On some inequalities for functions with nondecreasing increments of higher order

Abstract

We investigate a class of functions with nondecreasing increments of higher order. A generalization of Brunk’s theorem is proved for that class of functions. Also, we consider functions with nondecreasing increments of order three, we obtain the Levinson-type inequality, a generalization of Burkill-Mirsky-Pečarić’s results, and a result for the integral mean of a function with nondecreasing increments of higher order.

1 Introduction

Let ${\mathbb{R}}^k$ denote the k-dimensional vector lattice of points $\mathbf{x}=(x_1,\dots ,x_k)$, $x_i$ be real for $i=1,\dots ,k$, with the partial ordering $\mathbf{x}=(x_1,\dots ,x_k)\le \mathbf{y}=(y_1,\dots ,y_k)$ if and only if $x_i\le y_i$ for $i=1,\dots ,k$. We denote

$$a\mathbf{x}+b\mathbf{y}=(a{x}_1+b{y}_1,\dots ,a{x}_k+b{y}_k),$$

where $a,b\in \mathbb{R}$, and k-tuple $(0,\dots ,0)$ is denoted by 0.

For $\mathbf{a},\mathbf{b}\in {\mathbb{R}}^k$, $\mathbf{a}\le \mathbf{b}$, a set $\{\mathbf{x}\in {\mathbb{R}}^k:\mathbf{a}\le \mathbf{x}\le \mathbf{b}\}$ is called an interval $[\mathbf{a},\mathbf{b}]$. The following definition of a function with nondecreasing increments is given in [1].

Definition 1.1 A real-valued function f on an interval $\mathbf{I}\subset {\mathbb{R}}^k$ is said to have nondecreasing increments if

$$f(\mathbf{a}+\mathbf{h})-f(\mathbf{a})\le f(\mathbf{b}+\mathbf{h})-f(\mathbf{b}),$$

(1)

whenever $\mathbf{a}\in \mathbf{I}$, $\mathbf{b}+\mathbf{h}\in \mathbf{I}$, $\mathbf{0}\le \mathbf{h}\in {\mathbb{R}}^k$, $\mathbf{a}\le \mathbf{b}$.

In the same paper [1], Brunk gave some properties of that family of functions. The most remarkable result for functions with nondecreasing increments is the following Brunk theorem (see also [[2], p.266]).

Theorem 1.2 Let I be an interval in ${\mathbb{R}}^k$; $\mathbf{X}(t)=(X_1(t),\dots ,X_k(t))$ be a vector of functions where $X_i$’s ($1\le i\le k$), are nondecreasing and continuous from the right on $[a,b)$. Let H be continuous from the left and of bounded variation on $[a,b)$ with $H(a)=0$. Then

$${\int }_{[a,b)}f(\mathbf{X}(t))dH(t)\ge 0$$

holds for every continuous function $f:\mathbf{I}\to \mathbb{R}$ with nondecreasing increments if and only if

and

$${\int }_{[a,t]}H(u)d\mathbf{X}(u)\ge 0\mathit{\text{for}}[a,t]\subset [a,b],$$

where $\int Hd\mathbf{X}=(\int Hd{X}_1,\dots ,\int Hd{X}_k)$.

More results about functions with nondecreasing increments can be found in papers [3] and [4]. The following theorem is the Jensen-Steffensen type inequality for a function with nondecreasing increments and it is proved in [4].

Theorem 1.3 Let $G:[a,b]\to \mathbb{R}$ be a function of bounded variation such that

$$G(a)\le G(x)\le G(b),G(b)>G(a),$$

(2)

and let $\mathbf{X}(t)$ be a continuous nondecreasing map from the real interval $[a,b]$ to the interval $\mathbf{I}\subset {\mathbb{R}}^k$. If $f:\mathbf{I}\to \mathbb{R}$ is a continuous function with nondecreasing increments, then

$$f\left(\frac{{\int }_a^b\mathbf{X}(t)dG(t)}{{\int }_a^{b}dG(t)}\right)\le \frac{{\int }_a^{b}f(\mathbf{X}(t))dG(t)}{{\int }_a^{b}dG(t)},$$

(3)

where ${\int }_a^b\mathbf{X}dG$ is the vector $({\int }_a^b{X}_{1}dG,\dots ,{\int }_a^b{X}_{k}dG)$.

The following theorem gives us a Jensen-type inequality for a function with nondecreasing increments when the finite sequence of k-tuples $({\mathbf{X}}_{\mathbf{1}},\dots ,{\mathbf{X}}_{\mathbf{n}})$ is monotone in means [3]. It is a Pečarić’s generalization of Burkill-Mirsky’s result. Firstly, let us describe a monotonicity in means. Let $p_i$, $i=1,\dots ,n$, be positive numbers, $[\mathbf{a},\mathbf{b}]$ be an interval in ${\mathbb{R}}^k$. A finite sequence $({\mathbf{X}}_{\mathbf{1}},\dots ,{\mathbf{X}}_{\mathbf{n}})\in {[\mathbf{a},\mathbf{b}]}^n$ is said to be nondecreasing in means with respect to weights $\mathbf{p}=(p_1,\dots ,p_n)$ if

$${\mathbf{X}}_{\mathbf{1}}\le A_2(\mathbf{X};\mathbf{p})\le \cdots \le A_n(\mathbf{X};\mathbf{p}),$$

(4)

where

$$A_j(\mathbf{X};\mathbf{p})=\frac{1}{P_j}\sum _{i=1}^j{p}_i{\mathbf{X}}_{\mathbf{i}},P_j=\sum _{i=1}^n{p}_i.$$

If inequalities are reversed in (4), then $({\mathbf{X}}_{\mathbf{1}},\dots ,{\mathbf{X}}_{\mathbf{n}})$ is nonincreasing in means.

Theorem 1.4 Let I be an interval in ${\mathbb{R}}^k$, $f:\mathbf{I}\to \mathbb{R}$ be a continuous function with nondecreasing increments and let $p_1,\dots ,p_n$ be positive numbers. If

$$({\mathbf{X}}_{\mathbf{1}},\dots ,{\mathbf{X}}_{\mathbf{n}})({\mathbf{X}}_{\mathbf{i}}\in \mathbf{I};i=1,\dots ,n)$$

is nondecreasing or nonincreasing in means with respect to weights $\mathbf{p}=(p_1,\dots ,p_n)$, then the Jensen-type inequality

$$f\left(\frac{1}{P_n}\sum _{i=1}^n{p}_i{\mathbf{X}}_{\mathbf{i}}\right)\le \frac{1}{P_n}\sum _{i=1}^n{p}_{i}f({\mathbf{X}}_{\mathbf{i}})$$

holds.

In this paper, we extend the idea of functions with nondecreasing increments. Namely, we define a new class of functions with nondecreasing increments of higher order and prove a result similar to the above-mentioned Brunk theorem. In the third section, we consider functions with nondecreasing increments of order three. Finally, in the last section, a result for an arithmetic integral mean of a function with nondecreasing increments of higher order is given.

2 Functions with nondecreasing increments of order n

Let I be an interval from ${\mathbb{R}}^k$. Let us write

$${\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{1}}}f(\mathbf{x})=f(\mathbf{x}+{\mathbf{h}}_{\mathbf{1}})-f(\mathbf{x})$$

and inductively,

$${\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{1}}}{\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{2}}}\cdots {\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{n}}}f(\mathbf{x})={\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{1}}}({\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{2}}}\cdots {\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{n}}}f(\mathbf{x})),$$

where $\mathbf{x},\mathbf{x}+{\mathbf{h}}_{\mathbf{1}}+\cdots +{\mathbf{h}}_{\mathbf{n}}\in \mathbf{I}$, $\mathbf{0}\le {\mathbf{h}}_{\mathbf{i}}\in {\mathbb{R}}^k$ ($i=1,\dots ,n$). Using this notation with $\mathbf{h}={\mathbf{h}}_{\mathbf{1}}$, $\mathbf{s}={\mathbf{h}}_{\mathbf{2}}$, $\mathbf{b}=\mathbf{a}+\mathbf{s}$, a condition (1) from the definition of a function with nondecreasing increments becomes

$${\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{1}}}{\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{2}}}f(\mathbf{a})\ge 0.$$

Let us extend that definition to the following.

Definition 2.1 A real-valued function f on an interval $\mathbf{I}\subset {\mathbb{R}}^k$ is a function with nondecreasing increments of order n if

$${\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{1}}}\cdots {\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{n}}}f(\mathbf{x})\ge 0,$$

whenever $\mathbf{x},\mathbf{x}+{\mathbf{h}}_{\mathbf{1}}+\cdots +{\mathbf{h}}_{\mathbf{n}}\in \mathbf{I}$, $\mathbf{0}\le {\mathbf{h}}_{\mathbf{i}}\in {\mathbb{R}}^k$ ($i=1,\dots ,n$).

Brunk observed that even if $k=1$ and $n=2$, this does not imply continuity (see [1]). Indeed, every solution of Cauchy’s equation $f(\mathbf{x}+\mathbf{y})=f(\mathbf{x})+f(\mathbf{y})$ is a function with nondecreasing increments of order n with null increments, i.e., ${\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{1}}}\cdots {\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{n}}}f(\mathbf{x})=0$. If the n th partial derivatives $f_{i_1\cdots i_n}(\mathbf{x})=\frac{{\partial }^n}{\partial x_{i_1}\cdots x_{i_n}}f(\mathbf{x})$ exist, they are nonnegative. If f is a continuous function with nondecreasing increments of order n, it may be approximated uniformly on I by polynomials having nonnegative n th partial derivatives. To see this, let us set, for convenience, $\mathbf{I}=[\mathbf{0},\mathbf{1}]$, $\mathbf{1}=(1,\dots ,1)$. It is known that the Bernstein polynomials

$$\sum _{i_1=0}^{n_1}\cdots \sum _{i_k=0}^{n_k}f(\frac{i_1}{n_1},\dots ,\frac{i_k}{n_k})\prod _{j=1}^k\left(\genfrac{}{}{0ex}{}{n_j}{i_j}\right)x_j^{i_j}{(1-x_j)}^{n_j-i_j}$$

converge uniformly to f on I as $n_1\to \mathrm{\infty },\dots ,n_k\to \mathrm{\infty }$, if f is continuous. Furthermore, if f is a function with nondecreasing increments of order n, these polynomials have nonnegative n th partial derivatives, as may be shown by repeated application of the formula (see [1])

$$\frac{d}{dx}\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right)a_i{x}^i{(1-x)}^{n-i}=n\sum _{i=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{i}\right)(a_{i+1}-a_i)x^i{(1-x)}^{n-1-i}.$$

The aim of the rest of this section is to prove a result similar to Theorem 1.2. Let us introduce some further notations.

Let $p_1,\dots ,p_r$ be positive integers and let $p_1+\cdots +p_r=w$. Let ${(i_1^{p_1}\cdots i_r^{p_r})}_p$ be a set of all permutations with repetitions whose elements are from the multiset

$$S=\{\underset{p_1\text{-times}}{\underbrace{i_1,\dots ,i_1}},\underset{p_2\text{-times}}{\underbrace{i_2,\dots ,i_2}},\dots ,\underset{p_r\text{-times}}{\underbrace{i_r,\dots ,i_r}}\},i_1<\cdots <i_r,i_1,\dots ,i_r\in \{1,\dots ,k\}.$$

There are $\frac{w!}{p_1!p_2!\cdots p_r!}$ elements in the class ${(i_1^{p_1}\cdots i_r^{p_r})}_p$.

For $0<p_1\le p_2\le \cdots \le p_r$, $p_1+\cdots +p_r=w$, let ${(p_1\cdots p_r)}_c$ be a set whose elements are described in the following way. We say that permutation $j_1\cdots j_w$ belongs to the set ${(p_1\cdots p_r)}_c$ iff there exist $i_1,i_2,\dots ,i_r\in \{1,2,\dots ,k\}$, $i_1<i_2<\cdots <i_r$ and permutation σ of the multiset $\{p_1\cdots p_r\}$ such that $j_1\cdots j_w\in {(i_1^{\sigma (p_1)}\cdots i_r^{\sigma (p_r)})}_p$. Family of all classes ${(p_1\cdots p_r)}_c$ is denoted with $C_w^k$.

For illustration, we describe the above notation on one example. Let $k=5$ and $w=4$. Classes ${(p_1\cdots p_r)}_c$ are the following: ${(1,1,1,1)}_c$, ${(1,1,2)}_c$, ${(1,3)}_c$, ${(2,2)}_c$ and ${(4)}_c$. Let us describe the elements of the set ${(1,1,2)}_c$. There are three different permutations of the multiset $\{1,1,2\}$. These are

$$\left(\begin{array}{ccc}1& 1& 2\\ 1& 1& 2\end{array}\right),\left(\begin{array}{ccc}1& 1& 2\\ 1& 2& 1\end{array}\right),\left(\begin{array}{ccc}1& 1& 2\\ 2& 1& 1\end{array}\right).$$

So, ${(i_1^{\sigma (p_1)}\cdots i_r^{\sigma (p_r)})}_p$ are ${(i_1,i_2,i_3,i_3)}_p$, ${(i_1,i_2,i_2,i_3)}_p$, ${(i_1,i_1,i_2,i_3)}_p$, where $i_1<i_2<i_3$ and $i_1,i_2,i_3\in \{1,2,3,4,5\}$. If, for example, ${(i_1,i_2,i_3,i_3)}_p={(2,3,5,5)}_p$, then it contains all permutations with repetitions of elements $2,3,5,5$, i.e., ${(2,3,5,5)}_p=\{2355,2535,2533,\dots ,5532\}$ and it has $\frac{4!}{2!}=12$ elements.

In the following text, H is a function of bounded variation on $[a,b]$ with $H(a)=0$ and $i_1,i_2,\dots ,i_n\in \{1,2,\dots ,k\}$. Let $K_{i_1\cdots i_n}^n$ be a function such that

$$K_{i_1\cdots i_n}^n(t)={\int }_a^t{K}_{i_1\cdots i_{n-1}}^{n-1}(x_n)d{X}_{i_n}(x_n)\text{for }n\ge 2$$

and

$$K_{i_1}^1(t)={\int }_a^{t}H(x_1)d{X}_{i_1}(x_1).$$

Further we write

where S is a multiset with elements from $\{1,2,\dots ,k\}$.

It is obvious that

$$d\{\prod (S)(x)\}=-\sum _{j\in S}d{X}_j(x)\prod (S\setminus \{j\})(x),$$

and

$$d{K}_{i_1\cdots i_n}^n(t)=K_{i_1\cdots i_{n-1}}^{n-1}(t)d{X}_{i_n}(t).$$

Now, the following result holds.

Lemma 2.2 Let w be a fixed positive integer. Then

holds for every $m\in \{1,2,\dots ,w\}$.

Proof We prove it using induction by m. For $m=1$, using integration by parts, we have

$$\begin{array}{rcl}{\int }_a^t\prod (\{i_1,\dots ,i_w\})(x)dH(x)& =& -{\int }_a^{t}H(x)d(\prod (\{i_1,\dots ,i_w\})(x))\\ =& {\int }_a^{t}H(x)\sum _{j_1=1}^{w}d{X}_{j_1}(x)\prod (\{i_1,\dots ,i_m\}\setminus \{i_{j_1}\})(x)\\ =& \sum _{j_1=1}^w{\int }_a^t\prod (\{i_1,\dots ,i_w\}\setminus \{i_{j_1}\})(x)d{K}_{i_{j_1}}^1(x).\end{array}$$

Let us suppose that the statement holds for $m-1$ and let us apply integration by parts on the right-hand side of the formula.

 □

Especially for $m=w$, we have

$$\begin{array}{rcl}{\int }_a^t\prod (\{i_1,\dots ,i_w\})(x)dH(x)& =& \sum _{j_1=1}^w\cdots \underset{k<w}{\overset{w}{\underset{j_w\ne j_k}{\sum _{j_w=1}}}}{\int }_a^{t}d{K}_{i_{j_1}\cdots i_{j_w}}^w(x)\\ =& \sum _{j_1=1}^w\cdots \underset{k<w}{\overset{w}{\underset{j_w\ne j_k}{\sum _{j_w=1}}}}{K}_{i_{j_1}\cdots i_{j_w}}^w(t)\\ =& p_1!p_2!\cdots p_r!\sum _{i_{j_1}\cdots i_{j_w}\in {(i_1^{p_1}\cdots i_r^{p_r})}_p}{K}_{i_{j_1}\cdots i_{j_w}}^w(t),\end{array}$$

(5)

where $\{i_{j_1},\dots ,i_{j_w}\}=\{\underset{p_1\text{-times}}{\underbrace{i_1,\dots ,i_1}},\dots ,\underset{p_r\text{-times}}{\underbrace{i_r,\dots ,i_r}}\}$, $i_1<i_2<\cdots <i_r$; $i_1,i_2,\dots ,i_r\in \{1,2,\dots ,k\}$, $p_1+\cdots +p_r=w$.

Example 2.3 If $w=3$, $i_1=i_2=1$, $i_3=2$, then

$$\begin{array}{rcl}{\int }_a^t\prod (\{1,1,2\})(x)dH(x)& =& \sum _{j_1=1}^3\underset{j_{2\ne }{j}_1}{\overset{3}{\sum _{j_2=1}}}\underset{j_3\ne j_1,j_2}{\overset{3}{\sum _{j_3=1}}}{K}_{i_{j_1}{i}_{j_2}{i}_{j_3}}^3(t)\\ =& 2!1!(K_{112}^3+K_{121}^3+K_{211}^3).\end{array}$$

Furthermore, if we suppose

$${\int }_a^b{X}_{j_1}(u)\cdots X_{j_s}(u)dH(u)=0(j_1,\dots ,j_s\in \{1,\dots ,k\},s=0,\dots ,w),$$

then

$$\begin{array}{rcl}{p}_1!\cdots p_r!\sum K_{i_{j_1}\cdots i_{j_w}}^w(b)& =& {\int }_a^b\prod (\{i_1,\dots ,i_w\})(x)dH(x)\\ =& \sum {(-1)}^s{\int }_a^b{X}_{j_1}(x)\cdots X_{j_s}(x)X_{j_{s+1}}(b)\cdots X_{j_w}(b)dH(x)=0.\end{array}$$

(6)

Theorem 2.4 Let $\mathbf{X}:[a,b]\to \mathbf{I}\subset {\mathbb{R}}^k$ be a continuous function. Let H be a function of bounded variation on $[a,b]$ with $H(a)=H(b)=0$ and let f have continuous $(n-1)$th partial derivatives, $n\ge 2$. Then the following statement holds: if

$${\int }_a^b{X}_{i_1}(u)\cdots X_{i_m}(u)dH(u)=0(i_1,\dots ,i_m\in \{1,\dots ,k\},m=1,2,\dots ,n-1),$$

then

$$\begin{array}{rcl}{\int }_a^{b}f(\mathbf{X}(t))dH(t)& =& {(-1)}^{n-1}\sum _{{(p_1\cdots p_r)}_c\in C_{n-1}^k}\frac{1}{p_1!\cdots p_r!}\\ \times \sum _{{(i_1^{p_1}\cdots i_r^{p_r})}_p\subseteq {(p_1\cdots p_r)}_c}{\int }_a^b{f}_{\underset{p_1\text{-}\mathit{times}}{\underbrace{i_1\cdots i_1}}\cdots \underset{p_r\text{-}\mathit{times}}{\underbrace{i_r\cdots i_r}}}(\mathbf{X}(t))\\ \times d({\int }_a^t\prod \left(\{i_1^{p_1},\dots ,i_r^{p_r}\}\right)(x)dH(x)).\end{array}$$

(7)

Proof For $n=2$, we have

$$\begin{array}{rcl}{\int }_a^{b}f(\mathbf{X}(t))dH(t)& =& -\sum _{i=1}^k{\int }_a^b{f}_i(\mathbf{X}(t))H(t)d{X}_i(t)\\ =& -\sum _{i=1}^k{\int }_a^b{f}_i(\mathbf{X}(t))d{K}_i^1(t)\\ =& -\sum _{i=1}^k{\int }_a^b{f}_i(\mathbf{X}(t))d({\int }_a^{t}H(x)d{X}_i(x))\\ =& -\sum _{i=1}^k{\int }_a^b{f}_i(\mathbf{X}(t))d({\int }_a^{t}H(x)d(X_i(x)-X_i(t)))\\ =& \sum _{i=1}^k{\int }_a^b{f}_i(\mathbf{X}(t))d({\int }_a^{t}H(x)d(X_i(t)-X_i(x)))\\ =& -\sum _{i=1}^k{\int }_a^b{f}_i(\mathbf{X}(t))d({\int }_a^t(X_i(t)-X_i(x))dH(x))\\ =& -\sum _{i=1}^k{\int }_a^b{f}_i(\mathbf{X}(t))d({\int }_a^t\prod (\{i\})(x)dH(x)).\end{array}$$

If we have ${\int }_a^b{X}_{i_1}(u)\cdots X_{i_m}(u)dH(u)=0$ for $m=1,2,\dots ,n-2$, $i_1,\dots ,i_m\in \{1,\dots ,k\}$ and if we suppose that (7) holds for $(n-1)$, then

by (5) and (6). □

Theorem 2.5 Let X be a nondecreasing continuous map from the real interval $[a,b]$ into an interval $\mathbf{I}\subset {\mathbb{R}}^k$, and let H be a function of bounded variation on $[a,b]$ with $H(a)=0$. Then

$${\int }_a^{b}f(\mathbf{X}(t))dH(t)\ge 0$$

(8)

for every continuous function f with nondecreasing increments of order n on I if and only if

(9)

(10)

for $i_1,\dots ,i_m\in \{1,\dots k\}$, $m=1,2,\dots ,n-1$ and

$${(-1)}^n{\int }_a^t\prod (\{i_1,\dots ,i_{n-1}\})(u)dH(u)\ge 0$$

(11)

for all $t\in [a,b]$, $i_1,\dots ,i_{n-1}\in \{1,\dots k\}$.

Proof Necessity: The validity of (8) for constant functions $f=1$ and $f=-1$ implies (9). From (8) for $f(x)=x_{i_1}\cdots x_{i_s}$ and $f(x)=-x_{i_1}\cdots x_{i_s}$ ($s=1,\dots ,n-1$), we have (10).

Inequality (11) is obtained from (8) on setting, for fixed $t\in [a,b]$ and fixed $i_1\cdots i_{n-1}\in \{1,\dots ,k\}$,

$$f(x)=-{[x_{i_1}-X_{i_1}(t)]}^{-}\cdots {[x_{i_{n-1}}-X_{i_{n-1}(t)}]}^{-},\text{where }{c}^{-}=min\{c,0\},(c\in \mathbb{R}).$$

Sufficiency: Since f may be approximated uniformly on I by functions with continuous nonnegative n th partial derivatives, we may assume that the n th partials $f_{i_1\cdots i_n}$ exist and are continuous and nonnegative. By Theorem 2.4 and (10), we have

By (11), each term in the sum is nonnegative so that (8) is verified. □

3 Functions with nondecreasing increments of order three

3.1 On inequalities of Levinson type

Levinson [5] proved that if a real-valued function f defined on $[0,2a]\subset \mathbb{R}$ has a nonnegative third derivative, then

(12)

for $0<x_k<a$, $y_k=2a-x_k$, $p_k>0$ ($1\le k\le n$), $P_n={\sum }_{k=1}^n{p}_k$.

If $a=\frac{1}{2}$, $p_1=\cdots =p_n=1$ and $f(x)=logx$, then Levinson’s inequality (12) becomes the famous Ky-Fan inequality

$$\frac{G_n}{G_n^{\prime }}\le \frac{A_n}{A_n^{\prime }},$$

where $A_n=\frac{1}{n}{\sum }_{k=1}^n{x}_k$, $A_n^{\prime }=\frac{1}{n}{\sum }_{k=1}^n(1-x_k)$, $G_n={({\prod }_{k=1}^n{x}_k)}^{1/n}$ and $G_n^{\prime }={({\prod }_{k=1}^n(1-x_k))}^{1/n}$.

In [6] Pečarić showed that instead of variables the sum of which is equal to 2a, we can use variables the difference of which is constant, and that result becomes a source of some further generalizations [[2], pp.74, 75]. In fact, he proved that if f is a real-valued 3-convex function on $[a,b]$ and $x_k$, $y_k$ ($1\le k\le n$), 2n points on $[a,b]$ such that

$$y_1-x_1=y_2-x_2=\cdots =y_n-x_n>0$$

and $p_k>0$ ($1\le k\le n$), then (12) is valid.

The following theorem is a generalization of the Levinson inequality.

Theorem 3.1 Let $G:[a,b]\to \mathbb{R}$ be a function of bounded variation such that (2) holds, and let $\mathbf{X}(t)$ be a continuous and nondecreasing map from $[a,b]\subset \mathbb{R}$ to an interval $\mathbf{I}=[\mathbf{0},\mathbf{d}]\subset {\mathbb{R}}^k$, $\mathbf{d}>\mathbf{0}$. If f is a continuous function with nondecreasing increments of order three on $\mathbf{J}=[\mathbf{0},2\mathbf{d}]$, then

Proof If f is a function with nondecreasing increments of order three on J, then

$${\mathrm{\Delta }}_{\mathbf{h}}{\mathrm{\Delta }}_{\mathbf{t}}{\mathrm{\Delta }}_{\mathbf{s}}f(\mathbf{x})\ge 0(\mathbf{x},\mathbf{x}+\mathbf{h}+\mathbf{t}+\mathbf{s}\in \mathbf{J},\mathbf{0}\le \mathbf{h},\mathbf{t},\mathbf{s}\in {\mathbb{R}}^k),$$

i.e.,

$${\mathrm{\Delta }}_{\mathbf{h}}{\mathrm{\Delta }}_{\mathbf{t}}(f(\mathbf{x}+\mathbf{s})-f(\mathbf{x}))\ge 0.$$

(13)

If $\mathbf{x}\in \mathbf{I}$ and $\mathbf{s}=2\mathbf{d}-2\mathbf{x}$, we have

$${\mathrm{\Delta }}_{\mathbf{h}}{\mathrm{\Delta }}_{\mathbf{t}}(f(2\mathbf{d}-\mathbf{x})-f(\mathbf{x}))\ge 0,$$

i.e., the function $\mathbf{x}\mapsto f(2\mathbf{d}-\mathbf{x})-f(\mathbf{x})$ is a function with nondecreasing increments of order two, i.e., it is a function with nondecreasing increments. Now, using Theorem 1.3, we obtain Theorem 3.1. □

Theorem 3.2 Let $G:[a,b]\to \mathbb{R}$ be a function of bounded variation such that (2) holds, and let f be a continuous function with nondecreasing increments of order three on $[\mathbf{c},\mathbf{d}]\subset {\mathbb{R}}^k$. Let $\mathbf{0}<\mathbf{a}<\mathbf{d}-\mathbf{c}$. If $\mathbf{X}(t):[a,b]\to [\mathbf{c},\mathbf{d}-\mathbf{a}]$ is a continuous and nondecreasing map, then

Proof Using (13) for $\mathbf{s}=\mathbf{a}=\text{constant}\in {\mathbb{R}}^k$, we have that the function $\mathbf{x}\mapsto f(\mathbf{a}+\mathbf{x})-f(\mathbf{x})$ is a function with nondecreasing increments, so from Theorem 1.3, we obtain Theorem 3.2. For $k=1$, we have a result from [6]. □

Corollary 3.3 (i) Let X satisfy the assumptions of Theorem  3.1. Then

$$\begin{array}{rcl}0& \le & {({\int }_a^{b}dG(t))}^{k-1}{\int }_a^b\prod _{i=1}^k{X}_i(t)dG(t)-\prod _{i=1}^k{\int }_a^b{X}_i(t)dG(t)\\ \le & {({\int }_a^{b}dG(t))}^{k-1}{\int }_a^b\prod _{i=1}^k(2d_i-X_i(t))dG(t)-\prod _{i=1}^k{\int }_a^b(2d_i-X_i(t))dG(t).\end{array}$$

(ii) If X satisfies the assumptions of Theorem  3.2, then

$$\begin{array}{rcl}0& \le & {({\int }_a^{b}dG(t))}^{k-1}{\int }_a^b\prod _{i=1}^k{X}_i(t)dG(t)-\prod _{i=1}^k{\int }_a^b{X}_i(t)dG(t)\\ \le & {({\int }_a^{b}dG(t))}^{k-1}{\int }_a^b\prod _{i=1}^k(a_i+X_i(t))dG(t)-\prod _{i=1}^k{\int }_a^b(a_i+X_i(t))dG(t),\end{array}$$

where all components of X are nonnegative.

Proof The function $f(\mathbf{x})=x_1\cdots x_k$ is a function with nondecreasing increments of orders two and three for $x_i\ge 0$ ($i=1,\dots ,k$). So, using Theorems 1.3, 3.1, and 3.2, we obtain Corollary 3.3. □

3.2 Generalization of Burkill-Mirsky-Pečarić result

In this subsection, we consider a sequence of k-tuples $({\mathbf{X}}_{\mathbf{1}},\dots ,{\mathbf{X}}_{\mathbf{n}})$ which is monotone in means.

Theorem 3.4 Let f be a continuous function with nondecreasing increments of order three on $\mathbf{J}=[\mathbf{0},2\mathbf{d}]$, $\mathbf{d}>\mathbf{0}$, and let $p_1,\dots ,p_n$ be positive numbers. If

$$({\mathbf{X}}_{\mathbf{1}},\dots ,{\mathbf{X}}_{\mathbf{n}})({\mathbf{X}}_{\mathbf{i}}\in \mathbf{I}=[\mathbf{0},\mathbf{d}])$$

is nondecreasing or nonincreasing in means with respect to positive weights $p_i$ ($i=1,\dots ,n$), then

$$\frac{1}{P_n}\sum _{i=1}^n{p}_{i}f({\mathbf{X}}_{\mathbf{i}})-f\left(\frac{1}{P_n}\sum _{i=1}^n{p}_i{\mathbf{X}}_{\mathbf{i}}\right)\le \frac{1}{P_n}\sum _{i=1}^n{p}_{i}f(2\mathbf{d}-{\mathbf{X}}_{\mathbf{i}})-f(\frac{1}{P_n}\sum _{i=1}^n{p}_i(2\mathbf{d}-{\mathbf{X}}_{\mathbf{i}}))$$

holds.

Proof Since f is a function with nondecreasing increments of order three on J, so a function $\mathbf{x}\mapsto f(2\mathbf{d}-\mathbf{x})-f(\mathbf{x})$ is a function with nondecreasing increments. Then by Theorem 1.4, we obtain the required result. □

Theorem 3.5 Let f be a continuous function with nondecreasing increments of order three on $\mathbf{J}=[\mathbf{c},\mathbf{d}]$ and let $p_1,\dots ,p_n$ be positive numbers. Let $\mathbf{0}<\mathbf{a}<\mathbf{d}-\mathbf{c}$. If

$$({\mathbf{X}}_{\mathbf{1}},\dots ,{\mathbf{X}}_{\mathbf{n}})({\mathbf{X}}_{\mathbf{i}}\in \mathbf{I}=[\mathbf{c},\mathbf{d}-\mathbf{a}])$$

is nondecreasing or nonincreasing in means with respect to positive weights $p_i$ ($i=1,\dots ,n$), then

$$\frac{1}{P_n}\sum _{i=1}^n{p}_{i}f({\mathbf{X}}_{\mathbf{i}})-f\left(\frac{1}{P_n}\sum _{i=1}^n{p}_i{\mathbf{X}}_{\mathbf{i}}\right)\le \frac{1}{P_n}\sum _{i=1}^n{p}_{i}f(\mathbf{a}+{\mathbf{X}}_{\mathbf{i}})-f(\frac{1}{P_n}\sum _{i=1}^n{p}_i(\mathbf{a}+{\mathbf{X}}_{\mathbf{i}}))$$

holds.

Proof By following the proof of Theorem 3.2, we obtain Theorem 3.5 by simply replacing ‘Theorem 1.3’ by ‘Theorem 1.4’ in the proof of Theorem 3.2. □

Corollary 3.6 (i) Let X satisfy the assumptions of Theorem  3.4. Then

$$\begin{array}{rcl}0& \le & P_n^{k-1}\sum _{i=1}^n{p}_i^k\left(\prod _{j=1}^k{x}_{ij}\right)-\prod _{j=1}^k\left(\sum _{i=1}^n{p}_i{x}_{ij}\right)\\ \le & P_n^{k-1}\sum _{i=1}^n{p}_i^k(\prod _{j=1}^k(2d_j-x_{ij}))-\prod _{j=1}^k(\sum _{i=1}^n{p}_i(2d_j-x_{ij})).\end{array}$$

(ii) If X satisfies the assumptions of Theorem  3.5. Then

$$\begin{array}{rcl}0& \le & P_n^{k-1}\sum _{i=1}^n{p}_i^k\left(\prod _{j=1}^k{x}_{ij}\right)-\prod _{j=1}^k\left(\sum _{i=1}^n{p}_i{x}_{ij}\right)\\ \le & P_n^{k-1}\sum _{i=1}^n{p}_i^k(\prod _{j=1}^k(a_j+x_{ij}))-\prod _{j=1}^k(\sum _{i=1}^n{p}_i(a_j+x_{ij})),\end{array}$$

where all components of X are nonnegative.

Proof We consider again the function $f(\mathbf{x})=x_1\cdots x_k$ which is a function with nondecreasing increments of orders two and three for $x_i\ge 0$ ($i=1,\dots ,k$). So, using Theorems 1.4, 3.4, and 3.5, we obtain Corollary 3.6. □

4 Arithmetic integral mean

It is known that if $f:[0,a]\to \mathbb{R}$, $a>0$, is nonnegative and nondecreasing, then the function F,

$$F(x)=\frac{1}{x}{\int }_0^{x}f(u)du,$$

is also a nondecreasing function on $[0,a]$. Let us observe that F is an arithmetic integral mean of a function f on an interval $[0,a]$. This result was generalized in [7] considering a real-valued function f for which ${\mathrm{\Delta }}_h^{m}f(x)\ge 0$ holds for any $h>0$. ${\mathrm{\Delta }}_h^m$ is defined as follows: ${\mathrm{\Delta }}_h^{0}f(x)=f(x)$, ${\mathrm{\Delta }}_h^{m}f(x)={\mathrm{\Delta }}_h^{m-1}f(x+h)-{\mathrm{\Delta }}_h^{m-1}f(x)$.

Here, we extend the above-mentioned result to functions with nondecreasing increments of higher order.

Theorem 4.1 Let the function $f:[\mathbf{a},\mathbf{b}]\to \mathbb{R}$ be continuous and with nondecreasing increments of order n. Then the function

$$F(\mathbf{x})={(\prod _{i=1}^k(x_i-a_i))}^{-1}{\int }_{a_1}^{x_1}\cdots {\int }_{a_k}^{x_k}f(\mathbf{u})\mathbf{du},$$

where $\mathbf{u}=(u_1,\dots ,u_k)$ and $\mathbf{du}=d{u}_1\cdots d{u}_k$, is a function with nondecreasing increments of order n on $[\mathbf{a},\mathbf{b}]$.

Proof Let $\mathbf{x}>\mathbf{a}=(a_1,\dots ,a_k)$. Then

$$F(\mathbf{x})={\int }_0^1\cdots {\int }_0^{1}f(\mathbf{a}+\mathbf{s}(\mathbf{x}-\mathbf{a}))\mathbf{ds},$$

where we used the substitutions $u_i=a_i+s_i(x_i-a_i)$ ($1\le i\le k$, $0\le s_i\le 1$), and where $\mathbf{a}+\mathbf{s}(\mathbf{x}-\mathbf{a})=(a_1+s_1(x_1-a_1),\dots ,a_k+s_k(x_k-a_k))$, $\mathbf{ds}=d{s}_1\cdots d{s}_k$. Now, we have

$$\begin{array}{rcl}{\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{1}}}\cdots {\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{n}}}F(\mathbf{x})& =& {\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{1}}}\cdots {\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{n}}}{\int }_0^1\cdots {\int }_0^{1}f(\mathbf{a}+\mathbf{s}(\mathbf{x}-\mathbf{a}))\mathbf{ds}\\ =& {\int }_0^1\cdots {\int }_0^1{\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{1}}}\cdots {\mathrm{\Delta }}_{{\mathbf{h}}_{\mathbf{n}}}f(\mathbf{a}+\mathbf{s}(\mathbf{x}-\mathbf{a}))\mathbf{ds}\ge 0\end{array}$$

because if $f(\mathbf{x})$ is a function with nondecreasing increments of order n, then the function $f(\mathbf{a}+\mathbf{s}(\mathbf{x}-\mathbf{a}))$ is also a function with nondecreasing increments of order n. □