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Journal of Inequalities and Applications

Erschienen in:

Open Access 01.12.2012 | Research

On a Hilbert-type inequality with a homogeneous kernel in ℝ² and its equivalent form

verfasst von: Bing He

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2012

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Abstract

By using the way of weight functions and the technique of real analysis, a new integral inequality with a homogeneous kernel and the best constant factor in ℝ² is given. The equivalent form and the reverses are considered.

Mathematics Subject Classification (2000): 26D15.

Competing interests

The authors declare that they have no competing interests.

1. Introduction

One hundred years ago, Hilbert proved the following classic inequality [1]

\sum_{n} {\sum_{m} \frac{a_{m} b_{n}}{m + n} \leq π {(\sum_{n} α_{n}^{2})}^{1 / 2} (\sum_{n} b_{n}^{2})}^{1 / 2} .

(1.1)

The inequality (1.1) may be classified into several types (discrete and integral etc.), which is of great importance in analysis and its applications [1, 2]. Ever since the advent of inequality (1.1), all kinds of improvements and extensions can be seen in [3‐12]. Note that the kernel of (1.1) is homogeneous of degree -1. In 2009, [13] reviews the negative degree homogeneous kernel of the parameterized Hilbert-type inequalities.

In recent years, many authors have started on Hilbert-type inequality of 0-degree homo-geneous kernel and non-homogeneous kernel. They even established inequalities in ℝ². In 2008, Yang [14] obtained the improved inequality as follows: If p, r > 1, (1/p) + (1/q) = 1, (1/r) + (1/s) = 1, 0 < λ < 1 and the right-hand side integrals are convergent, then

\begin{gathered} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{f (x) g (y)}{{|x + y|}^{λ}} d x d y \\ < k_{λ} (r) {(\int_{- \infty}^{\infty} {|x|}^{p (1 - \frac{λ}{r})} f^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {|x|}^{q (1 - \frac{λ}{s})} g^{q} (x) d x)}^{1 / q}, \end{gathered}

(1.2)

where the constant factor

k_{λ} (r) = B (\frac{λ}{r}, \frac{λ}{s}) + B (1 - λ, \frac{λ}{r}) + B (1 - λ, \frac{λ}{s})

is the best possible.

Motivated by (1.2) and the technique of real analysis, we establish a new inequality in ℝ² with a homogeneous kernel of 0-degree. Furthermore, the equivalent form and the corresponding reverse inequalities are also considered.

In what follows, α₁, α₂ will be real numbers such that 0 < α₁ < α₂ < π.

2. Lemmas

LEMMA 2.1. If

k : = 2 ln (4 cos \frac{α_{1}}{2} sin \frac{α_{2}}{2}) - α_{1} cot α_{1} + (π - α_{2}) cot α_{2}

, the weight function

ϖ (x) : = \int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} \cdot \frac{1}{|y|} d y, x \in (- \infty, \infty),

(2.1)

then for all x ∈ (-∞, 0) ∪ (0, ∞)

ϖ (x) = k .

(2.2)

Proof. If x ∈ (-∞,0), then

ϖ (x) = \int_{- \infty}^{0} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} \cdot \frac{1}{- y} d y + \int_{0}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} \cdot \frac{1}{y} d y .

Letting u = y/x for the first integrals and u = -y/x for the second integrals gives

\begin{aligned} ϖ (x) & = \int_{0}^{\infty} min_{i \in {1, 2}} \frac{min {1, u^{2}}}{u^{2} + 2 u cos α_{i} + 1} \cdot \frac{1}{u} d u + \int_{0}^{\infty} min_{i \in {1, 2}} \frac{min {1, u^{2}}}{u^{2} - 2 u cos α_{i} + 1} \cdot \frac{1}{u} d u \\ = \int_{0}^{1} \frac{u}{u^{2} + 2 u cos α_{1} + 1} d u + \int_{1}^{\infty} \frac{u^{- 1}}{u^{2} + 2 u cos α_{1} + 1} d u \\ + \int_{0}^{1} \frac{u}{u^{2} - 2 u cos α_{2} + 1} d u + \int_{1}^{\infty} \frac{u^{- 1}}{u^{2} - 2 u cos α_{2} + 1} d u . \\ = \int_{0}^{1} \frac{u}{u^{2} + 2 u cos α_{1} + 1} d u + \int_{0}^{1} \frac{u}{u^{2} + 2 u cos α_{1} + 1} d u \\ + \int_{0}^{1} \frac{u}{u^{2} - 2 u cos α_{2} + 1} d u + \int_{0}^{1} \frac{u}{u^{2} - 2 u cos α_{2} + 1} d u \\ = 2 (\int_{0}^{1} \frac{u}{u^{2} + 2 u cos α_{1} + 1} d u + \int_{0}^{1} \frac{u}{u^{2} - 2 u cos α_{2} + 1} d u) \\ = 2 [(ln 2 cos \frac{α_{1}}{2} - \frac{α_{1}}{2} cot α_{1}) + (ln 2 sin \frac{α_{2}}{2} - \frac{α_{2}}{2} cot α_{2} + \frac{π}{2} cot α_{2})] \\ = 2 ln (4 cos \frac{α_{1}}{2} sin \frac{α_{2}}{2}) - α_{1} cot α_{1} + (π - α_{2}) cot α_{2} = k . \end{aligned}

(2.3)

Similarly, ϖ(x) = k for x ∈ (0, ∞). Hence (2.2) is valid for x ∈ (-∞, 0) ∪ (0, ∞). □

Note. (i) It is obvious that ϖ(0) = 0. (ii) If α₁ = α₂ = α, then

min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} = \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α + y^{2}},

and ϖ(x) = 2 ln (2 sin α) + (π - 2α) cot α.

LEMMA 2.2. If

p > 1, \frac{1}{p} + \frac{1}{q} = 1

and f(x) is a nonnegative measurable function in (-∞,∞), then for all x ∈ (-∞, 0) ∪ (0, ∞)

\begin{gathered} J : = \int_{- \infty}^{\infty} {|y|}^{- 1} {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} f (x) d x)}^{p} d y \\ \leq k^{p} \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x . \end{gathered}

(2.4)

Proof. By Hölder's inequality with weight [15] and Lemma 2.1, we obtain

\begin{gathered} {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} f (x) d x)}^{p} \\ = {[\int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} (\frac{{|x|}^{1 / q}}{{|y|}^{1 / p}} f (x)) (\frac{{|y|}^{1 / p}}{{|x|}^{1 / q}}) d x]}^{p} \\ \leq \int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} \frac{{|x|}^{p - 1}}{|y|} f^{p} (x) d x \\ \times {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} \frac{{|y|}^{q - 1}}{|x|} d x)}^{p - 1} \\ = k^{p - 1} |y| \int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} \frac{{|x|}^{p - 1}}{|y|} f^{p} (x) d x . \end{gathered}

(2.5)

By Fubini theorem, we find

\begin{aligned} J & \leq k^{p - 1} \int_{- \infty}^{\infty} (\int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} \frac{{|x|}^{p - 1}}{|y|} f^{p} (x) d x) d y \\ = k^{p - 1} \int_{- \infty}^{\infty} (\int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} \frac{{|x|}^{p - 1}}{|y|} d y) f^{p} (x) d x \\ = k^{p} \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x . \end{aligned}

□

LEMMA 2.3. If

0 < p < 1, \frac{1}{p} + \frac{1}{q} = 1

and g(x) is a nonnegative measurable function in (-∞,∞), then for all x ∈ (-∞, 0) ∪ (0, ∞)

J \geq k^{p} \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x,

(2.6)

\begin{gathered} L : = \int_{- \infty}^{\infty} {|x|}^{- 1} {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} g (y) d y)}^{q} d x \\ \leq k^{q} \int_{- \infty}^{\infty} {|y|}^{q - 1} g^{q} (y) d y, \end{gathered}

(2.7)

where

k = 2 ln (4 cos \frac{α_{1}}{2} sin \frac{α_{2}}{2}) - α_{1} cot α_{1} + (π - α_{2}) cot α_{2}

Proof. It can be completed similarly by following the proof of Lemma 2.2 as long as applying the reverse Hölder's inequality [15], hence we omit the details. Since q < 0, thus (2.7) takes the positive inequality. □

3. Main results and applications

THEOREM 3.1. If

p > 1, \frac{1}{p} + \frac{1}{q} = 1, f, g \geq 0

such that

0 < \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x < \infty

and

0 < \int_{- \infty}^{\infty} {|x|}^{q - 1} g^{q} (x) d x < \infty

, then we obtain the following equivalent inequalities

\begin{gathered} I : = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} f (x) g (y) d x d y \\ < k {(\int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {|x|}^{q - 1} g^{q} (x) d x)}^{1 / q}, \end{gathered}

(3.1)

\begin{gathered} J = \int_{- \infty}^{\infty} {|y|}^{- 1} {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} f (x) d x)}^{p} d y \\ < k^{p} \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x, \end{gathered}

(3.2)

where the constant factors

k = 2 ln (4 cos \frac{α_{1}}{2} sin \frac{α_{2}}{2}) - α_{1} cot α_{1} + (π - α_{2}) cot α_{2}

and k^p are both the best possible.

Proof. If (2.5) takes the form of equality for some y ∈ (-∞,0) ∪ (0,∞), then there exist constants A and B such that they are not all zero and

A \frac{{|x|}^{p - 1}}{|y|} \cdot f^{p} (x) = B \frac{{|y|}^{q - 1}}{|x|} a .e . in(- \infty, \infty) \times (- \infty, \infty) .

i.e., A|x|^pf^p(x) = B|y|^qa.e. in (-∞,∞) × (-∞, ∞). We conform that A ≠ 0 (otherwise B = A = 0). Then

{|x|}^{p - 1} f^{p} (x) = \frac{B {|y|}^{q}}{A |x|}

a.e. in(-∞,∞), which contradicts the fact that

0 < \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x < \infty

. Hence (2.5) takes a strict inequality and the same as (2.4), thus (3.2) is valid.

By Hölder's inequality with weight [15], we find

\begin{gathered} I = \int_{- \infty}^{\infty} ({|y|}^{- 1 + (1 / q)} \int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} f (x) d x) {|y|}^{1 - (1 / q)} g (y) d y \\ \leq J^{1 / p} {(\int_{- \infty}^{\infty} {|y|}^{q - 1} g^{q} (y) d y)}^{1 / q} . \end{gathered}

(3.3)

By (3.2), we obtain (3.1). On the other hand, suppose that (3.1) is valid. Let

g (y) : = {|y|}^{- 1} {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} f (x) d x)}^{p - 1},

then

J = \int_{- \infty}^{\infty} {|y|}^{q - 1} g^{q} (y) d y

. In view of (2.4), J < ∞. If J = 0, then (3.2) is naturally valid; if J > 0, by (3.1), then

\begin{gathered} 0 < \int_{- \infty}^{\infty} {|y|}^{q - 1} g^{q} (y) d y = J = I \\ < k {(\int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {|x|}^{q - 1} g^{q} (x) d x)}^{1 / q} \end{gathered}

(3.4)

J^{1 / p} = {(\int_{- \infty}^{\infty} {|y|}^{q - 1} g^{q} (y) d y)}^{1 / p} < k {(\int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x)}^{1 / p} .

(3.5)

Hence we obtain (3.2). Thus (3.2) and (3.1) are equivalent.

For any ε > 0, suppose that

\tilde{f} (x) = \{\begin{matrix} x^{- 1 - \frac{2 ε}{p}}, & x \in [1, \infty), \\ 0, & x \in (- 1, 1), \\ {(- x)}^{- 1 - \frac{2 ε}{p}}, & x \in (- \infty, - 1], \end{matrix} \tilde{g} (x) = \{\begin{matrix} x^{- 1 - \frac{2 ε}{q}}, & x \in [1, \infty), \\ 0, & x \in (- 1, 1), \\ {(- x)}^{- 1 - \frac{2 ε}{q}}, & x \in (- \infty, - 1] . \end{matrix}

Then we get the following inequality

H (ε) : = {(\int_{- \infty}^{\infty} {|x|}^{p - 1} {\tilde{f}}^{p} (x) d x)}^{\frac{1}{p}} {(\int_{- \infty}^{\infty} {|x|}^{q - 1} {\tilde{g}}^{q} (x) d x)}^{\frac{1}{q}} = \frac{1}{ε},

(3.6)

I (ε) : = \int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} \tilde{f} (x) \tilde{g} (y) d x d y = I_{1} + I_{2} + I_{3} + I_{4},

(3.7)

where

\begin{gathered} I_{1} : = \int_{- \infty}^{- 1} {(- y)}^{- 1 - \frac{2 ε}{q}} (\int_{- \infty}^{- 1} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} {(- x)}^{- 1 - \frac{2 ε}{p}} d x) d y, \\ I_{2} : = \int_{- \infty}^{- 1} {(- y)}^{- 1 - \frac{2 ε}{q}} (\int_{1}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} x^{- 1 - \frac{2 ε}{p}} d x) d y, \\ I_{3} : = \int_{1}^{\infty} y^{- 1 - \frac{2 ε}{q}} (\int_{- \infty}^{- 1} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} {(- x)}^{- 1 - \frac{2 ε}{p}} d x) d y, \\ I_{4} : = \int_{1}^{\infty} y^{- 1 - \frac{2 ε}{q}} (\int_{1}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} x^{- 1 - \frac{2 ε}{p}} d x) d y . \end{gathered}

By Fubini theorem [16], it follows

\begin{gathered} I_{1} = I_{4} = \int_{1}^{\infty} y^{- 1 - \frac{2 ε}{q}} (\int_{1}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} x^{- 1 - \frac{2 ε}{p}} d x) d y \\ = \int_{1}^{\infty} y^{- 1 - 2 ε} (\int_{\frac{1}{y}}^{\infty} min_{i \in {1, 2}} \frac{min {u^{2}, 1}}{u^{2} + 2 u cos α_{i} + 1} u^{- 1 - \frac{2 ε}{p}} d u) d y (u = x / y) \\ = \int_{1}^{\infty} y^{- 1 - 2 ε} (\int_{\frac{1}{y}}^{1} \frac{u^{2}}{u^{2} + 2 u cos α_{1} + 1} u^{- 1 - \frac{2 ε}{p}} d u) d y \\ + \int_{1}^{\infty} y^{- 1 - 2 ε} (\int_{1}^{\infty} \frac{1}{u^{2} + 2 u cos α_{1} + 1} u^{- 1 - \frac{2 ε}{p}} d u) d y \\ = \int_{0}^{1} (\int_{\frac{1}{u}}^{\infty} y^{- 1 - 2 ε} d y) \frac{u^{1 - \frac{2 ε}{p}}}{u^{2} + 2 u cos α_{1} + 1} d u + \frac{1}{2 ε} \int_{1}^{\infty} \frac{u^{- 1 - \frac{2 ε}{p}}}{u^{2} + 2 u cos α_{1} + 1} d u \\ = \frac{1}{2 ε} (\int_{0}^{1} \frac{u^{1 + \frac{2 ε}{q}}}{u^{2} + 2 u cos α_{1} + 1} d u + \int_{1}^{\infty} \frac{1}{u^{2} + 2 u cos α_{1} + 1} u^{- 1 - \frac{2 ε}{p}} d u) . \\ I_{2} = I_{3} = \int_{1}^{\infty} y^{- 1 - \frac{2 ε}{q}} (\int_{1}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} - 2 x y cos α_{i} + y^{2}} x^{- 1 - \frac{2 ε}{p}} d x) d y \\ = \frac{1}{2 ε} (\int_{0}^{1} \frac{u^{1 + \frac{2 ε}{q}}}{u^{2} - 2 u cos α_{2} + 1} d u + \int_{1}^{\infty} \frac{1}{u^{2} - 2 u cos α_{2} + 1} u^{- 1 - \frac{2 ε}{p}} d u) . \end{gathered}

If the constant factor k in (3.1) is not the best possible, then there exists a constant 0 < M ≤ k, such that

\begin{gathered} I : = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} f (x) g (y) d x d y \\ < M {(\int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {|x|}^{q - 1} g^{q} (x) d x)}^{1 / q}, \end{gathered}

In view of (3.6) and (3.7), we obtain

\begin{gathered} \int_{0}^{1} \frac{u^{1 + \frac{2 ε}{q}}}{u^{2} + 2 u cos α_{1} + 1} d u + \int_{1}^{\infty} \frac{1}{u^{2} + 2 u cos α_{1} + 1} u^{- 1 - \frac{2 ε}{p}} d u + \int_{0}^{1} \frac{u^{1 + \frac{2 ε}{q}}}{u^{2} - 2 u cos α_{2} + 1} d u \\ + \int_{1}^{\infty} \frac{1}{u^{2} - 2 u cos α_{2} + 1} u^{- 1 - \frac{2 ε}{p}} d u = ε I (ε) < ε k H (ε) = k \end{gathered}

(3.8)

By (3.8), (2.3) and Fatou lemma [16], we find

\begin{gathered} k = \int_{0}^{1} \frac{u}{u^{2} + 2 u cos α_{1} + 1} d u + \int_{1}^{\infty} \frac{u^{- 1}}{u^{2} + 2 u cos α_{1} + 1} d u \\ + \int_{0}^{1} \frac{u}{u^{2} - 2 u cos α_{2} + 1} d u + \int_{1}^{\infty} \frac{u^{- 1}}{u^{2} - 2 u cos α_{2} + 1} d u \\ = \int_{0}^{1} lim_{ε \to 0^{+}} \frac{u^{1 + \frac{2 ε}{q}}}{u^{2} + 2 u cos α_{i} + 1} d u + \int_{1}^{\infty} lim_{ε \to 0^{+}} \frac{1}{u^{2} + 2 u cos α_{i} + 1} u^{^{- 1 - \frac{2 ε}{p}}} d u \\ + \int_{0}^{1} lim_{ε \to 0^{+}} \frac{u^{1 + \frac{2 ε}{q}}}{u^{2} - 2 u cos α_{i} + 1} d u + \int_{1}^{\infty} lim_{ε \to 0^{+}} \frac{1}{u^{2} - 2 u cos α_{i} + 1} u^{- 1 - \frac{2 ε}{p}} d u \\ \leq \frac{lim}{ε \to 0^{+}} (\int_{0}^{1} \frac{u^{1 + \frac{2 ε}{q}}}{u^{2} + 2 u cos α_{i} + 1} d u + \int_{1}^{\infty} \frac{1}{u^{2} + 2 u cos α_{i} + 1} u^{- 1 - \frac{2 ε}{p}} d u \\ + \int_{0}^{1} \frac{u^{1 + \frac{2 ε}{q}}}{u^{2} - 2 u cos α_{i} + 1} d u + \int_{1}^{\infty} \frac{1}{u^{2} - 2 u cos α_{i} + 1} u^{- 1 - \frac{2 ε}{p}} d u) \leq M . \end{gathered}

Hence k is the best value of (3.1). We conform that k^pis also the best value of (3.2). Otherwise, we can get a contradiction by (3.3) that (3.1) is not the best possible. □

THEOREM 3.2. If

0 < p < 1, \frac{1}{p} + \frac{1}{q} = 1, f, g \geq 0

such that

0 < \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x < \infty

and

0 < \int_{- \infty}^{\infty} {|x|}^{q - 1} g^{q} (x) d x < \infty

, then we have the following equivalent inequalities

\begin{gathered} I = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} f (x) g (y) d x d y \\ < k {(\int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {|x|}^{q - 1} g^{q} (x) d x)}^{1 / q}, \end{gathered}

(3.9)

\begin{gathered} J = \int_{- \infty}^{\infty} {|y|}^{- 1} {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} f (x) d x)}^{p} d y \\ > k^{p} \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x, \end{gathered}

(3.10)

\begin{gathered} L : = \int_{- \infty}^{\infty} {|x|}^{- 1} {(\int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} g (y) d y)}^{q} d x \\ < k^{q} \int_{- \infty}^{\infty} {|y|}^{q - 1} g^{q} (y) d y, \end{gathered}

(3.11)

where the constant factors

k = 2 ln (4 cos \frac{α_{1}}{2} sin \frac{α_{2}}{2}) - α_{1} cot α_{1} + (π - α_{2}) cot α_{2}

, both k^pand k^qare the best possible.

Proof. By Lemma 2.3, similar to the proof of (3.2), we obtain that (3.10) and (3.11) are valid. In view of the reverse equality of (3.3), (3.9) is valid too. On the other hand, suppose that (3.9) is valid, let g(y) defined as Theorem 3.1, it is obvious J > 0. If J = ∞, then (3.10) is valid naturally; if 0 < J < ∞, then by (3.9), we find

\begin{gathered} \int_{- \infty}^{\infty} {|y|}^{q - 1} g^{q} (y) d y = J = I \\ > k {(\int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {|x|}^{q - 1} g^{q} (x) d x)}^{1 / q}, \\ J^{1 / p} = {(\int_{- \infty}^{\infty} {|y|}^{q - 1} g^{q} (y) d y)}^{1 / p} > k {(\int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x)}^{1 / p} . \end{gathered}

(3.12)

Hence we obtain (3.10). Thus (3.10) and (3.9) are equivalent.

(3.11) and (3.9) are equivalent. In fact, we have proved (3.11) is valid above. On the other hand, suppose that (3.11) is valid, by the reverse Hölder's inequality with weight [15], we obtain

\begin{gathered} I = \int_{- \infty}^{\infty} ({|x|}^{1 - (1 / p)} f (x) d x) ({|x|}^{- 1 + (1 / p)} \int_{- \infty}^{\infty} min_{i \in {1, 2}} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α_{i} + y^{2}} g (y) d y) \\ \geq L^{1 / q} {(\int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d y)}^{1 / p} . \end{gathered}

(3.13)

By (3.11), we obtain (3.9), and it is equivalent between (3.11) and (3.9). Thus (3.9), (3.10), and (3.11) are equivalent.

k is the best value of (3.9). In fact, If there exists a constant M ≥ k, such that (3.9) is still valid as we replace k by M. By the reverse inequality of (3.8), we obtain

\begin{gathered} \int_{0}^{1} (\frac{1}{u^{2} + 2 u cos α_{1} + 1} + \frac{1}{u^{2} - 2 u cos α_{2} + 1}) u^{1 + \frac{2 ε}{q}} d u \\ + \int_{1}^{\infty} (\frac{1}{u^{2} + 2 u cos α_{1} + 1} + \frac{1}{u^{2} - 2 u cos α_{2} + 1}) u^{- 1 - \frac{2 ε}{p}} d u > k \end{gathered}

(3.14)

Suppose that

0 < ε_{0} < \frac{|q|}{2}

such that

\frac{2 ε_{0}}{q} + 1 > 0

. Letting 0 < ε ≤ ε₀ gives

u^{\frac{2 ε}{q}} \leq u^{\frac{2 ε_{0}}{q}} (u \in (0, 1])

and

\int_{0}^{1} (\frac{1}{u^{2} + 2 u cos α_{1} + 1} + \frac{1}{u^{2} - 2 u cos α_{2} + 1}) u^{1 + \frac{2 ε_{0}}{q}} d u \leq \frac{k}{2} \int_{0}^{1} u^{\frac{2 ε_{0}}{q}} = \frac{k}{2} \cdot \frac{1}{1 + (2 ε_{0}) / q} .

By Lebesgue control convergent theorem [16], it follows

\begin{gathered} \int_{0}^{1} (\frac{1}{u^{2} + 2 u cos α_{1} + 1} + \frac{1}{u^{2} - 2 u cos α_{2} + 1}) u^{1 + \frac{2 ε}{q}} d u \\ = \int_{0}^{1} (\frac{1}{u^{2} + 2 u cos α_{1} + 1} + \frac{1}{u^{2} - 2 u cos α_{2} + 1}) u d u + o (1) (ε \to 0^{+}) . \end{gathered}

Then by Levi theorem [16], we obtain

\begin{gathered} \int_{1}^{\infty} (\frac{1}{u^{2} + 2 u cos α_{1} + 1} + \frac{1}{u^{2} - 2 u cos α_{2} + 1}) u^{- 1 - \frac{2 ε}{p}} d u \\ = \int_{1}^{\infty} (\frac{1}{u^{2} + 2 u cos α_{1} + 1} + \frac{1}{u^{2} - 2 u cos α_{2} + 1}) \frac{1}{u} d u + \tilde{o} (1) (ε \to 0^{+}) . \end{gathered}

By (3.14), if follows that k ≥ M for ε → 0⁺. Hence k is the best value of (3.9). Furthermore, the constant factors in (3.10) and (3.11) are both the best value too. Otherwise, by (3.3) or (3.13), we may get a contradiction that the constant factor in (3.9) is not the best possible. □

By Note (ii), Theorems 3.1 and 3.2, it follows that

COROLLARY 3.3. If

p > 1, \frac{1}{p} + \frac{1}{q} = 1, f, g \geq 0

such that

0 < \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x < \infty

and

0 < \int_{- \infty}^{\infty} {|x|}^{q - 1} g^{q} (x) d x < \infty

, then we obtain the following equivalent inequalities

\begin{gathered} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{min {x^{2}, y^{2}}}{x^{2} + 2 x y cos α + y^{2}} f (x) g (y) d x d y \\ < k_{1} {(\int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {|x|}^{q - 1} g^{q} (x) d x)}^{1 / q}, \end{gathered}

(3.15)

\int_{- \infty}^{\infty} {|y|}^{- 1} {(\int_{- \infty}^{\infty} \frac{min \{x^{2}, y^{2}\}}{x^{2} + 2 x y cos α + y^{2}} f (x) d x)}^{p} d y < k_{1}^{p} \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x,

(3.16)

where the constant factors k₁ = 2 ln (2 sin α) + (π - 2α) cot α and

k_{1}^{p}

are both the best possible. In particular, for α = π/3 or 2π/3, it reduces to

\begin{gathered} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{min \{x^{2}, y^{2}\}}{x^{2} \pm x y + y^{2}} f (x) g (y) d x d y \\ < (ln 3 + \frac{\sqrt{3} π}{9}) {(\int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {|x|}^{q - 1} g^{q} (x) d x)}^{1 / q}, \end{gathered}

(3.17)

\int_{- \infty}^{\infty} {|y|}^{- 1} {(\int_{- \infty}^{\infty} \frac{min \{x^{2}, y^{2}\}}{x^{2} \pm x y + y^{2}} f (x) d x)}^{p} d y < {(ln 3 + \frac{\sqrt{3} π}{9})}^{p} \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x .

(3.18)

COROLLARY 3.4. If

0 < p < 1, \frac{1}{p} + \frac{1}{q} = 1, f, g \geq 0

such that

0 < \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x < \infty

and

0 < \int_{- \infty}^{\infty} {|x|}^{q - 1} g^{q} (x) d x < \infty

, then we have the following equivalent inequalities

\begin{gathered} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{min \{x^{2}, y^{2}\}}{x^{2} + 2 x y cos α + y^{2}} f (x) g (y) d x d y \\ > k_{1} {(\int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {|x|}^{q - 1} g^{q} (x) d x)}^{1 / q}, \end{gathered}

(3.19)

\int_{- \infty}^{\infty} {|y|}^{- 1} {(\int_{- \infty}^{\infty} \frac{min \{x^{2}, y^{2}\}}{x^{2} + 2 x y cos α + y^{2}} f (x) d x)}^{p} d y > k_{1}^{p} \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x,

(3.20)

\int_{- \infty}^{\infty} {|x|}^{- 1} {(\int_{- \infty}^{\infty} \frac{min \{x^{2}, y^{2}\}}{x^{2} + 2 x y cos α + y^{2}} g (y) dy)}^{q} d x < k_{1}^{q} \int_{- \infty}^{\infty} {|y|}^{q - 1} g^{q} (y) d y,

(3.21)

where the constant factors k₁ = 2 ln (2 sin α) + (π - 2α) cot α,

k_{1}^{p}

and

k_{1}^{q}

are both the best possible. In particular, for α = π/3 or 2π/3, it reduces to

\begin{gathered} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{min \{x^{2}, y^{2}\}}{x^{2} \pm x y + y^{2}} f (x) g (y) d x d y \\ > (ln 3 + \frac{\sqrt{3} π}{9}) {(\int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x)}^{1 / p} {(\int_{- \infty}^{\infty} {|x|}^{q - 1} g^{q} (x) d x)}^{1 / q}, \end{gathered}

(3.22)

\int_{- \infty}^{\infty} {|y|}^{- 1} {(\int_{- \infty}^{\infty} \frac{min \{x^{2}, y^{2}\}}{x^{2} \pm x y + y^{2}} f (x) d x)}^{p} d y > {(ln 3 + \frac{\sqrt{3} π}{9})}^{p} \int_{- \infty}^{\infty} {|x|}^{p - 1} f^{p} (x) d x,

(3.23)

\int_{- \infty}^{\infty} {|x|}^{- 1} {(\int_{- \infty}^{\infty} \frac{min \{x^{2}, y^{2}\}}{x^{2} \pm x y + y^{2}} g (y) d y)}^{q} d x < {(ln 3 + \frac{\sqrt{3} π}{9})}^{q} \int_{- \infty}^{\infty} {|y|}^{q - 1} g^{q} (y) d y .

(3.24)

Acknowledgements

The study was partially supported by the Emphases Natural Science Foundation of Guangdong Institution of Higher Learning, College and University (No. 05Z026).

Open AccessThis 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.

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Titel: On a Hilbert-type inequality with a homogeneous kernel in ℝ2 and its equivalent form
verfasst von: Bing He
Publikationsdatum: 01.12.2012
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2012
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1186/1029-242X-2012-94

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On a Hilbert-type inequality with a homogeneous kernel in ℝ² and its equivalent form

Abstract

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1. Introduction

2. Lemmas

3. Main results and applications

Acknowledgements

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Springer Professional

Abstract

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1. Introduction

2. Lemmas

3. Main results and applications

Acknowledgements

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