1 Introduction
For fixed
\(m\geq2\), let
\(\mathbf{x}=(x_{1},\ldots,x_{m})\) and
\(\mathbf{y}=(y_{1},\ldots,y_{m})\) denote two
m-tuples. Let
$$\begin{aligned}& x_{[1]} \geq x_{[2]}\geq\cdots\geq x_{[m]},\qquad y_{[1]}\geq y_{[2]}\geq\cdots\geq y_{[m]}, \\& x_{(1)} \leq x_{(2)}\leq\cdots\leq x_{(m)},\qquad y_{(1)}\leq y_{(2)}\leq\cdots\leq y_{(m)} \end{aligned}$$
be their ordered components. We say that
x
majorizes
y or
y is
majorized by
x and write
$$\mathbf{y}\prec\mathbf{x}$$
if
$$ \begin{aligned} &\sum_{i=1}^{k} y_{[i]} \leq \sum_{i=1}^{k} x_{[i]}, \quad k=1,\ldots,m-1, \\ &\sum_{i=1}^{m} y_{i} = \sum_{i=1}^{m} x_{i}. \end{aligned} $$
(1.1)
Note that (
1.1) is equivalent to
$$\sum_{i=m-k+1}^{m} y_{(i)}\leq \sum _{i=m-k+1}^{m} x_{(i)}, \quad k=1, \ldots,m-1. $$
The following notion of Schur-convexity generalizes the definition of a convex function via the notion of majorization.
A function
\(F:S\subseteq\mathbb{R}^{m}\rightarrow\mathbb{R}\) is called
Schur-
convex on
S if
$$ F(\mathbf{y})\leq F(\mathbf{x}) $$
(1.2)
for every
\(\mathbf{x}, \mathbf{y}\in S\) such that
$$\mathbf{y}\prec\mathbf{x}.$$
A relation between a one-dimensional convex function and an
m-dimensional Schur-convex function is included in the following
majorization theorem proved by Hardy
et al. (see [
1], [
2], p.333).
The following theorem gives a weighted generalization of the majorization theorem (see [
3], [
2], p.323).
The Jensen inequality in the form
$$ f \Biggl( \frac{1}{P_{m}}\sum_{i=1}^{m}p_{i}x_{i} \Biggr) \leq\frac {1}{P_{m}}\sum_{i=1}^{m}p_{i}f ( x_{i} ) $$
(1.3)
for a convex function
f, where
\(\mathbf{p}= ( p_{1},\ldots ,p_{m} ) \) is a nonnegative
m-tuple such that
\(P_{m}=\sum_{i=1}^{m}p_{i}>0\), can be obtained as a special case of the previous result putting
\(y_{1}=y_{2}=\cdots=y_{m}=\frac{1}{P_{m}}\sum_{i=1}^{m}p_{i}x_{i}\).
A natural problem of interest is the extension of the notation from m-tuples (vectors) to \(m\times l\) matrices \(\mathbf{A}=(a_{ij})\in\mathcal{M}_{ml}(\mathbb{R})\). Thus we introduce the notion of row stochastic and double stochastic matrices.
A matrix \(\mathbf{A}=(a_{ij})\in\mathcal{M}_{ml}(\mathbb{R})\) is called row stochastic if all of its entries are greater or equal to zero, i.e.
\(a_{ij}\geq0\) for \(i=1,\ldots,m\), \(j=1,\ldots,l\) and the sum of the entries in each row is equal to 1, i.e.
\(\sum_{j=1}^{l} a_{ij}=1\) for \(i=1,\ldots,m\). A square matrix \(\mathbf{A}=(a_{ij})\in \mathcal{M}_{l}(\mathbb{R})\) is called double stochastic if all of its entries are greater or equal to zero (nonnegative), i.e.
\(a_{ij}\geq0\) for \(i,j=1,\ldots,l\), and the sum of the entries in each column and each row is equal to 1, i.e.
\(\sum_{i=1}^{l} a_{ij}=1\) for \(j=1,\ldots,l\) and \(\sum_{j=1}^{l} a_{ij}=1\) for \(i=1,\ldots,l\).
It is well known that, for
\(\mathbf{x},\mathbf{y}\in\mathbb{R}^{l}\),
$$\mathbf{y}\prec\mathbf{x}\quad \text{if and only if}\quad \mathbf {y}= \mathbf{xA}$$
for some double stochastic matrix
\(\mathbf{A}\in\mathcal{M}_{l}(\mathbb {R})\).
The next generalization was obtained by Sherman (see [
4,
5]).
In 1929, Lidstone [
6] introduced a generalization of Taylor’s series: it approximates a given function in the neighborhood of two points instead of one. This series includes the polynomials later called Lidstone’s polynomials. These polynomials have been studied in the work of Boas [
7], Poritsky [
8], Widder [
9], and others. See also [
10].
In [
9], Widder proved the following fundamental lemma.
The Lidstone polynomial can be expressed in terms of
\(G_{n}(t,s)\) as
$$ \Lambda_{n}(t)=\int _{0}^{1} G_{n}(t,s)s\, ds. $$
(1.9)
To complete the Introduction, we state a definition of the divided differences and
n-convexity (see for example [
2]).
This definition may be extended to include the case in which some or all the points coincide. Assuming that
\(f^{(j-1)}(x)\) exists, we define
$$ [\underset{j\text{-times}}{\underbrace{x,\ldots,x}};f]=\frac {f^{(j-1)}(x)}{(j-1)!}. $$
(1.10)
From Definition
3, it follows that 2-convex functions are just convex functions. Furthermore, 1-convex functions are increasing functions and 0-convex functions are nonnegative functions.
2 Main results
First we prove an identity related to a generalization of Sherman’s inequality using Lidstone’s interpolating polynomial.
Using the previous result, we get the following generalizations of Sherman’s theorem for 2n-convex functions.
Under the assumptions of Sherman’s theorem the following generalizations are valid.
3 Bounds for identities related to generalizations of Sherman’s inequality
For two Lebesgue integrable functions
\(f,g:[\alpha,\beta]\rightarrow \mathbb{R}\), we consider the Čebyšev functional:
$$T(f,g):=\frac{1}{\beta-\alpha} \int _{\alpha}^{\beta }f(t)g(t)\, dt-\frac {1}{\beta-\alpha} \int _{\alpha}^{\beta}f(t)\, dt\cdot\frac{1}{\beta -\alpha } \int _{\alpha}^{\beta}g(t)\, dt. $$
We use the following two theorems, proved in [
11], to obtain generalizations of the results from the previous section.
For the sake of simplicity and to avoid an overload of notations, we define two functions as follows.
Let
\(\mathbf{x}=(x_{1},\ldots,x_{l})\in{}[\alpha,\beta]^{l}\),
\(\mathbf{y}=(y_{1},\ldots,y_{m})\in{}[\alpha,\beta]^{m}\),
\(\mathbf{a}=(a_{1},\ldots,a_{l})\in\mathbb{R}^{l}\),
\(\mathbf{b}=(b_{1},\ldots, b_{m})\in\mathbb{R}^{m}\) and the function
\(G_{n}\) be defined as in (
1.7) and (
1.8). The function
\(\mathcal{R}:[\alpha,\beta]\rightarrow\mathbb{R}\) is defined by
$$ \mathcal{R}(t)= \sum_{j=1}^{l} a_{j}G_{n} \biggl( \frac{x_{j}-\alpha}{\beta-\alpha},\frac{t-\alpha}{\beta-\alpha} \biggr) -\sum_{i=1}^{m} b_{i}G_{n} \biggl( \frac{y_{i}-\alpha}{\beta-\alpha},\frac{t-\alpha}{\beta-\alpha} \biggr) . $$
(3.3)
Using the Čebyšev functional we obtain a bound for the identity (
2.1) related to a generalization of Sherman’s inequality.
Using Theorem
8 we obtain the Grüss type inequality.
We present the Ostrowsky type inequality related to generalizations of Sherman’s inequality.
4 Mean value theorems and exponential convexity
In this section, we present mean-value theorems of Lagrange and Cauchy type using results from the previous section. We also use the so-called
exponential convexity method, established in [
12], in order to interpret our results in the form of exponentially convex functions or in the special case logarithmically convex functions. For some related results see also [
13,
14].
Motivated by the inequality (
2.5), we define the linear functional as follows.
Under the assumptions of Theorem
5, equipped with condition (
2.4), we define
$$\begin{aligned} \begin{aligned}[b] A(\phi) ={}& \sum_{j=1}^{l} a_{j}\phi(x_{j})- \sum_{i=1}^{m} b_{i}\phi(y_{i}) \\ &{}- \sum_{k=0}^{n-1} (\beta- \alpha)^{2k}\phi^{(2k)}(\alpha) \Biggl[ \sum _{j=1}^{l} a_{j}\Lambda_{k} \biggl( \frac{\beta-x_{j}}{\beta-\alpha} \biggr) - \sum_{i=1}^{m} b_{i}\Lambda_{k} \biggl( \frac{\beta-y_{i}}{\beta-\alpha} \biggr) \Biggr] \\ &{}- \sum_{k=0}^{n-1} (\beta- \alpha)^{2k}\phi^{(2k)}(\beta) \Biggl[ \sum _{j=1}^{l} a_{j}\Lambda_{k} \biggl( \frac{x_{j}-\alpha}{\beta-\alpha} \biggr) - \sum_{i=1}^{m} b_{i}\Lambda_{k} \biggl( \frac{y_{i}-\alpha}{\beta-\alpha} \biggr) \Biggr] . \end{aligned} \end{aligned}$$
(4.1)
Throughout the rest of this paper, I denotes an open interval in \(\mathbb{R}\).
The notation of
n-exponential convexity is introduced in [
14].
The following two lemmas are equivalent to the definition of convexity (see [
2], p.2).
In order to obtain results regarding the exponential convexity, we define the families of functions as follows.
For every choice of
\(2l+1\) mutually different points
\(z_{0},z_{1} ,\ldots,z_{2l}\in{}[\alpha,\beta]\) we define
-
\(\mathcal{F}_{1}\) = {\(f_{t}:[\alpha,\beta]\rightarrow\mathbb{R}:t\in I\) and \(t\mapsto{}[ z_{0},z_{1},\ldots,z_{2l};f_{t}]\) is n-exponentially convex in the Jensen sense on I};
-
\(\mathcal{F}_{2}\) = {\(f_{t}:[\alpha,\beta]\rightarrow\mathbb{R}:t\in I\) and \(t\mapsto{}[ z_{0},z_{1},\ldots,z_{2l};f_{t}]\) is exponentially convex in the Jensen sense on I};
-
\(\mathcal{F}_{3}\) = {\(f_{t}:[\alpha,\beta]\rightarrow\mathbb{R}:t\in I\) and \(t\mapsto{}[ z_{0},z_{1},\ldots,z_{2l};f_{t}]\) is 2-exponentially convex in the Jensen sense on I}.
The following corollary is an easy consequence of the previous theorem.
5 Applications to means
Using some families of convex functions which are given below, we construct different examples of exponentially convex functions. As consequences, applying the mean-value theorem of Cauchy type from the previous section to these special families of functions, we establish new classes of two-parameter Cauchy type means that are symmetric and have monotone properties over both parameters.
Throughout this section id denotes the identity function, i.e.
\(\mathrm{id}(x)=x\) for each \(x\in\mathbb{R}\).
6 Conclusions
In this paper we give generalizations of Sherman’s theorem from which a majorization theorem follows as a special case. Our results hold for real, not necessarily nonnegative entries of vectors a, b and matrix A, as is the case of Sherman’s theorem, and for 2n-convex functions which are in a special case convex in the usual sense. The methods used are based on classical real analysis and the application of Lidstone’s interpolating polynomials and exponential convexity method and can be extended to the investigation of other inequalities.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to deriving all the results of this article, and read and approved the final manuscript.