & A (x, p) & 1 \\ A (ϕ (x), p) & A (x, p) & 1 \\ ϕ (x_{j}) & x_{j} & 1 \end{array} | \\ - \sum_{i = 1}^{n} \frac{p_{i}}{x_{i} - A (x, p)} | \begin{array}{c} ϕ (A (x, p)) & A (x, p) & 1 \\ ϕ (x_{i}) & x_{i} & 1 \\ A (ϕ (x), p) & A (x, p) & 1 \end{array} |] \\ = \frac{1}{2} [\sum_{j = 1}^{n} \frac{p_{j}}{x_{j} - A (x, p)} | \begin{array}{c} ϕ (A (x, p)) - A (ϕ (x), p) & 0 & 0 \\ A (ϕ (x), p) & A (x, p) & 1 \\ ϕ (x_{j}) & x_{j} & 1 \end{array} | \\ - \sum_{i = 1}^{n} \frac{p_{i}}{x_{i} - A (x, p)} | \begin{array}{c} ϕ (A (x, p)) - A (ϕ (x), p) & 0 & 0 \\ ϕ (x_{i}) & x_{i} & 1 \\ A (ϕ (x), p) & A (x, p) & 1 \end{array} |] \\ = \frac{1}{2} {\sum_{j = 1}^{n} \frac{p_{j}}{x_{j} - A (x, p)} [ϕ (A (x, p)) - A (ϕ (x), p)] [A (x, p) - x_{j}] \\ - \sum_{i = 1}^{n} \frac{p_{i}}{x_{i} - A (x, p)} [ϕ (A (x, p)) - A (ϕ (x), p)] [x_{i} - A (x, p)]} \\ = \frac{1}{2} {\sum_{j = 1}^{n} p_{j} [A (ϕ (x), p) - ϕ (A (x, p))] + \sum_{i = 1}^{n} p_{i} [A (ϕ (x), p) - ϕ (A (x, p))]} \\ = A (ϕ (x), p) - ϕ (A (x, p)) . \end{aligned}

Therefore, (8) holds. The proof is complete. □

Remark 1 The well-known Jensen inequality can be described as follows [20‐22]: If the function

ϕ : I \to R

satisfies

ϕ^{''} (t) \geq 0

for all t in the interval I, then for any

x \in I^{n}

and

p \in S^{n}

, we have

A (ϕ (x), p) \geq ϕ (A (x, p)) .

(9)

The above proof may be regarded as a constructive proof of (9).

Remark 2 We remark that the Dresher variance mean

V_{r, s} (x, p)

extends the variance mean

V_{r, 2} (x, p)

(see [2], p.664, and [13, 21]), and Lemma 1 is a generalization of (2.23) of [19].

Lemma 2 If

x \in {(0, \infty)}^{n}

p \in S^{n}

and

(r, s) \in R^{2}

, then

min {x} = min {x_{1}, \dots, x_{n}} \leq V_{r, s} (x, p) \leq max {x_{1}, \dots, x_{n}} = max {x} .

(10)

Proof If x is a constant vector, our assertion is clearly true. Let x be a non-constant vector, that is, there exist

i, j \in N_{n}

such that

x_{i} \neq x_{j}

. Note that

V_{r, s} (x, p) = V_{s, r} (x, p)

and

V_{r, s} (x, p)

is continuous with respect to

(r, s)

R^{2}

, we may then assume that

r (r - 1) s (s - 1) \neq 0 and r - s > 0 .

In (8), let

ϕ : (0, \infty) \to R

be defined by

ϕ (t) = t^{γ}

, where

γ (γ - 1) \neq 0

. Then we obtain

{Var}_{γ} (x, p) = 2 \sum_{1 \leq i < j \leq n} p_{i} p_{j} {\int \int_{Φ} {[w_{i, j} (x, p, t_{1}, t_{2})]}^{γ - 2} d t_{1} d t_{2}} {(x_{i} - x_{j})}^{2} .

(11)

Since

min {x} = min {x_{1}, \dots, x_{n}} \leq w_{i, j} (x, p, t_{1}, t_{2}) \leq max {x_{1}, \dots, x_{n}} = max {x},

(12)

by (11), (12),

r - s > 0

and the fact that

\begin{aligned} V_{r, s} (x, p) \\ = {[\frac{{Var}_{r} (x, p)}{{Var}_{s} (x)}]}^{\frac{1}{r - s}} \\ = {[\frac{\sum_{1 \leq i < j \leq n} p_{i} p_{j} {(x_{i} - x_{j})}^{2} \int \int_{Φ} w_{i, j}^{r - 2} (x, p, t_{1}, t_{2}) d t_{1} d t_{2}}{\sum_{1 \leq i < j \leq n} p_{i} p_{j} {(x_{i} - x_{j})}^{2} \int \int_{Φ} w_{i, j}^{s - 2} (x, p, t_{1}, t_{2}) d t_{1} d t_{2}}]}^{\frac{1}{r - s}} \\ = {[\frac{\sum_{1 \leq i < j \leq n} p_{i} p_{j} {(x_{i} - x_{j})}^{2} \int \int_{Φ} w_{i, j}^{s - 2} (x, p, t_{1}, t_{2}) \times w_{i, j}^{r - s} (x, p, t_{1}, t_{2}) d t_{1} d t_{2}}{\sum_{1 \leq i < j \leq n} p_{i} p_{j} {(x_{i} - x_{j})}^{2} \int \int_{Φ} w_{i, j}^{s - 2} (x, p, t_{1}, t_{2}) d t_{1} d t_{2}}]}^{\frac{1}{r - s}}, \end{aligned}

(13)

we obtain (10). This concludes the proof. □

We may now turn to the proof of Theorem 1.

Proof First, we may assume that f is a nonconstant function and that

r (r - 1) s (s - 1) \neq 0, r - s > 0 .

Let

T = {Δ Ω_{1}, Δ Ω_{2}, \dots, Δ Ω_{n}}

be a partition of Ω, and let

∥ T ∥ = max_{1 \leq i \leq n} max_{X, Y \in Δ Ω_{i}} {∥ X - Y ∥}

be the ‘norm’ of the partition T, where

∥ X - Y ∥ = \sqrt{(X - Y) \cdot (X - Y)}

is the length of the vector

X - Y

. Pick any

ξ_{i} \in Δ Ω_{i}

for each

i = 1, 2, \dots, n

, set

ξ = (ξ_{1}, ξ_{2}, \dots, ξ_{n}), f (ξ) = (f (ξ_{1}), f (ξ_{2}), \dots, f (ξ_{n})),

and

\bar{p} (ξ) = ({\bar{p}}_{1} (ξ), {\bar{p}}_{2} (ξ), \dots, {\bar{p}}_{n} (ξ)) = \frac{(p (ξ_{1}) | Δ Ω_{1} |, p (ξ_{2}) | Δ Ω_{2} |, \dots, p (ξ_{n}) | Δ Ω_{n} |)}{\sum_{i = 1}^{n} p (ξ_{i}) | Δ Ω_{i} |},

then

lim_{∥ T ∥ \to 0} \sum_{i = 1}^{n} p (ξ_{i}) | Δ Ω_{i} | = \int_{Ω} p d ω = 1,

where

| Δ Ω_{i} |

is the m-dimensional volume of

Δ Ω_{i}

for

i = 1, 2, \dots, n

Furthermore, when

γ (γ - 1) \neq 0

, we have

\begin{array}{rcl} {Var}_{γ} (f, p) & = & \frac{2}{γ (γ - 1)} [lim_{∥ T ∥ \to 0} \sum_{i = 1}^{n} p (ξ_{i}) f^{γ} (ξ_{i}) | Δ Ω_{i} | - {(lim_{∥ T ∥ \to 0} \sum_{i = 1}^{n} p (ξ_{i}) f (ξ_{i}) | Δ Ω_{i} |)}^{γ}] \\ = & \frac{2}{γ (γ - 1)} [(lim_{∥ T ∥ \to 0} \sum_{i = 1}^{n} p (ξ_{i}) | Δ Ω_{i} |) (lim_{∥ T ∥ \to 0} \sum_{i = 1}^{n} {\bar{p}}_{i} (ξ) f^{γ} (ξ_{i})) \\ - {(lim_{∥ T ∥ \to 0} \sum_{i = 1}^{n} p (ξ_{i}) | Δ Ω_{i} |)}^{γ} {(lim_{∥ T ∥ \to 0} \sum_{i = 1}^{n} {\bar{p}}_{i} (ξ) f (ξ_{i}))}^{γ}] \\ = & \frac{2}{γ (γ - 1)} [lim_{∥ T ∥ \to 0} \sum_{i = 1}^{n} {\bar{p}}_{i} (ξ) f^{γ} (ξ_{i}) - {(lim_{∥ T ∥ \to 0} \sum_{i = 1}^{n} {\bar{p}}_{i} (ξ) f (ξ_{i}))}^{γ}] \\ = & lim_{∥ T ∥ \to 0} \frac{2}{γ (γ - 1)} [\sum_{i = 1}^{n} {\bar{p}}_{i} (ξ) f^{r} (ξ_{i}) - {(\sum_{i = 1}^{n} {\bar{p}}_{i} (ξ) f (ξ_{i}))}^{γ}] \\ = & lim_{∥ T ∥ \to 0} {Var}_{γ} (f (ξ), \bar{p} (ξ)) . \end{array}

(14)

By (14), we obtain

\begin{aligned} V_{r, s} (f, p) & = {[\frac{{Var}_{r} (f, p)}{{Var}_{s} (f, p)}]}^{\frac{1}{r - s}} = lim_{∥ T ∥ \to 0} {[\frac{{Var}_{r} (f (ξ), \bar{p} (ξ))}{{Var}_{s} (f (ξ), \bar{p} (ξ))}]}^{\frac{1}{r - s}} \\ = lim_{∥ T ∥ \to 0} V_{r, s} (f (ξ), \bar{p} (ξ)) . \end{aligned}

(15)

By Lemma 2, we have

min {f (ξ)} \leq V_{r, s} (f (ξ), \bar{p} (ξ)) \leq max {f (ξ)} .

(16)

From (15) and (16), we obtain

\begin{array}{rcl} min_{t \in Ω} {f (t)} & = & lim_{∥ T ∥ \to 0} min {f (ξ)} \leq lim_{∥ T ∥ \to 0} V_{r, s} (f (ξ), \bar{p} (ξ)) \\ = & V_{r, s} (f, p) \leq lim_{∥ T ∥ \to 0} max {f (ξ)} = max_{t \in Ω} {f (t)} . \end{array}

This completes the proof of Theorem 1. □

Remark 3 By [21], if the function

ϕ : I \to R

has the property that

ϕ^{''} : I \to R

is a continuous and convex function, then for any

x \in I^{n}

and

p \in S^{n}

, we obtain

\begin{array}{rcl} ϕ^{''} (V_{3, 2} (x, p)) & \leq & \frac{2 [A (ϕ (x), p) - ϕ (A (x, p))]}{{Var}_{2} (x, p)} \\ \leq & \frac{1}{3} {max_{1 \leq i \leq n} {ϕ^{''} (x_{i})} + A (ϕ^{''} (x), p) + ϕ^{''} (A (x, p))} . \end{array}

(17)

Thus, according to the proof of Theorem 1, we may see that: If the function

f : Ω \to (0, \infty)

is continuous and the function

ϕ : f (Ω) \to R

has the property that

ϕ^{''} : f (Ω) \to R

is a continuous convex function, then

\begin{array}{rcl} ϕ^{''} (V_{3, 2} (f, p)) & \leq & \frac{2 [E (ϕ \circ f, p) - ϕ (E (f, p))]}{{Var}_{2} (f, p)} \\ \leq & \frac{1}{3} {max_{t \in Ω} {ϕ^{''} (f (t))} + E (ϕ^{''} \circ f, p) + ϕ^{''} (E (f, p))}, \end{array}

(18)

where

ϕ \circ f

is the composite of ϕ and f. Therefore, the Dresher variance mean

V_{r, s} (f, p)

has a wide mathematical background.

3 Proof of Theorem 2

In this section, we use the same notations as in the previous section. In addition, for fixed

γ, r, s \in R

, if

x \in {(0, \infty)}^{n}

and

p \in S^{n}

, then the γ-order power mean of x with respect to p (see, e.g., [1‐9]) is defined by

M^{[γ]} (x, p) = {\begin{matrix} {[A (x^{γ}, p)]}^{\frac{1}{γ}}, & γ \neq 0, \\ exp A (ln x, p), & γ = 0, \end{matrix}

and the two-parameter Dresher mean of x (see [10, 11]) with respect to p is defined by

D_{r, s} (x, p) = {\begin{matrix} {[\frac{A (x^{r}, p)}{A (x^{s}, p)}]}^{\frac{1}{r - s}}, & r \neq s, \\ exp [\frac{A (x^{s} ln x, p)}{A (x^{s}, p)}], & r = s . \end{matrix}

We have the following well-known power mean inequality [1‐9]: If

α < β

, then

M^{[α]} (x, p) \leq M^{[β]} (x, p) .

We also have the following result (see [2], p.74, and [10, 11]).

Lemma 3 (Dresher inequality)

x \in {(0, \infty)}^{n}

p \in S^{n}

and

(r, s), (r^{*}, s^{*}) \in R^{2}

, then the inequality

D_{r, s} (x, p) \geq D_{r^{*}, s^{*}} (x, p)

(19)

holds if and only if

max {r, s} \geq max {r^{*}, s^{*}} and min {r, s} \geq min {r^{*}, s^{*}} .

(20)

Proof Indeed, if (20) hold, since

D_{r, s} (x, p) = D_{s, r} (x, p)

, we may assume that

r \geq r^{*}

and

s \geq s^{*}

. By the power mean inequality, we have

\begin{array}{rcl} D_{r, s} (x, p) & = & M^{[r - s]} (x, \frac{x^{s} p}{A (x^{s}, p)}) \geq M^{[r^{*} - s]} (x, \frac{x^{s} p}{A (x^{s}, p)}) \\ = & M^{[s - r^{*}]} (x, \frac{x^{r^{*}} p}{A (x^{r^{*}}, p)}) \geq M^{[s^{*} - r^{*}]} (x, \frac{x^{r^{*}} p}{A (x^{r^{*}}, p)}) \\ = & D_{r^{*}, s^{*}} (x, p) . \end{array}

If (19) holds, by [2], p.74 and [10, 11], (20) hold. □

Lemma 4 Let

x \in {(0, \infty)}^{n}

p \in S^{n}

and

(r, s), (r^{*}, s^{*}) \in R^{2}

. If (20) holds, then

V_{r, s} (x, p) \geq V_{r^{*}, s^{*}} (x, p) .

Proof If x is a constant n-vector, our assertion clearly holds. We may, therefore, assume that there exist

i, j \in N_{n}

such that

x_{i} \neq x_{j}

. We may further assume that

r (r - 1) s (s - 1) (r - s) \neq 0 .

Let

G = {Δ Φ_{1}, Δ Φ_{2}, \dots, Δ Φ_{l}}

be a partition of

Φ = {(t_{1}, t_{2}) \in {[0, \infty)}^{2} | t_{1} + t_{2} \leq 1}

. Let the area of each

Δ Φ_{i}

be denoted by

| Δ Φ_{i} |

, and let

∥ G ∥ = max_{1 \leq k \leq l} max_{x, y \in Δ Φ_{k}} {∥ x - y ∥}

be the ‘norm’ of the partition, then for any

(ξ_{k, 1}, ξ_{k, 2}) \in Δ Φ_{k}

, we have

\int \int_{Φ} {[w_{i, j} (x, p, t_{1}, t_{2})]}^{r - 2} d t_{1} d t_{2} = lim_{∥ G ∥ \to 0} \sum_{k = 1}^{l} {[w_{i, j} (x, p, ξ_{k, 1}, ξ_{k, 2})]}^{r - 2} | Δ Φ_{k} | .

By (11), when

γ (γ - 1) \neq 0

, we have

\begin{array}{rcl} {Var}_{γ} (x, p) & = & 2 \sum_{1 \leq i < j \leq n} p_{i} p_{j} {(x_{i} - x_{j})}^{2} \int \int_{Φ} {[w_{i, j} (x, p, t_{1}, t_{2})]}^{γ - 2} d t_{1} d t_{2} \\ = & 2 \sum_{1 \leq i < j \leq n} p_{i} p_{j} {(x_{i} - x_{j})}^{2} lim_{∥ G ∥ \to 0} \sum_{k = 1}^{l} {[w_{i, j} (x, p, ξ_{k, 1}, ξ_{k, 2})]}^{γ - 2} | Δ Φ_{k} | \\ = & lim_{∥ G ∥ \to 0} {2 \sum_{1 \leq i < j \leq n} p_{i} p_{j} {(x_{i} - x_{j})}^{2} \sum_{k = 1}^{l} {[w_{i, j} (x, p, ξ_{k, 1}, ξ_{k, 2})]}^{γ - 2} | Δ Φ_{k} |} \\ = & lim_{∥ G ∥ \to 0} {\sum_{1 \leq i < j \leq n, 1 \leq k \leq l} 2 p_{i} p_{j} | Δ Φ_{k} | {(x_{i} - x_{j})}^{2} {[w_{i, j} (x, p, ξ_{k, 1}, ξ_{k, 2})]}^{γ - 2}} . \end{array}

(21)

By (21) and Lemma 3, we then see that

\begin{array}{rcl} V_{r, s} (x, p) & = & {[\frac{{Var}_{r} (x, p)}{{Var}_{s} (x, p)}]}^{\frac{1}{r - s}} \\ = & lim_{∥ G ∥ \to 0} {\frac{\sum_{1 \leq i < j \leq n, 1 \leq k \leq l} p_{i} p_{j} | Δ Φ_{k} | {(x_{i} - x_{j})}^{2} {[w_{i, j} (x, p, ξ_{k, 1}, ξ_{k, 2})]}^{r - 2}}{\sum_{1 \leq i < j \leq n, 1 \leq k \leq l} p_{i} p_{j} | Δ Φ_{k} | {(x_{i} - x_{j})}^{2} {[w_{i, j} (x, p, ξ_{k, 1}, ξ_{k, 2})]}^{s - 2}}}^{\frac{1}{(r - 2) - (s - 2)}} \\ \geq & lim_{∥ G ∥ \to 0} {\frac{\sum_{1 \leq i < j \leq n, 1 \leq k \leq l} p_{i} p_{j} | Δ Φ_{k} | {(x_{i} - x_{j})}^{2} {[w_{i, j} (x, p, ξ_{k, 1}, ξ_{k, 2})]}^{r^{*} - 2}}{\sum_{1 \leq i < j \leq n, 1 \leq k \leq l} p_{i} p_{j} | Δ Φ_{k} | {(x_{i} - x_{j})}^{2} {[w_{i, j} (x, p, ξ_{k, 1}, ξ_{k, 2})]}^{s^{*} - 2}}}^{\frac{1}{(r^{*} - 2) - (s^{*} - 2)}} \\ = & V_{r^{*}, s^{*}} (x, p) . \end{array}

This ends the proof. □

We may now easily obtain the proof of Theorem 2.

Proof Indeed, by (20), (15) and Lemma 4, we get that

V_{r, s} (f, p) = lim_{∥ T ∥ \to 0} V_{r, s} (f (ξ), \bar{p} (ξ)) \geq lim_{∥ T ∥ \to 0} V_{r^{*}, s^{*}} (f (ξ), \bar{p} (ξ)) = V_{r^{*}, s^{*}} (f, p) .

This completes the proof of Theorem 2. □

4 Proof of Theorem 3

In this section, we use the same notations as in the previous two sections. In addition, let

I_{n} = (1, 1, \dots, 1)

be an n-vector and

Q_{+} = {\frac{s}{r} | r \in {1, 2, 3, \dots}, s \in {0, 1, 2, \dots}},

let

S_{n}

be the

(n - 1)

-dimensional simplex that

S_{n} = {x \in {(0, \infty)}^{n} | \sum_{i = 1}^{n} x_{i} = n},

and let

F_{n} (x) = \sum_{i = 1}^{n} x_{i}^{γ} ln x_{i} - \frac{1}{γ - 1} (\sum_{i = 1}^{n} x_{i}^{γ} - n)

be defined on

S_{n}

Lemma 5 Let

γ \in (1, \infty)

. If x is a relative extremum point of the function

F_{n} : S_{n} \to R

, then there exist

k \in N_{n}

and

u, v \in (0, n)

such that

k u + (n - k) v = n,

(22)

and

F_{n} (x) = k u^{k} ln u + (n - k) v^{k} ln v - \frac{1}{γ - 1} [k u^{k} + (n - k) v^{k} - n] .

(23)

Proof Consider the Lagrange function

L (x) = F_{n} (x) + μ (\sum_{i = 1}^{n} x_{i} - n),

with

\frac{\partial L}{\partial x_{j}} = x_{j}^{γ - 1} (γ ln x_{j} + 1) - \frac{γ}{γ - 1} x_{j}^{γ - 1} + μ = x_{j}^{γ - 1} L (x_{j}) = 0, j = 1, 2, \dots, n,

where the function

L : (0, n) \to R

is defined by

L (t) = γ ln t + μ t^{1 - γ} - \frac{1}{γ - 1} .

Then

x_{j} \in (0, n), L (x_{j}) = 0, j = 1, 2, \dots, n .

(24)

Note that

L^{'} (t) = \frac{γ}{t} - \frac{γ - 1}{t^{γ}} μ = \frac{γ}{t^{γ}} (t^{γ - 1} - \frac{γ - 1}{γ} μ) .

Hence, the function

L : (0, n) \to R

has at most one extreme point, and

L (t)

has at most two roots in

(0, n)

. By (24), we have

| {x_{1}, x_{2}, \dots, x_{n}} | \leq 2,

where

| {x_{1}, x_{2}, \dots, x_{n}} |

denotes the count of elements in the set

{x_{1}, x_{2}, \dots, x_{n}}

. Since

F_{n} : S_{n} \to R

is a symmetric function, we may assume that there exists

k \in N_{n}

such that

\begin{matrix} x_{1} = x_{2} = \dots = x_{k} = u, \\ x_{k + 1} = x_{k + 2} = \dots = x_{n} = v . \end{matrix}

That is, (23) and (22) hold. The proof is complete. □

Lemma 6 Let

γ \in (1, \infty)

. If x is a relative extremum point of the function

F_{n} : S_{n} \to R

, then

F_{n} (x) \geq 0 .

(25)

Proof By Lemma 5, there exist

k \in N_{n}

and

u, v \in (0, n)

such that (23) and (22) hold. If

u = v = 1

, then (25) holds. We may, therefore, assume without loss of generality that

0 < u < 1 < v

. From (22), we see that

\frac{k}{n} = \frac{1 - v}{u - v} .

(26)

Putting (26) into (23), we obtain that

\begin{array}{rcl} F_{n} (x) & = & n {\frac{k}{n} u^{γ} ln u + (1 - \frac{k}{n}) v^{γ} ln v - \frac{1}{γ - 1} [\frac{k}{n} u^{γ} + (1 - \frac{k}{n}) v^{γ} - 1]} \\ = & n {\frac{1 - v}{u - v} u^{γ} ln u + \frac{u - 1}{u - v} v^{γ} ln v - \frac{1}{γ - 1} [\frac{1 - v}{u - v} u^{γ} + \frac{u - 1}{u - v} v^{γ} - 1]} \\ = & n {\frac{1 - v}{u - v} u^{γ} ln u + \frac{u - 1}{u - v} v^{γ} ln v - \frac{1}{γ - 1} [\frac{1 - v}{u - v} (u^{γ} - 1) + \frac{u - 1}{u - v} (v^{γ} - 1)]} \\ = & \frac{n (1 - v) (u - 1)}{u - v} {\frac{u^{γ} ln u}{u - 1} + \frac{v^{γ} ln v}{1 - v} - \frac{1}{γ - 1} [\frac{u^{γ} - 1}{u - 1} + \frac{v^{γ} - 1}{1 - v}]} \\ = & \frac{n (1 - u) (v - 1)}{v - u} {\frac{v^{γ} ln v}{v - 1} - \frac{u^{γ} ln u}{u - 1} - \frac{1}{γ - 1} [\frac{v^{γ} - 1}{v - 1} - \frac{u^{γ} - 1}{u - 1}]} \\ = & \frac{n (1 - u) (v - 1)}{(γ - 1) (v - u)} [ψ (v, γ) - ψ (u, γ)], \end{array}

(27)

where the auxiliary function

ψ : (0, \infty) \times (1, \infty) \to R

is defined by

ψ (t, γ) = \frac{(γ - 1) t^{γ} ln t - (t^{γ} - 1)}{t - 1} .

Since

\frac{n (1 - u) (v - 1)}{(γ - 1) (v - u)} > 0

, by (27), inequality (25) is equivalent to the following inequality:

ψ (v, γ) \geq ψ (u, γ), \forall u \in (0, 1), \forall v \in (1, \infty), \forall γ \in (1, \infty) .

(28)

By the software Mathematica, we can depict the image of the function

ψ : (0, 2) \times {\frac{3}{2}} \to R

in Figure 1, and the image of the function

ψ : (0, 2) \times (1, 2] \to R

in Figure 2.

Now, let us prove the following inequalities:

ψ (u, γ) < - 1, \forall u \in (0, 1), \forall γ \in (1, \infty);

(29)

ψ (v, γ) > - 1, \forall v \in (1, \infty), \forall γ \in (1, \infty) .

(30)

By Cauchy’s mean value theorem, there exists

ξ \in (u, 1)

such that

\begin{array}{rcl} ψ (u, γ) & = & \frac{(γ - 1) u^{γ} ln u - (u^{γ} - 1)}{u - 1} = \frac{(γ - 1) ln u - 1 + u^{- γ}}{u^{1 - γ} - u^{- γ}} \\ = & \frac{\partial [(γ - 1) ln u - 1 + u^{- γ}] / \partial u}{\partial (u^{1 - γ} - u^{- γ}) / \partial u} |_{u = ξ} = \frac{(γ - 1) ξ^{- 1} - γ ξ^{- γ - 1}}{(1 - γ) ξ^{- γ} + γ ξ^{- γ - 1}} \\ = & \frac{(γ - 1) ξ^{γ} - γ}{- (γ - 1) ξ + γ} < \frac{(γ - 1) ξ - γ}{- (γ - 1) ξ + γ} = - 1 . \end{array}

Therefore, inequality (29) holds.

Next, note that

\begin{array}{rcl} ψ (v, γ) > - 1 & \Leftrightarrow & \frac{(γ - 1) v^{γ} ln v - (v^{γ} - 1)}{v - 1} > - 1 \\ \Leftrightarrow & (γ - 1) v^{γ} ln v - v^{γ} + v > 0 \\ \Leftrightarrow & ψ_{*} (v, γ) = (γ - 1) ln v - 1 + v^{1 - γ} > 0 . \end{array}

(31)

By Lagrange’s mean value theorem, there exists

ξ_{*} \in (1, v)

such that

\begin{array}{rcl} ψ_{*} (v, γ) & = & ψ_{*} (v, γ) - ψ_{*} (1, γ) \\ = & (v - 1) \frac{\partial ψ_{*} (v, γ)}{\partial v} |_{v = ξ_{*}} \\ = & (v - 1) [(γ - 1) ξ_{*}^{- 1} + (1 - γ) ξ_{*}^{- γ}] \\ = & (γ - 1) (v - 1) ξ_{*}^{- γ} (ξ_{*}^{γ - 1} - 1) > 0 . \end{array}

Hence, (31) holds. It then follows that inequality (30) holds.

By inequalities (29) and (30), we may easily obtain inequality (28). This ends the proof of Lemma 6. □

Lemma 7 If

γ \in (1, \infty)

, then for any

x \in S_{n}

, inequality (25) holds.

Proof We proceed by induction.

(A)

Suppose

n = 2

. By the well-known non-linear programming (maximum) principle and Lemma 6, we only need to prove that

lim_{x_{1} \to 0, (x_{1}, x_{2}) \in S_{2}} F_{2} (x_{1}, x_{2}) \geq 0 and lim_{x_{2} \to 0, (x_{1}, x_{2}) \in S_{2}} F_{2} (x_{1}, x_{2}) \geq 0 .

We show the first inequality, the second being similar.

Indeed, it follows from Lagrange’s mean value theorem that there exists

γ_{*} \in (1, γ)

such that

\begin{array}{rcl} lim_{x_{1} \to 0, (x_{1}, x_{2}) \in S_{2}} F_{2} (x_{1}, x_{2}) & = & 2^{γ} ln 2 - \frac{2^{γ} - 2}{γ - 1} \\ = & 2^{γ} \frac{(γ - 1) ln 2 - 1 + 2^{1 - γ}}{γ - 1} \\ = & 2^{γ} \frac{d [(γ - 1) ln 2 - 1 + 2^{1 - γ}]}{d γ} |_{γ = γ_{*}} \\ = & 2^{γ} ln 2 (1 - 2^{1 - γ_{*}}) \\ > & 0 . \end{array}

(B)

Assume by induction that the function

F_{n - 1} : S_{n - 1} \to R

satisfies

F_{n - 1} (y) \geq 0

for all

y \in S_{n - 1}

. We prove inequality (25) as follows. By Lemma 6, we only need to prove that

lim_{x_{1} \to 0, x \in S_{n}} F_{n} (x) \geq 0, \dots, lim_{x_{n} \to 0, x \in S_{n}} F_{n} (x) \geq 0 .

We will only show the last inequality. If we set

y = (y_{1}, y_{2}, \dots, y_{n - 1}) = \frac{n - 1}{n} (x_{1}, x_{2}, \dots, x_{n - 1}),

then

y \in S_{n - 1}

. By Lagrange’s mean value theorem, there exists

γ_{*} \in (1, γ)

such that

\begin{aligned} \frac{1}{γ - 1} [(γ - 1) (ln \frac{n}{n - 1}) - 1 + {(\frac{n}{n - 1})}^{1 - γ}] \\ = \frac{\partial}{\partial γ} [(γ - 1) (ln \frac{n}{n - 1}) - 1 + {(\frac{n}{n - 1})}^{1 - γ}] |_{γ = γ_{*}} \\ = (ln \frac{n}{n - 1}) [1 - {(\frac{n}{n - 1})}^{1 - γ_{*}}] . \end{aligned}

Thus, by the power mean inequality

\sum_{i = 1}^{n - 1} y_{i}^{γ} \geq (n - 1) {(\frac{1}{n - 1} \sum_{i = 1}^{n - 1} y_{i})}^{γ} = n - 1

and induction hypothesis, we see that

\begin{aligned} lim_{x_{n} \to 0, x \in S_{n}} F_{n} (x) \\ = \sum_{i = 1}^{n - 1} x_{i}^{γ} ln x_{i} - \frac{1}{γ - 1} (\sum_{i = 1}^{n} x_{i}^{γ} - n) \\ = \sum_{i = 1}^{n - 1} {(\frac{n}{n - 1} y_{i})}^{γ} ln (\frac{n}{n - 1} y_{i}) - \frac{1}{γ - 1} [\sum_{i = 1}^{n - 1} {(\frac{n}{n - 1} y_{i})}^{γ} - n] \\ = {(\frac{n}{n - 1})}^{γ} {(ln \frac{n}{n - 1}) \sum_{i = 1}^{n - 1} y_{i}^{γ} + \sum_{i = 1}^{n - 1} y_{i}^{γ} ln y_{i} - \frac{1}{γ - 1} [\sum_{i = 1}^{n - 1} y_{i}^{γ} - n {(\frac{n}{n - 1})}^{γ}]} \\ = {(\frac{n}{n - 1})}^{γ} {(ln \frac{n}{n - 1}) \sum_{i = 1}^{n - 1} y_{i}^{r} + F_{n - 1} (y) - \frac{1}{γ - 1} [n - 1 - n {(\frac{n - 1}{n})}^{γ}]} \\ \geq {(\frac{n}{n - 1})}^{γ} {(ln \frac{n}{n - 1}) (n - 1) - \frac{1}{γ - 1} [n - 1 - n {(\frac{n}{n - 1})}^{γ}]} \\ = (n - 1) {(\frac{n}{n - 1})}^{γ} \frac{1}{γ - 1} [(γ - 1) (ln \frac{n}{n - 1}) - 1 + {(\frac{n - 1}{n})}^{γ - 1}] \\ = (n - 1) {(\frac{n}{n - 1})}^{γ} \frac{1}{γ - 1} [(γ - 1) (ln \frac{n}{n - 1}) - 1 + {(\frac{n - 1}{n})}^{1 - γ}] \\ = (n - 1) {(\frac{n}{n - 1})}^{γ} (ln \frac{n}{n - 1}) [1 - {(\frac{n}{n - 1})}^{1 - γ_{*}}] > 0 . \end{aligned}

This ends the proof of Lemma 7. □

Lemma 8 Let

x \in {(0, \infty)}^{n}

and

p \in S^{n}

. If

r > s \geq 1

, then

V_{r, s} (x, p) \geq {(\frac{s}{r})}^{\frac{1}{r - s}} A (x, p),

(32)

and the coefficient

{(\frac{s}{r})}^{\frac{1}{r - s}}

in (32) is the best constant.

Proof We may assume that there exist

i, j \in N_{n}

such that

x_{i} \neq x_{j}

. By continuity considerations, we may also assume that

r > s > 1

(A)

Suppose

p = n^{- 1} I_{n}

. Then (32) can be rewritten as

V_{r, s} (x, n^{- 1} I_{n}) \geq {(\frac{s}{r})}^{\frac{1}{r - s}} A (x, n^{- 1} I_{n}),

{[\frac{{Var}_{r} (x, n^{- 1} I_{n})}{{Var}_{s} (x, n^{- 1} I_{n}))}]}^{\frac{1}{r - s}} \geq {(\frac{s}{r})}^{\frac{1}{r - s}} A (x, n^{- 1} I_{n}),

{[\frac{s (s - 1)}{r (r - 1)} \frac{A (x^{r}, n^{- 1} I_{n}) - A^{r} (x, n^{- 1} I_{n})}{A (x^{s}, n^{- 1} I_{n}) - A^{s} (x, n^{- 1} I_{n})}]}^{\frac{1}{r - s}} \geq {(\frac{s}{r})}^{\frac{1}{r - s}} A (x, n^{- 1} I_{n}) .

That is,

F_{r} (x, n^{- 1} I_{n}) \geq F_{s} (x, n^{- 1} I_{n}),

(33)

where we have introduced the auxiliary function

F_{γ} (x, p) = ln \frac{A (x^{γ}, p) - A^{γ} (x, p)}{(γ - 1) A^{γ} (x, p)}, γ > 1 .

Since, for any

t \in (0, \infty)

, we have

F_{γ} (t x, n^{- 1} I_{n}) = F_{γ} (x, n^{- 1} I_{n})

, we may assume that

x \in S_{n}

. By Lemma 7, we have

\begin{array}{rcl} \frac{\partial F_{γ} (x, n^{- 1} I_{n})}{\partial γ} & = & \frac{\partial}{\partial γ} [ln (\frac{1}{n} \sum_{i = 1}^{n} x_{i}^{γ} - 1) - ln (γ - 1)] \\ = & \frac{n^{- 1} \sum_{i = 1}^{n} x_{i}^{γ} ln x_{i}}{n^{- 1} \sum_{i = 1}^{n} x_{i}^{γ} - 1} - \frac{1}{γ - 1} \\ = & \frac{\sum_{i = 1}^{n} x_{i}^{γ} ln x_{i} - {(γ - 1)}^{- 1} (\sum_{i = 1}^{n} x_{i}^{γ} - n)}{\sum_{i = 1}^{n} x_{i}^{γ} - n} \\ = & \frac{F_{n} (x)}{\sum_{i = 1}^{n} x_{i}^{γ} - n} \geq 0 . \end{array}

Hence, for a fixed

x \in {(0, \infty)}^{n}

F_{γ} (x, n^{- 1} I_{n})

is increasing with respect to γ in

(1, \infty)

. Thus, by

r > s > 1

, we obtain (33) and (32).

(B)

Suppose

p \neq n^{- 1} I_{n}

, but

p \in Q_{+}^{n}

. Then there exists

N \in {1, 2, 3, \dots}

such that

N p_{i} \in {0, 1, 2, \dots}

for

i = 1, \dots, n

. Setting

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-366/MediaObjects/13660_2013_Article_804_Equbv_HTML.gif

and

p_{*} = m^{- 1} I_{m},

then

x_{*} \in {(0, \infty)}^{m}

p_{*} \in S^{m}

. Inequality (32) can then be rewritten as

V_{r, s} (x_{*}, p_{*}) \geq {(\frac{s}{r})}^{\frac{1}{r - s}} A (x_{*}, p_{*}) .

(34)

According to the result in (A), inequality (34) holds.

(C)

Suppose

p \neq n^{- 1} I_{n}

and

p \in S^{n} ∖ Q_{+}^{n}

. Then it is easy to see that there exists a sequence

{p^{(k)}}_{k = 1}^{\infty} \subset Q_{+}^{n}

such that

{lim}_{k \to \infty} p^{(k)} = p

. According to the result in (B), we get

V_{r, s} (x, p^{(k)}) \geq {(\frac{s}{r})}^{\frac{1}{r - s}} A (x, p^{(k)}), \forall k \in {1, 2, 3, \dots} .

Therefore

V_{r, s} (x, p) = lim_{k \to \infty} V_{r, s} (x, p^{(k)}) \geq {(\frac{s}{r})}^{\frac{1}{r - s}} lim_{k \to \infty} A (x, p^{(k)}) = {(\frac{s}{r})}^{\frac{1}{r - s}} A (x, p) .

Next, we show that the coefficient

{(\frac{s}{r})}^{\frac{1}{r - s}}

is the best constant in (32). Assume that the inequality

V_{r, s} (x, p) \geq C_{r, s} A (x, p)

(35)

holds. Setting

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-366/MediaObjects/13660_2013_Article_804_Equbz_HTML.gif

and

p = n^{- 1} I_{n}

in (35), we obtain

\begin{aligned} {[\frac{s (s - 1)}{r (r - 1)} \frac{\frac{n - 1}{n} - {(\frac{n - 1}{n})}^{r}}{\frac{n - 1}{n} - {(\frac{n - 1}{n})}^{s}}]}^{\frac{1}{r - s}} \geq C_{r, s} \frac{n - 1}{n} \\ \Leftrightarrow {[\frac{s (s - 1)}{r (r - 1)} \frac{1 - {(\frac{n - 1}{n})}^{r - 1}}{1 - {(\frac{n - 1}{n})}^{s - 1}}]}^{\frac{1}{r - s}} \geq C_{r, s} \frac{n - 1}{n} . \end{aligned}

(36)

In (36), by letting

n \to \infty

, we obtain

{(\frac{s}{r})}^{\frac{1}{r - s}} \geq C_{r, s} .

Hence, the coefficient

{(\frac{s}{r})}^{\frac{1}{r - s}}

is the best constant in (32). The proof is complete. □

Remark 4 If

x \in {(0, \infty)}^{n}

p \in S^{n}

and

r > s > 1

, then there cannot be any

θ, C_{r, s} : θ \in (0, \infty)

and

C_{r, s} \in (0, \infty)

such that

V_{r, s} (x, p) \leq C_{r, s} M^{[θ]} (x, p) .

(37)

Indeed, if there exist

θ \in (0, \infty)

and

C_{r, s} \in (0, \infty)

such that (37) holds, then by setting

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-366/MediaObjects/13660_2013_Article_804_Equcb_HTML.gif

and

p = n^{- 1} I_{n}

in (37), we see that

{[\frac{s (s - 1)}{r (r - 1)} \frac{1 - n^{1 - r}}{1 - n^{1 - s}}]}^{\frac{1}{r - s}} \leq C_{r, s} n^{- \frac{1}{θ}}

which implies

{[\frac{s (s - 1)}{r (r - 1)}]}^{\frac{1}{r - s}} = lim_{n \to \infty} {[\frac{s (s - 1)}{r (r - 1)} \frac{1 - n^{1 - r}}{1 - n^{1 - s}}]}^{\frac{1}{r - s}} \leq lim_{n \to \infty} C_{r, s} n^{- \frac{1}{θ}} = 0,

(38)

which is a contradiction.

Remark 5 The method of the proof of Lemma 8 is referred to as the descending method in [6, 7, 13, 23‐27], but the details in this paper are different.

We now return to the proof of Theorem 3.

Proof By (15) and Lemma 8, we obtain

V_{r, s} (f, p) = lim_{∥ T ∥ \to 0} V_{r, s} (f (ξ), \bar{p} (ξ)) \geq {(\frac{s}{r})}^{\frac{1}{r - s}} lim_{∥ T ∥ \to 0} A (f (ξ), \bar{p} (ξ)) = {(\frac{s}{r})}^{\frac{1}{r - s}} E (f, p) .

Thus, inequality (4) holds. Furthermore, by Lemma 8, the coefficient

{(\frac{s}{r})}^{\frac{1}{r - s}}

is the best constant. This completes the proof of Theorem 3. □

5 Applications in space science

It is well known that there are nine planets in the solar system, i.e., Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto. In this paper, we also believe that Pluto is a planet in the solar system. In space science, we always consider the gravity to the Earth from other planets in the solar system (see Figure 3).

We can build the mathematical model of the problem. Let the masses of these planets be

m_{0}, m_{1}, \dots, m_{n}

, where

m_{0}

denotes the mass of the Earth and

1 \leq n \leq 8

. At moment

T_{0}

, in

R^{3}

, let the coordinate of the center of the Earth be

o = (0, 0, 0)

and the center of the i th planet be

p_{i} = (p_{i 1}, p_{i 2}, p_{i 3})

, and the distance between

p_{i}

and o be

∥ p_{i} ∥ = \sqrt{p_{i 1}^{2} + p_{i 2}^{2} + p_{i 3}^{2}}

, where

i = 1, 2, \dots, n

. By the famous law of gravitation, the gravity to the Earth o from the planets

p_{1}, p_{2}, \dots, p_{n}

F : = G_{0} m_{0} \sum_{i = 1}^{n} \frac{m_{i} p_{i}}{{∥ p_{i} ∥}^{3}},

where

G_{0}

is the gravitational constant in the solar system. Assume the coordinate of the center of the sun is

g = (g_{1}, g_{2}, g_{3})

, then there exists a ball

B (g, r)

such that the planets

p_{1}, p_{2}, \dots, p_{n}

move in this ball. In other words, at any moment, we have

p_{i} \in B (g, r), i = 1, 2, \dots, n,

where r is the radius of the ball

B (g, r)

We denote by

θ_{i, j} : = ∠ (p_{i}, p_{j})

the angle between the vectors

\vec{o p_{i}}

and

\vec{o p_{j}}

, where

1 \leq i \neq j \leq n

. This angle is also considered as the observation angle between two planets

p_{i}

p_{j}

from the Earth o, which can be measured from the Earth by telescope.

Without loss of generality, we suppose that

n \geq 2

G_{0} = 1

m_{0} = 1

and

\sum_{i = 1}^{n} m_{i} = 1

in this paper.

We can generalize the above problem to an Euclidean space. Let

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-366/MediaObjects/13660_2013_Article_804_IEq176_HTML.gif

be an Euclidean space. For two vectors

α \in E

and

β \in E

, the inner product of α, β and the norm of α are denoted by

〈 α, β 〉

and

∥ α ∥ = \sqrt{〈 α, α 〉}

, respectively. The angle between α and β is denoted by

∠ (α, β) : = arccos \frac{〈 α, β 〉}{∥ α ∥ \cdot ∥ β ∥} \in [0, π],

where α and β are nonzero vectors.

Let

g \in E

, we say the set

B (g, r) : = {x \in E | ∥ x - g ∥ \leq r}

is a closed sphere and the set

S (g, r) : = {x \in E | ∥ x - g ∥ = r}

is a spherical, where

r \in R_{+ +} = (0, + \infty)

Now let us define the planet system and the λ-gravity function.

Let

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-366/MediaObjects/13660_2013_Article_804_IEq183_HTML.gif

be an Euclidean space, the dimension of

dim E \geq 3

P = (p_{1}, p_{2}, \dots, p_{n})

and

m = (m_{1}, m_{2}, \dots, m_{n})

be the sequences of

and

R_{+ +}

, respectively, and

B (g, r)

be a closed sphere in

. The set

PS {P, m, B (g, r)}_{E}^{n} : = {P, m, B (g, r)}

is called the planet system if the following three conditions hold:

(H1)

∥ p_{i} ∥ > 0

i = 1, 2, \dots, n

;

(H2)

p_{i} \in B (g, r)

i = 1, 2, \dots, n

;

(H3)

\sum_{i = 1}^{n} m_{i} = 1

Let

PS {P, m, B (g, r)}_{E}^{n}

be a planet system. The function

F_{λ} : E^{n} \to E, F_{λ} (P) = \sum_{i = 1}^{n} \frac{m_{i} p_{i}}{{∥ p_{i} ∥}^{λ + 1}}

is called a λ-gravity function of the planet system

PS {P, m, B (g, r)}_{E}^{n}

, and

F_{0}

is the gravity kernel of

F_{λ}

, where

λ \in [0, + \infty)

Write

p_{i} = ∥ p_{i} ∥ e_{i}; θ_{i, j} : = ∠ (p_{i}, p_{j}) \in [0, π]; θ_{i} : = ∠ (p_{i}, g) \in [0, π] .

The matrix

A = {[〈 e_{i}, e_{j} 〉]}_{n \times n} = {[cos θ_{i, j}]}_{n \times n}

is called an observation matrix of the planet system

PS {P, m, B (g, r)}_{E}^{n}

θ_{i, j}

(

j \neq i

1 \leq i

j \leq n

) and

θ_{i}

(

1 \leq i \leq n

) are called the observation angle and the center observation angle of the planet system, respectively. For the planet system, we always consider that the observation matrix A is a constant matrix in this paper.

It is worth noting that the norm

∥ F_{0} ∥

of the gravity kernel

F_{0} = F_{0} (P)

is independent of

∥ p_{1} ∥, ∥ p_{2} ∥, \dots, ∥ p_{n} ∥

; furthermore,

0 \leq ∥ F_{0} ∥ = \sqrt{m A m^{T}} \leq 1,

(39)

where

m^{T}

is the transpose of the row vector m, and

m A m^{T} = \sum_{1 \leq i, j \leq n} m_{i} m_{j} cos θ_{i, j}

is a quadratic.

In fact, from

F_{0} = \sum_{i = 1}^{n} m_{i} e_{i}

and

〈 e_{i}, e_{j} 〉 = cos θ_{i j}

, we have that

\begin{matrix} 0 \leq ∥ F_{0} ∥ = \sqrt{〈 F_{0}, F_{0} 〉} = \sqrt{\sum_{1 \leq i, j \leq n} m_{i} m_{j} 〈 e_{i}, e_{j} 〉} = \sqrt{m A m^{T}}; \\ ∥ F_{0} ∥ = ∥ \sum_{i = 1}^{n} m_{i} e_{i} ∥ \leq \sum_{i = 1}^{n} ∥ m_{i} e_{i} ∥ = 1 . \end{matrix}

Let

PS {P, m, B (g, 1)}_{E}^{n}

be a planet system. By the above definitions, the gravity to the Earth o from n planets

p_{1}, p_{2}, \dots, p_{n}

in the solar system is

F_{2} (P)

, and

∥ F_{2} (P) ∥

is the norm of

F_{2} (P)

. If we take a point

q_{i}

in the ray

o p_{i}

such that

∥ \vec{o q_{i}} ∥ = 1

, and place a planet at

q_{i}

with mass

m_{i}

for

i = 1, 2, \dots, n

, then the gravity of these n planets

q_{1}, q_{2}, \dots, q_{n}

to the Earth o is

F_{0} (P)

Let

PS {P, m, B (g, 1)}_{E}^{n}

be a planet system. If we believe that g is a molecule and

p_{1}, p_{2}, \dots, p_{n}

are atoms of g, then the gravity to another atom o from n atoms

p_{1}, p_{2}, \dots, p_{n}

of g is

F_{2} (P)

In the solar system, the gravity of n planets

p_{1}, p_{2}, \dots, p_{n}

to the planet o is

F_{2} (P)

, while for other galaxy in the universe, the gravity may be

F_{λ} (P)

, where

λ \in (0, 2) \cup (2, + \infty)

Let

PS {P, m, B (g, r)}_{E}^{n}

be a planet system. Then the function

f_{λ} : E^{n} \to R_{+ +}, f_{λ} (P) = \sum_{i = 1}^{n} \frac{m_{i}}{{∥ p_{i} ∥}^{λ}}

is called an absolute λ-gravity function of the planet system

PS {P, m, B (g, r)}_{E}^{n}

, where

λ \in [0, + \infty)

Let P be a planetary sequence in the solar system. Then

\frac{1}{n} f_{2} (P)

is the average value of the gravities of the planets

p_{1}, p_{2}, \dots, p_{n}

to the Earth o.

Let P be a planetary sequence in the solar system. If we think that

m_{i}

is the radiation energy of the planet

p_{i}

, then, according to optical laws, the radiant energy received by the Earth o is

c \frac{m_{i}}{{∥ p_{i} ∥}^{2}}

i = 1, 2, \dots, n

, and the total radiation energy received by the Earth o is

c f_{2} (P)

, where

c > 0

is a constant.

By Minkowski’s inequality (see [28])

∥ x + y ∥ \leq ∥ x ∥ + ∥ y ∥, \forall x, y \in E,

we know that if

F_{λ} (P)

and

f_{λ} (P)

are a λ-gravity function and an absolute λ-gravity function of the planet system

PS {P, m, B (g, r)}_{E}^{n}

, respectively, then we have

∥ F_{λ} (P) ∥ \leq f_{λ} (P), \forall λ \in [0, + \infty) .

(40)

Now, we will define absolute λ-gravity variance and λ-gravity variance. To this end, we need the following preliminaries.

Two vectors x and y in

are said to be in the same (opposite) direction if (i)

x = 0

y = 0

, or (ii)

x \neq 0

and

y \neq 0

and x is a positive (respectively negative) constant multiple of y. Two vectors x and y in the same (opposite) direction are indicated by

x ↑ y

(respectively

x ↓ y

We say that the set

S : = S (0, 1)

is a unit sphere in

For each

α \in S

, we say that the set

Π_{α} : = {γ \in E | 〈 γ, α 〉 = 1}

is the tangent plane to the unit sphere

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-366/MediaObjects/13660_2013_Article_804_IEq244_HTML.gif

at the vector α. It is obvious that

γ \in Π_{α} \Leftrightarrow 〈 γ - α, α 〉 = 0 \Leftrightarrow α ⊥ γ - α .

Assume that

α, β \in S

, and that

α \pm β \neq 0

. We then say that the set

α β : = {Ϝ_{α, β} (t) | t \in (- \infty, \infty)},

where

Ϝ_{α, β} (t) = \frac{(1 - t) α + t β}{∥ (1 - t) α + t β ∥},

are straight lines on the unit sphere

, and that the sets

\begin{aligned} [α β] : = {Ϝ_{α, β} (t) | λ \in [0, 1]}, (α β] : = {Ϝ_{α, β} (t) | t \in (0, 1]}, \\ [α β) : = {Ϝ_{α, β} (t) | t \in [0, 1)}, (α β) : = {Ϝ_{α, β} (t) | t \in (0, 1)}, \end{aligned}

are the straight line segments on the sphere

, and that

∥ α β ∥ : = ∠ (α, β) = arccos 〈 α, β 〉

is the length of these line segments.

It is easy to see that

α + β \neq 0

implies

(1 - t) α + t β \neq 0

. Thus, we may easily get the existence and uniqueness of these line segments. Similarly,

α \pm β \neq 0

implies that

∥ α β ∥ \in (0, π)

Assuming that

γ \in Π_{α} \cap Π_{β}

, and that α, β, γ are linearly dependent vectors, we say that

γ - α

is the tangent vector to the line segment

[α β)

at α. By definition, we see that there exist

u, v \in R

such that

γ = u α + v β .

Therefore

\begin{matrix} 1 = 〈 γ, α 〉 = u 〈 α, α 〉 + v 〈 α, β 〉 = u + v 〈 α, β 〉, \\ 1 = 〈 γ, β 〉 = u 〈 α, β 〉 + v 〈 β, β 〉 = u 〈 α, β 〉 + v . \end{matrix}

We infer from

〈 α, β 〉 \in (- 1, 1)

that

u = v = \frac{1}{1 + 〈 α, β 〉}, γ = \frac{α + β}{1 + 〈 α, β 〉}, \frac{γ - α}{∥ γ - α ∥} = \frac{β - 〈 α, β 〉 α}{∥ β - 〈 α, β 〉 α ∥} .

We define also the tangent vector of

[α β)

at α by

β - 〈 α, β 〉 α

. The tangent vector

β - 〈 α, β 〉 α

enjoys the following properties: If

γ \in (α β)

, then

(γ - 〈 α, γ 〉 α) ↑ (β - 〈 α, β 〉 α) .

(41)

In fact, there exists

t \in (0, 1)

such that

γ = Ϝ_{α, β} (t) = \frac{(1 - t) α + t β}{∥ (1 - t) α + t β ∥} .

Since

〈 α, α 〉 = 1

, we see that

\begin{array}{rcl} (γ - 〈 α, γ 〉 α) & = & \frac{(1 - t) α + t β}{∥ (1 - t) α + t β ∥} - 〈 α, \frac{(1 - t) α + t β}{∥ (1 - t) α + t β ∥} 〉 α \\ = & \frac{(1 - t) α + t β - 〈 α, (1 - t) α + t β 〉 α}{∥ (1 - λ) α + t β ∥} \\ ↑ & [(1 - t) α + t β - 〈 α, (1 - t) α + t β 〉 α] \\ = & t (β - 〈 α, β 〉 α) \\ ↑ & (β - 〈 α, β 〉 α) . \end{array}

The angle between two line segments

[α β)

[α γ)

on the unit sphere

is defined as

∠ ([α β), [α γ)) : = ∠ (β - 〈 α, β 〉 α, γ - 〈 α, γ 〉 α) .

[α β)

[β γ)

[γ α)

are three straight line segments on the unit sphere

, and

α \notin β γ, β \notin γ α, γ \notin α β

, then we say that the set

△ α β γ : = [α β) \cup [β γ) \cup [γ α)

is a spherical triangle. Write

A = ∠ ([α β), [α γ))

B = ∠ ([β γ), [β α))

C = ∠ ([γ α), [γ β))

a = ∥ β γ ∥

b = ∥ γ α ∥

c = ∥ α β ∥

. Then we obtain that

\begin{array}{rcl} cos A & = & cos ∠ (β - 〈 α, β 〉 α, γ - 〈 α, γ 〉 α) \\ = & \frac{〈 β - 〈 α, β 〉 α, γ - 〈 α, γ 〉 α 〉}{∥ β - 〈 α, β 〉 α ∥ \cdot ∥ γ - 〈 α, γ 〉 α ∥} \\ = & \frac{〈 β, γ 〉 - 〈 α, β 〉 〈 α, γ 〉}{\sqrt{1 - {〈 α, β 〉}^{2}} \cdot \sqrt{1 - {〈 α, γ 〉}^{2}}} \\ = & \frac{cos a - cos b cos c}{sin b sin c} . \end{array}

Thus, we may get the law of cosine for spherical triangle

cos a = cos b cos c + sin b sin c cos A .

(42)

By (42) we may get the law of cosine for spherical triangle

cos A = - cos B cos C + sin B sin C cos a .

(43)

By (42) we may get the law of sine for spherical triangle

\frac{sin a}{sin A} = \frac{sin b}{sin B} = \frac{sin c}{sin C} .

(44)

cos A > - 1

cos a < 1

and (42)-(43), we get

a, b, c \in (0, π), b + c > a,

∠ (β, γ) < ∠ (γ, α) + ∠ (α, β),

(45)

and

A, B, C \in (0, π), A + B + C > π .

(46)

Lemma 9 Let

be an Euclidean space, let the dimension of

satisfy

dim E \geq 3

, and let

B (g, 1)

be a closed sphere in

. If

∥ g ∥ > 1

, then

max_{α, β \in B (g, 1)} {∠ (α, β)} = 2 arcsin \frac{1}{∥ g ∥} .

(47)

Proof From

α \in B (g, 1)

we get

\begin{matrix} 〈 α - g, α - g 〉 \leq 1, \\ {∥ α ∥}^{2} - 2 〈 g, α 〉 + {∥ g ∥}^{2} \leq 1, \\ cos ∠ (α, g) = \frac{〈 g, α 〉}{∥ α ∥ ∥ g ∥} \geq \frac{{∥ α ∥}^{2} + {∥ g ∥}^{2} - 1}{2 ∥ α ∥ ∥ g ∥} \geq \frac{\sqrt{{∥ g ∥}^{2} - 1}}{∥ g ∥} . \end{matrix}

Thus,

∠ (α, g) \leq arcsin \frac{1}{∥ g ∥} .

(48)

Similarly, from

β \in B (g, 1)

we have

∠ (β, g) \leq arcsin \frac{1}{∥ g ∥} .

(49)

If we set

α^{*} = \frac{α}{∥ α ∥}, β^{*} = \frac{β}{∥ β ∥}, g^{*} = \frac{g}{∥ g ∥},

then

α^{*}, α^{*}, g^{*} \in S

. According to inequalities (48), (49) and (45), we get

∠ (α, β) = ∠ (α^{*}, β^{*}) \leq ∠ (α^{*}, g^{*}) + ∠ (g^{*}, β^{*}) = ∠ (α, g) + ∠ (g, β) \leq 2 arcsin \frac{1}{∥ g ∥},

hence

∠ (α, β) \leq 2 arcsin \frac{1}{∥ g ∥} .

(50)

Now we discuss the conditions such that the equality in inequality (50) holds. From the above analysis, we know that these conditions are:

(a)

α \in B (g, 1)

and

∥ α ∥ = \sqrt{{∥ g ∥}^{2} - 1}

;

(b)

β \in B (g, 1)

and

∥ β ∥ = \sqrt{{∥ g ∥}^{2} - 1}

;

(c)

g^{*} \in [α^{*} β^{*}]

and

∠ (α^{*}, g^{*}) = ∠ (g^{*}, β^{*}) = arcsin \frac{1}{∥ g ∥}

From (a) and (b) we know that the condition (c) can be rewritten as

(c^∗)

\frac{g}{∥ g ∥} = \frac{α + β}{∥ α + β ∥}

g ↑ α + β

Based on the above analysis, we know that the equality in inequality (50) can hold. Therefore, equality (47) holds. The lemma is proved. □

It is worth noting that if

PS {P, m, B (g, 1)}_{E}^{n}

is a planet system and

∥ g ∥ \geq \sqrt{2}

, according to Lemma 9 and

p_{i}, p_{j} \in B (g, 1)

, then we have that

0 \leq θ_{i, j} = ∠ (p_{i}, p_{j}) \leq max_{α, β \in B (g, 1)} {∠ (α, β)} = 2 arcsin \frac{1}{∥ g ∥} \leq \frac{π}{2}

(51)

for any

i, j = 1, 2, \dots, n

, and

{∥ F_{λ} (P) ∥}^{2} = \sum_{1 \leq i, j \leq 1} m_{i} m_{j} cos θ_{i, j} {(\frac{1}{{∥ p_{i} ∥}^{2} {∥ p_{j} ∥}^{2}})}^{\frac{λ}{2}},

(52)

{∥ F_{2} (P) ∥}^{λ} = {[\sum_{1 \leq i, j \leq 1} m_{i} m_{j} cos θ_{i, j} (\frac{1}{{∥ p_{i} ∥}^{2} {∥ p_{j} ∥}^{2}})]}^{\frac{λ}{2}},

(53)

where

m_{i} m_{j} cos θ_{i, j} \geq 0, i, j = 1, 2, \dots, n .

(54)

Now, we will define absolute λ-gravity variance and λ-gravity variance.

Let

PS {P, m, B (g, r)}_{E}^{n}

be a planet system, and let the functions

f_{λ} (P)

F_{λ} (P)

be an absolute λ-gravity function and a λ-gravity function of the planet system, respectively. We say the functions

{Var}_{λ} (P) = {Var}_{\frac{λ}{2}} (p^{- 2}, m) = \frac{8}{λ (λ - 2)} [f_{λ} (P) - f_{2}^{\frac{λ}{2}} (P)], 0 < λ \neq 2,

and

{Var}_{λ}^{*} (P) = \frac{8}{λ (λ - 2)} [{(\frac{∥ F_{λ} (P) ∥}{∥ F_{0} (P) ∥})}^{2} - {(\frac{∥ F_{2} (P) ∥}{∥ F_{0} (P) ∥})}^{λ}], 0 < λ \neq 2,

are absolute λ-gravity variance and λ-gravity variance of the planet system, respectively, where

p = (∥ p_{1} ∥, ∥ p_{2} ∥, \dots, ∥ p_{n} ∥), p^{- 2} = ({∥ p_{1} ∥}^{- 2}, {∥ p_{2} ∥}^{- 2}, \dots, {∥ p_{n} ∥}^{- 2}) .

Let

PS {P, m, B (g, 1)}_{E}^{n}

be a planet system, and

0 < λ

μ \neq 2

λ \neq μ

. By Lemma 2, we have

\frac{1}{{max}_{1 \leq i \leq n} {{∥ p_{i} ∥}^{2}}} \leq {[\frac{{Var}_{λ} (P)}{{Var}_{μ} (P)}]}^{\frac{2}{λ - μ}} \leq \frac{1}{{min}_{1 \leq i \leq n} {{∥ p_{i} ∥}^{2}}} .

(55)

∥ g ∥ \geq \sqrt{2}

, according to (51)-(54) and Lemma 2, we have

\frac{1}{{max}_{1 \leq i \leq n} {{∥ p_{i} ∥}^{4}}} \leq {[\frac{{Var}_{λ}^{*} (P)}{{Var}_{μ}^{*} (P)}]}^{\frac{2}{λ - μ}} \leq \frac{1}{{min}_{1 \leq i \leq n} {{∥ p_{i} ∥}^{4}}} .

(56)

Let P be a planetary sequence in the solar system, and the gravity of the planet

p_{i}

to the Earth o is

F (p_{i})

i = 1, 2, \dots, n

. Then inequalities (55) and (56) can be rewritten as

min_{1 \leq i \leq n} {\frac{∥ F (p_{i}) ∥}{m_{i}}} \leq {[\frac{{Var}_{λ} (P)}{{Var}_{μ} (P)}]}^{\frac{2}{λ - μ}} \leq max_{1 \leq i \leq n} {\frac{∥ F (p_{i}) ∥}{m_{i}}}

(57)

and

min_{1 \leq i \leq n} {\frac{∥ F (p_{i}) ∥}{m_{i}}} \leq {[\frac{{Var}_{λ}^{*} (P)}{{Var}_{μ}^{*} (P)}]}^{\frac{1}{λ - μ}} \leq max_{1 \leq i \leq n} {\frac{∥ F (p_{i}) ∥}{m_{i}}},

(58)

respectively.

Let

PS {P, m, B (g, 1)}_{E}^{n}

be a planet system. By Lemma 8, if

λ > μ > 2

, then

\frac{{Var}_{λ} (P)}{{Var}_{μ} (P)} \geq \frac{μ}{λ} {[f_{2} (P)]}^{\frac{λ - μ}{2}} .

(59)

λ > μ > 2

and

∥ g ∥ \geq \sqrt{2}

, according to (51)-(54) and Lemma 8, we have

\frac{{Var}_{λ}^{*} (P)}{{Var}_{μ}^{*} (P)} \geq \frac{μ}{λ} {[\frac{∥ F_{2} (P) ∥}{∥ F_{0} (P) ∥}]}^{λ - μ},

(60)

where the coefficient

μ / λ

is the best constant in (59) and (60).

Remark 6 For some new literature related to space science, please see [4] and [25].

Acknowledgements

This work is supported by the Natural Science Foundation of China (No. 10671136) and the Natural Science Foundation of Sichuan Province Education Department (No. 07ZA207).

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in this paper. All authors read and approved the final manuscript.

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Titel: Inequalities involving Dresher variance mean
verfasst von: Jiajin Wen
Tianyong Han
Sui Sun Cheng
Publikationsdatum: 01.12.2013
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2013
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1186/1029-242X-2013-366

Springer Professional

Inequalities involving Dresher variance mean

Abstract

Electronic supplementary material

Competing interests

Authors’ contributions

1 Introduction and main results

2 Proof of Theorem 1

3 Proof of Theorem 2

4 Proof of Theorem 3

5 Applications in space science

Acknowledgements

Competing interests

Authors’ contributions

Authors’ original submitted files for images

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Springer Professional

Abstract

Electronic supplementary material

Competing interests

Authors’ contributions

1 Introduction and main results

2 Proof of Theorem 1

3 Proof of Theorem 2

4 Proof of Theorem 3

5 Applications in space science

Acknowledgements

Competing interests

Authors’ contributions

Authors’ original submitted files for images

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