Majorization Inequalities for n-Convex Functions with Applications to 3-Convex Functions

Abstract

In this paper, we study majorization-type inequalities for n-convex (specifically 3-convex) functions. Numerous papers deal with such integral inequalities, in which n-convex functions are defined on compact intervals and nonnegative measures are used in the integrals. The main goal of this paper is to formulate similar results for noncompact intervals and signed measures. We follow a well-known method often used for compact intervals: approximation of n-convex functions with simple n-convex functions. After some preliminary results, we present new approximation theorems, some of which extend classical results, while others are completely unique approximations. Then we obtain some novel majorization-type inequalities, which can be applied under more general conditions than those currently known. Finally, we illustrate the applicability of our results by answering problems from different areas: discrete majorization-type inequalities, specifically one-dimensional inequality of Sherman for n-convex functions; characterization of Steffensen–Popoviciu measures for nonnegative, continuous, and increasing 3-convex functions; Hermite–Hadamard-type inequalities for 3-convex functions.

1. Introduction

Let $C\subset \mathbb{R}$ be an interval with nonempty interior, and let $f:C\to \mathbb{R}$ be a function. The divided differences of f of order $0,1,\dots ,n$ associated with a family $t_0,t_1,\dots ,t_n$ of $n+1$ distinct points from C are, respectively, defined by the formulae

$$\begin{gathered} \left[t_0;f\right]:=f\left(t_0\right), \\ \left[t_0,t_1;f\right]:={\displaystyle \frac{f\left(t_1\right)-f\left(t_0\right)}{t_1-t_0}}, \\ \dots \\ \left[t_0,t_1,\dots t_n;f\right]:={\displaystyle \frac{\left[t_1,\dots t_n;f\right]-\left[t_0,t_1,\dots t_{n-1};f\right]}{t_n-t_0}}={\displaystyle \sum _{i=0}^n}{\displaystyle \frac{f\left(t_i\right)}{{\displaystyle \prod _{\stackrel{j=0}{j\ne i}}^n}\left(t_i-t_j\right)}}. \end{gathered}$$

The function f is called n-convex if all divided differences of order n are nonnegative. This notion was introduced by E. Hopf [1] and T Popoviciu [2]. It is worth mentioning that the 0-convex functions are the nonnegative functions, the 1-convex functions are the increasing functions, and the 2-order convex functions are the usual convex functions.

In this paper, we study majorization-type inequalities for n-convex (specifically 3-convex) functions. Such inequalities have been studied by many authors, and they have important applications. We will mention a few recent papers that examine the topic from different perspectives. Further references can be found in these papers. In papers [3,4], majorization inequalities are obtained by using Green’s functions. The results are applied to inequalities in information theory. In [5] the authors present a new weighted majorization theorem for n-convex functions, by using a generalization of Taylor’s formula. Sherman’s result is extended to the class of n-strongly convex functions in [6]. Hermite–Hadamard-type inequalities for n-convex functions are studied in [7,8]. Papers [9,10] illustrate the usefulness of majorization theory and higher-order convexity in generalizing several classical inequalities as functional inequalities.

Whether considering earlier or the above-mentioned papers, n-convex functions defined on compact intervals are considered in majorization-type inequalities. An important reason for this is as follows: one of the most frequently used methods in the study of majorization inequalities is the approximation of n-convex functions by simple n-convex functions, and such results are known on compact intervals. Among majorization inequalities, integral inequalities play an important role; most of them use nonnegative measures. The use of signed measures can also be interesting, as illustrated by the applicability of the classical Steffensen–Popoviciu measures. Based on our previous remarks, the main goal of this paper is to obtain novel majorization-type integral inequalities for n-convex functions, where the functions are defined on arbitrary intervals and signed measures are used.

The paper is organized as follows.

In Section 2, we give some preliminary statements, some of which are interesting in their own right.

In Section 3, we present new approximation results for n-convex functions, which we will need in the proofs. It is known that n-convex functions defined on compact intervals can be approximated uniformly by simple n-convex functions (see [11,12]). We extend this result to n-convex functions defined on intervals of any type, using monotonic convergence instead of uniform convergence. Furthermore, we show that (nonnegative) increasing 3-convex functions defined on a compact interval can be approximated uniformly by elementary (nonnegative) increasing 3-convex functions. The precise definition of simple and elementary n-convex functions will be given later.

Section 4 contains new majorization results for n-convex (specifically 3-convex) functions. We obtain necessary and sufficient conditions for the inequality

$$\underset{X}{\int }f\circ \phi d\mu \le \underset{Y}{\int }f\circ \psi d\nu$$

(1)

to hold for any suitable n-convex function $f:C\to \mathbb{R}$, where $C\subset \mathbb{R}$ is an arbitrary interval, and $\mu$, $\nu$ are finite signed measures. If C is a compact interval and $\mu$, $\nu$ are finite measures, several papers deal with the problem, which is solved in these cases. A detailed analysis can be found in paper [13] for probability measures. If C is a compact interval, then we give necessary and sufficient conditions for inequality (1) to hold for any (nonnegative) increasing 3-convex function. I did not find similar results in the related literature.

Section 5 contains various applications.

First, we give the discrete version of our majorization result for n-convex functions. This generalizes the one-dimensional inequality of Sherman [14] and Theorem 9 of [15] to n-convex functions.

For convex functions, the concept of Steffensen–Popoviciu measure is well known and widely applicable. As a next application, we examine finite signed measures analogous to Steffensen–Popoviciu measures for 3-convex functions. We can partially solve the problem by characterizing those finite signed measures on the Borel subsets of $\left[a,b\right]$ for which

$${\displaystyle \underset{\left[a,b\right]}{\int }}fd\mu \ge 0$$

for every nonnegative, continuous, and increasing 3-convex function $f:\left[a,b\right]\to \mathbb{R}$.

Then we show that (with a few elementary exceptions) there exist unique $a<c<b$ and $0<\alpha <1$, and there exist unique $a<d<b$ and $0<\beta <1$ such that for every continuous 3-convex function $f:\left[a,b\right]\to \mathbb{R}$, the Hermite–Hadamard-type inequalities

$$\alpha f\left(a\right)+\left(1-\alpha \right)f\left(c\right)\le {\displaystyle \underset{\left[a,b\right]}{\int }}fd\mu \le \left(1-\beta \right)f\left(d\right)+\beta f\left(b\right)$$

(2)

hold, where $\mu$ is a probability measure on the Borel subsets of $\left[a,b\right]$. On the one hand, this solves Problem 3 of paper [16], and on the other hand, for $n=3$, it generalizes Corollary 2 of [7] and Theorem 3 of [8]. Finally, we give the form of inequality (2) analogous to Fejér’s inequality.

2. Preliminary Results

In the rest of the paper, $n\ge 2$ is an integer.

By $\mathbb{N}$$\left({\mathbb{N}}_{+}\right)$ we denote the set of nonnegative (positive) integers. The interior of an interval $C\subset \mathbb{R}$ is denoted by $C°$.

First, we give some basic statements about n-convex functions.

We need some equivalent reformulation of n-convexity.

Theorem 1 

(see [17]). Let $C\subset \mathbb{R}$ be an interval with nonempty interior, and let $f:C\to \mathbb{R}$ be a continuous function. The following statements are equivalent:

(a) 

The function f is n-convex.

(b) 

The function f is n-convex on $C°$.

(b) 

The function f is $\left(n-2\right)$-times differentiable on $C°$ and $f^{\left(n-2\right)}$ is a convex function on $C°$.

Let $\left[a,b\right]\subset \mathbb{R}$ be a compact interval with $a<b$, and let $f:\left[a,b\right]\to \mathbb{R}$ be a function. Given a family $t_0,t_1,\dots ,t_n$ of $n+1$ distinct points from $\left[a,b\right]$, there is a unique polynomial of degree at most n passing through the points $\left(t_0,f\left(t_0\right)\right)$, …, $\left(t_n,f\left(t_n\right)\right)$ given by

$${\pi }_n\left(t;t_0,\dots ,t_n;f\right):={\displaystyle \sum _{i=0}^n}f\left(t_i\right){\displaystyle \prod _{\stackrel{j=0}{j\ne i}}^n}{\displaystyle \frac{t-t_j}{t_i-t_j}},t\in \mathbb{R}.$$

This polynomial is known as the Lagrange interpolating polynomial.

The convexity of a function $f:\left[a,b\right]\to \mathbb{R}$ means geometrically that the points of the graph of $f{\mid }_{\left]t_1,t_2\right[}$ are under (or on) the chord joining the endpoints $\left(t_1,f\left(t_1\right)\right)$ and $\left(t_2,f\left(t_2\right)\right)$ for all $a\le t_1<t_2\le b$. The following result shows that n-convex functions have a similar geometric description.

Theorem 2 

(see Theorem 5 of [18]). Let $\left[a,b\right]\subset \mathbb{R}$ be a compact interval with $a<b$, and let $f:\left[a,b\right]\to \mathbb{R}$ be a function. The function f is n-convex if and only if for all sets of n distinct points $a\le t_1<\dots <t_n\le b$, the graph of f lies alternately above and below the curve $y={\pi }_{n-1}\left(t;t_1,\dots ,t_n;f\right)$, lying below if $t_{n-1}\le t\le t_n$. Further,

$${\pi }_{n-1}\left(t;t_1,\dots ,t_n;f\right)\le f\left(t\right),t_n\le t\le b;$$

(3)

and

$${\pi }_{n-1}\left(t;t_1,\dots ,t_n;f\right)\le f\left(t\right)\left(\ge f\left(t\right)\right),a\le t\le t_1,$$

(4)

n being even (odd).

The following consequence of the previous statement will be important later on.

Corollary 1. 

Let $\left[a,b\right]\subset \mathbb{R}$ be a compact interval with $a<b$, and let $f:\left[a,b\right]\to \mathbb{R}$ be a continuous n-convex function. Let $g:\left[a,b\right]\to \mathbb{R}$ be another function such that $g\left(t\right)=f\left(t\right)$ for all $t\in \left]a,b\right[$. Then g is n-convex if and only if $g\left(b\right)\ge f\left(b\right)$, and $g\left(a\right)\ge f\left(a\right)\left(\le f\left(a\right)\right)$, n being even (odd).

Proof. 

Assume that g is n-convex.

Let $a<t_1<\dots <t_n<b$, where $t_1,\dots ,t_{n-1}$ are fixed and $t_n$ can vary. Then by (3),

$${\pi }_{n-1}\left(b;t_1,\dots ,t_n;f\right)={\pi }_{n-1}\left(b;t_1,\dots ,t_n;g\right)\le g\left(b\right),$$

and by the continuity of f,

$$\underset{t_n\to b-}{lim}{\pi }_{n-1}\left(b;t_1,\dots ,t_n;f\right)=f\left(b\right).$$

Consequently, $g\left(b\right)\ge f\left(b\right)$.

By a similar method (using (4) instead of (3)), we obtain that $g\left(a\right)\ge f\left(a\right)$ if n is even and $g\left(a\right)\le f\left(a\right)$ if n is odd.

Conversely, assume that $g\left(b\right)\ge f\left(b\right)$, and $g\left(a\right)\ge f\left(a\right)$ if n is even and $g\left(a\right)\le f\left(a\right)$ if n is odd.

Let $a\le t_1<\dots <t_n\le b$. Under the given conditions, it is easy to check that if

$${\pi }_{n-1}\left(t;t_1,\dots ,t_n;f\right)\le f\left(t\right)$$

on a section, then

$${\pi }_{n-1}\left(t;t_1,\dots ,t_n;g\right)\le {\pi }_{n-1}\left(t;t_1,\dots ,t_n;f\right)$$

on the same section, and vice versa. From this, by means of the n-convexity of f and Theorem 2, we get that g is also n-convex.

The proof is complete. □

In the next part of this section, we define the concepts of simple n-convex and elementary increasing 3-convex functions.

To avoid misunderstandings, from now on

$${\left(t-w\right)}_{+}^m:={\left({\left(t-w\right)}_{+}\right)}^m,{\left(t-w\right)}_{-}^m:={\left({\left(t-w\right)}_{-}\right)}^m.$$

First, we introduce the definition of simple n-convex functions.

We use the convention that the sum over an empty set is defined as 0.

Definition 1. 

We say that the function $g:\mathbb{R}\to \mathbb{R}$ is a simple n-convex function if it has the form

$$g\left(t\right)={\displaystyle \sum _{i=0}^{n-1}}{\alpha }_i{t}^i+{\displaystyle \sum _{j=1}^p}{\beta }_j{\left(t-t_j\right)}_{+}^{n-1},$$

(5)

for suitable points $t_1<t_2<\dots <t_p$ from $\mathbb{R}$$\left(p\in \mathbb{N}\right)$ and suitable constants ${\alpha }_i\in \mathbb{R}$ $\left(i=0,\dots ,n-1\right)$ and ${\beta }_j>0$$\left(j=1,\dots ,p\right)$.

It follows from Theorem 1 that the function g is n-convex for any choice of parameters.

For future needs we give the following result.

Lemma 1. 

Let $c\in \mathbb{R}$ be fixed. Consider the simple n-convex function described in Definition 1. If the function ${\widehat{g}}_c:\mathbb{R}\to \mathbb{R}$ is defined by

$${\widehat{g}}_c\left(t\right):={\displaystyle \underset{c}{\overset{t}{\int }}}g\left(s\right)ds,$$

then ${\widehat{g}}_c$ is a simple $\left(n+1\right)$-convex function with the form

$${\widehat{g}}_c\left(t\right)={\displaystyle \sum _{i=0}^n}{\widehat{\alpha }}_i{t}^i+{\displaystyle \sum _{j=1}^p}{\widehat{\beta }}_j{\left(t-t_j\right)}_{+}^n,$$

(6)

where ${\widehat{\alpha }}_i\in \mathbb{R}$$\left(i=0,\dots ,n\right)$ and ${\widehat{\beta }}_j>0$$\left(j=1,\dots ,p\right)$.

Proof. 

Using the elementary facts that

$${\displaystyle \underset{c}{\overset{t}{\int }}}{\displaystyle \sum _{i=0}^{n-1}}{\alpha }_i{s}^{i}ds={\displaystyle \sum _{i=0}^{n-1}}{\displaystyle \frac{{\alpha }_i}{i+1}}\left(t^{i+1}-c^{i+1}\right),t\in C$$

and

$${\displaystyle \underset{c}{\overset{t}{\int }}}{\beta }_j{\left(s-t_j\right)}_{+}^{n-1}ds={\displaystyle \frac{{\beta }_j}{n}}{\left(t-t_j\right)}_{+}^n-{\displaystyle \frac{{\beta }_j}{n}}{\left(c-t_j\right)}_{+}^n,t\in C,$$

we obtain the result.

The proof is complete. □

We now introduce a special class of nonnegative 3-convex functions.

Definition 2. 

Let $a,b\in \mathbb{R}$, $a<b$. We say that the function $g:\left[a,b\right]\to \mathbb{R}$ is an elementary increasing 3-convex function on $\left[a,b\right]$ if it has the form

$$g\left(t\right)=\alpha \left(t-a\right)+\beta +{\displaystyle \sum _{j=1}^p}{\gamma }_j{\left(t-t_j\right)}_{+}^2+{\displaystyle \sum _{k=1}^q}{\delta }_k\left({\left(s_k-a\right)}^2-{\left(t-s_k\right)}_{-}^2\right)$$

for suitable points $t_1<t_2<\dots <t_p$, $s_1<s_2<\dots <s_q$ from $\left]a,b\right[$$\left(p,q\in \mathbb{N}\right)$ and suitable constants $\alpha \ge 0$, $\beta \in \mathbb{R}$ and ${\gamma }_j$, ${\delta }_k>0$ $\left(j=1,\dots ,p,k=1,\dots ,q\right)$.

It is obvious that g is continuous and increasing, and it follows from Theorem 1 that g is 3-convex for any choice of parameters. It is worth mentioning that if $\beta \ge 0$, then g is nonnegative.

Finally, we present some results that we will need to obtain new Hermite–Hadamard-type inequalities in the context of 3-convex functions.

Let $\left(X,\mathcal{A}\right)$ be a measurable space. The unit mass at $x\in X$ (the Dirac measure at x) is denoted by ${\epsilon }_x$.

The $\sigma$-algebra of Borel sets on an interval $C\subset \mathbb{R}$ is denoted by ${\mathcal{B}}_C$.

Lemma 2. 

Let $a,b\in \mathbb{R}$, $a<b$, and let μ be a probability measure on ${\mathcal{B}}_{\left[a,b\right]}$.

(a) If

$$u:={\displaystyle \underset{\left[a,b\right]}{\int }}sd\mu \left(s\right),v:={\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu \left(s\right),$$

then

$$u^2\le v\le au+bu-ba.$$

(b) If $\mu \ne {\epsilon }_t$ for some $a<t<b$ and $\mu \ne \lambda {\epsilon }_a+\left(1-\lambda \right){\epsilon }_b$ for some $0\le \lambda \le 1$, then

$$u^2<v<au+bu-ba.$$

Proof. 

(a) By the integral Jensen inequality (see [17]),

$$u^2\le v,$$

while by the right-hand side of the Hermite–Hadamard inequality (see [17]),

$$v\le {\displaystyle \frac{b-u}{b-a}}·a^2+{\displaystyle \frac{u-a}{b-a}}·b^2=au+bu-ba.$$

(b) It follows from the fact that the function $t\to t^2$$\left(t\in \left[a,b\right]\right)$ is strictly convex.

The proof is complete. □

Lemma 3. 

Let $a,b\in \mathbb{R}$, $a<b$. If u, $v\in \mathbb{R}$ for which

$$u^2<v<au+bu-ba,$$

(7)

then

(a) There exists exactly one $0<\alpha <1$ and $a<c<b$ such that

$$\left.\begin{array}{c}u=\alpha a+\left(1-\alpha \right)c\\ v=\alpha a^2+\left(1-\alpha \right)c^2\end{array}\right\}.$$

(8)

(b) There exists exactly one $0<\beta <1$ and $a<d<b$ such that

$$\left.\begin{array}{c}u=\left(1-\beta \right)d+\beta b\\ v=\left(1-\beta \right)d^2+\beta b^2\end{array}\right\}.$$

(9)

Proof. 

We first show that (7) implies

$$a<u<b,v>2au-a^2,v>2bu-b^2.$$

(10)

It follows from (7) that

$$u^2-u\left(a+b\right)+ba=\left(u-a\right)\left(u-b\right)<0,$$

which yields that $a<u<b$.

Since $v>u^2$,

$$v-\left(2au-a^2\right)>u^2-\left(2au-a^2\right)={\left(u-a\right)}^2\ge 0$$

and

$$v-\left(2bu-b^2\right)>u^2-\left(2bu-b^2\right)={\left(u-b\right)}^2\ge 0.$$

(a) Rearranging the equations, we obtain that

$$\left.\begin{array}{c}\alpha \left(c-a\right)=c-u\\ \alpha \left(c-a\right)\left(c+a\right)=c^2-v\end{array}\right\}.$$

It follows that

$$\left(c-u\right)\left(c+a\right)=c^2-v,$$

and therefore by $a<u$,

$$c={\displaystyle \frac{v-ua}{u-a}}.$$

(11)

It is easy to see that $a<c<b$ is equivalent to

$$2au-a^2<v<au+bu-ba,$$

and these inequalities hold by (7) and (10).

Since

$$\alpha ={\displaystyle \frac{c-u}{c-a}},$$

$a<u$ and $a<c$ imply $\alpha <1$. It remains to show that $0<\alpha$, that is $u<c$, which easily comes from (11), $a<u$ and $u^2<v$.

(b) It can be justified similarly to (a).

The proof is complete. □

Corollary 2. 

Let $a,b\in \mathbb{R}$, $a<b$. Let μ be a probability measure on ${\mathcal{B}}_{\left[a,b\right]}$ for which $\mu \ne {\epsilon }_t$ for any $a<t<b$ and $\mu \ne \lambda {\epsilon }_a+\left(1-\lambda \right){\epsilon }_b$ for any $0\le \lambda \le 1$. Then

(a) There exists a unique probability measure ${\nu }_1:=\alpha ·{\epsilon }_a+\left(1-\alpha \right)·{\epsilon }_c$$\left(a<c<b,0<\alpha <1\right)$ on ${\mathcal{B}}_{\left[a,b\right]}$ such that

$${\displaystyle \underset{\left[a,b\right]}{\int }}sd\mu \left(s\right)={\displaystyle \underset{\left[a,b\right]}{\int }}sd{\nu }_1\left(s\right),{\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu \left(s\right)={\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d{\nu }_1\left(s\right).$$

(b) There exists a unique probability measure ${\nu }_2:=\left(1-\beta \right)·{\epsilon }_d+\beta ·{\epsilon }_b$$\left(a<d<b,0<\beta <1\right)$ on ${\mathcal{B}}_{\left[a,b\right]}$ such that

$${\displaystyle \underset{\left[a,b\right]}{\int }}sd\mu \left(s\right)={\displaystyle \underset{\left[a,b\right]}{\int }}sd{\nu }_2\left(s\right),{\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu \left(s\right)={\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d{\nu }_2\left(s\right).$$

Proof. 

Let

$$u:={\displaystyle \underset{\left[a,b\right]}{\int }}sd\mu \left(s\right),v:={\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu \left(s\right).$$

By Lemma 2, the numbers u and v satisfy (7).

(a) The result follows from Lemma 3 (a) by choosing the unique solution $\left(\alpha ,c\right)$ of the system (8).

(b) The result follows from Lemma 3 (b) by choosing the unique solution $\left(\beta ,d\right)$ of the system (9).

The proof is complete. □

3. Approximation Results

The following approximation result is well known.

Theorem 3 

(see [11,12]). If $f:\left[a,b\right]\to \mathbb{R}$ is an n-convex function, then there exists a sequence ${\left(f_l\right)}_{l=1}^{\infty }$ of simple n-convex functions on $\left[a,b\right]$ such that f is the uniform limit of the sequence $\left(f_l\right)$.

In this part of the paper, we prove new approximation theorems. The results are interesting in themselves, and they also play a fundamental role in our further investigations.

Theorem 4. 

Let $C\subset \mathbb{R}$ be an interval with endpoints $-\infty \le a<b\le \infty$, let $f:C\to \mathbb{R}$ be a continuous n-convex function, and let $c\in C°$ be fixed. Then the function f is the pointwise limit of a sequence ${\left(f_l\right)}_{l=1}^{\infty }$ of simple n-convex functions whose elements have the form

$$g\left(t\right)={\displaystyle \sum _{i=0}^{n-1}}{\alpha }_i{t}^i+{\displaystyle \sum _{j=1}^p}{\beta }_j{\left(t-t_j\right)}_{+}^{n-1},t\in \mathbb{R}$$

(12)

for suitable points $t_1<t_2<\dots <t_p$ from $C°$$\left(p\in \mathbb{N}\right)$ and suitable constants ${\alpha }_i\in \mathbb{R}$ $\left(i=0,\dots ,n-1\right)$ and ${\beta }_j>0$$\left(j=1,\dots ,p\right)$. If n is even, then $\left(f_l\right)$ can be chosen to be increasing on C, and if n is odd, then $\left(f_l\right)$ can be chosen to be decreasing on $C\cap \left]-\infty ,c\right]$ and increasing on $C\cap \left[c,\infty \right[$.

Proof. 

The proof is divided into several parts.

(a) We first assume that C is an open interval.

By Theorem 4 of [19], the result is true for $n=2$. Suppose then that $n\ge 2$ is an integer for which the result holds, and let $f:C\to \mathbb{R}$ be an $\left(n+1\right)$-convex function.

It follows from Theorem 1 that f is differentiable and $f^{\prime }$ is an n-convex function. Two cases are distinguished.

(i) Assume n is even.

By the induction hypothesis, $f^{\prime }$ is the pointwise limit of an increasing sequence ${\left(f_l\right)}_{l=1}^{\infty }$ of simple n-convex functions whose elements have the form (12). We define ${\widehat{f}}_l:\mathbb{R}\to \mathbb{R}$ by

$${\widehat{f}}_l\left(t\right):={\displaystyle \underset{c}{\overset{t}{\int }}}{f}_l\left(s\right)ds,l\ge 1.$$

According to Lemma 1, ${\widehat{f}}_l$$\left(l\ge 1\right)$ is a simple $\left(n+1\right)$-convex function of the form (6). Since $\left(f_l\right)$ is an increasing sequence on C, it follows from the definition of $\left({\widehat{f}}_l\right)$ that it is decreasing on $C\cap \left]-\infty ,c\right]$ and increasing on $C\cap \left[c,\infty \right[$. Because $f_l\left(t\right)\stackrel{l\to \infty }{\to }f\left(t\right)$$\left(t\in C\right)$ pointwise, and all these functions are continuous, the monotone convergence theorem implies that

$${\widehat{f}}_l\left(t\right)={\displaystyle \underset{c}{\overset{t}{\int }}}{f}_l\left(s\right)ds\stackrel{l\to \infty }{\to }{\displaystyle \underset{c}{\overset{t}{\int }}}{f}^{\prime }\left(s\right)ds=f\left(t\right)-f\left(c\right),t\in C,$$

that is

$${\widehat{f}}_l\left(t\right)+f\left(c\right)\stackrel{l\to \infty }{\to }f\left(t\right),t\in C,$$

and this completes the proof of the considered case.

(ii) If n is odd, we can follow the argument in (i) with suitable modifications.

(b) Assume $a\in C$.

(i) If n is even, then by (a), the function $f{\mid }_{C°}$ is the pointwise limit of an increasing sequence ${\left(f_l\right)}_{l=1}^{\infty }$ of simple n-convex functions whose elements have the form (12). Since $f_l$$\left(l\in {\mathbb{N}}_{+}\right)$ is continuous, the sequence ${\left(f_l\left(a\right)\right)}_{l=1}^{\infty }$ is also increasing, and by the continuity of f at a, $d:=\underset{l\to \infty }{lim}{f}_l\left(a\right)\le f\left(a\right)$.

Since the limit of a pointwise convergent sequence of n-convex functions is also an n-convex function, the function

$$t\to \left\{\begin{array}{c}d,\mathrm{if}t=a\\ f\left(t\right),\mathrm{if}a<t\le c\end{array}\right.$$

is n-convex, and hence Corollary 1 implies that $d\ge f\left(a\right)$ that is $d=f\left(a\right)$.

(ii) We can think analogously if n is odd.

(c) If $b\in C$, we can follow a similar approach to (i) in part (b).

The proof is complete. □

Remark 1. 

(a) Dini’s theorem yields that the convergence of the sequence $\left(f_l\right)$ is also uniform on every compact subinterval of C.

(b) In the case of compact intervals, the uniform approximation of n-convex functions with “simple” n-convex functions (simple in some sense) (see [2,11]), as well as the uniform approximation with n-convex functions (see [20]), plays an important role in applications. Uniform approximation of n-convex functions with simple n-convex functions is generally not feasible on noncompact intervals. Theorem 4 guarantees monotonic convergence, which can replace uniform convergence in applications.

Now we give another approximation theorem for increasing 3-convex functions on compact intervals.

Theorem 5. 

Let $a,b\in \mathbb{R}$, $a<b$, and let $f:\left[a,b\right]\to \mathbb{R}$ be a continuous and increasing 3-convex function. Then

(a) The function f is the pointwise limit of an increasing sequence ${\left(f_l\right)}_{l=1}^{\infty }$ of elementary increasing 3-convex functions on $\left[a,b\right]$.

(b) If f is nonnegative, then it is the pointwise limit of an increasing sequence ${\left(f_l\right)}_{l=1}^{\infty }$ of nonnegative elementary increasing 3-convex functions on $\left[a,b\right]$.

Proof. 

(a) By Theorem 1, f is differentiable on $\left]a,b\right[$ and $f^{\prime }:\left]a,b\right[\to \mathbb{R}$ is a convex function. Since f is increasing, $f^{\prime }$ is nonnegative.

By Theorem 4 of [15], $f^{\prime }$ is the pointwise limit of an increasing sequence ${\left(g_l\right)}_{l=1}^{\infty }$ of piecewise linear convex functions whose elements have the form

$$g\left(t\right)=\alpha +\sum _{j=1}^p{\eta }_j{\left(t-t_j\right)}^{+}+\sum _{k=1}^q{\vartheta }_k{\left(t-s_k\right)}^{-},t\in \left]a,b\right[$$

for suitable points $t_1<t_2<\dots <t_p$, $s_1<s_2<\dots <s_q$ from $\left]a,b\right[$$\left(p,q\in \mathbb{N}\right)$ and suitable constants $\alpha \ge 0$ and ${\eta }_j$, ${\vartheta }_k>0$$\left(j=1,\dots ,p,k=1,\dots ,q\right)$.

A simple calculation confirms (see also Lemma 1) that

$$\begin{gathered} \widehat{g}\left(t\right):={\displaystyle \underset{a}{\overset{t}{\int }}}g\left(s\right)ds=\alpha \left(t-a\right)+\sum _{j=1}^p{\displaystyle \frac{{\eta }_j}{2}}{\left(t-t_j\right)}_{+}^2 \\ +\sum _{k=1}^q{\displaystyle \frac{{\vartheta }_k}{2}}\left({\left(s_k-a\right)}^2-{\left(t-s_k\right)}_{-}^2\right),t\in \left[a,b\right], \end{gathered}$$

and therefore $\widehat{g}$ is an elementary increasing 3-convex function on $\left[a,b\right]$.

Since f is not only continuous but absolutely continuous,

$$f\left(t\right)=f\left(a\right)+{\displaystyle \underset{a}{\overset{t}{\int }}}{f}^{\prime }\left(s\right)ds,t\in \left[a,b\right].$$

(13)

Let us define the new sequence ${\left(f_l\right)}_{l=1}^{\infty }$ as follows:

$$f_l\left(t\right):=f\left(a\right)+{\displaystyle \underset{a}{\overset{t}{\int }}}{g}_l\left(s\right)ds,t\in \left[a,b\right].$$

Based on the above, $\left(f_l\right)$ is a sequence of elementary increasing 3-convex functions on $\left[a,b\right]$. Since $\left(g_l\right)$ is an increasing sequence, $\left(f_l\right)$ is increasing too. The monotone convergence theorem and (13) imply that $\left(f_l\right)$ converges pointwise to f.

(b) This follows from (a) and from the fact that $f\left(a\right)\ge 0$.

The proof is complete. □

Remark 2. 

(a) It follows from Dini’s theorem that in the previous statement, the convergence of the sequence $\left(f_l\right)$ is also uniform on $\left[a,b\right]$.

(b) The approximability of nonnegative n-convex functions by “simple” and nonnegative n-convex functions is not only an interesting problem, but would also be useful in applications. Theorem 5 (b) is an attempt in this direction.

4. Majorization-Type Inequalities

Let $\left(X,\mathcal{A},\mu \right)$ and $\left(Y,\mathcal{B},\nu \right)$ be measure spaces, where $\mu$ and $\nu$ are finite signed measures. Let $C\subset \mathbb{R}$ be an interval with nonempty interior, and let $\phi :X\to C$, $\psi :Y\to C$ be measurable functions. We define $F_C^n\left(\phi ,\psi \right)$ as the set of all continuous n-convex functions $f:C\to \mathbb{R}$ such that $f\circ \phi \in L\left(\mu \right)$ and $f\circ \psi \in L\left(\nu \right)$.

The first result extends Theorem 6 (c₁) of [15] from convex functions to n-convex functions.

Theorem 6. 

Let $\left(X,\mathcal{A},\mu \right)$ and $\left(Y,\mathcal{B},\nu \right)$ be measure spaces, where μ and ν are finite signed measures. Let $C\subset \mathbb{R}$ be an interval with nonempty interior, and let $\phi :X\to C$, $\psi :Y\to C$ be functions such that ${\phi }^i\in L\left(\mu \right)$ and ${\psi }^i\in L\left(\nu \right)$ for every $i=1,\dots n-1$. Then

(a) For every $f\in F_C^n\left(\phi ,\psi \right)$ inequality

$$\underset{X}{\int }f\circ \phi d\mu \le \underset{Y}{\int }f\circ \psi d\nu$$

(14)

holds if and only if

$$\mu \left(X\right)=\nu \left(Y\right),\underset{X}{\int }{\phi }^{i}d\mu =\underset{Y}{\int }{\psi }^{i}d\nu ,i=1,\dots ,n-1$$

(15)

and

$$\underset{\left\{\phi \ge w\right\}}{\int }{\left(\phi -w\right)}^{n-1}d\mu \le \underset{\left\{\psi \ge w\right\}}{\int }{\left(\psi -w\right)}^{n-1}d\nu ,w\in C°.$$

(16)

(b) For every $f\in F_C^n\left(\psi \right)$ inequality

$$0\le \underset{Y}{\int }f\circ \psi d\nu$$

(17)

holds if and only if

$$0=\nu \left(Y\right),0=\underset{Y}{\int }{\psi }^{i}d\nu ,i=1,\dots ,n-1$$

and

$$0\le \underset{\left\{\psi \ge w\right\}}{\int }{\left(\psi -w\right)}^{n-1}d\nu ,w\in C°.$$

Proof. 

(a) Since $\mu$ and $\nu$ are finite, the constant functions $f_1$, $f_2:C\to \mathbb{R}$, $f_1\left(t\right):=1$ and $f_2\left(t\right):=-1$ belong to $F_C^n\left(\phi ,\psi \right)$, and hence (14) implies $\mu \left(X\right)=\nu \left(Y\right)$.

The functions

$$t\to t^i,t\to -t^i,t\in C,i=1,\dots n-1$$

and

$$t\to {\left(t-w\right)}_{+}^{n-1},t\in C,w\in C°$$

are n-convex, and since $\phi \in L\left(\mu \right)$, $\psi \in L\left(\nu \right)$ and $\mu$, $\nu$ are finite, they belong to $F_C^n\left(\phi ,\psi \right)$. Therefore (14) yields the second part of (15) and (16).

It follows that the conditions (15) and (16) are necessary.

We now show that these conditions are sufficient.

(i) Assume n is even.

By Theorem 4, f is the pointwise limit of an increasing sequence of elementary n-convex functions. If ${\left(f_l\right)}_{l=1}^{\infty }$ is such a sequence, then $\left(f_l\circ \phi \right)$ is also increasing and converges pointwise to $f\circ \phi$ on X. Similarly, $\left(f_l\circ \psi \right)$ is increasing too, and converges pointwise to $f\circ \psi$ on Y.

Theorem 4 also tells us that each element of $\left(f_l\right)$ have the form

$$g\left(t\right)={\displaystyle \sum _{i=0}^{n-1}}{\alpha }_i{t}^i+{\displaystyle \sum _{j=1}^p}{\beta }_j{\left(t-t_j\right)}_{+}^{n-1},t\in \mathbb{R}$$

(18)

for suitable points $t_1<t_2<\dots <t_p$ from $C°$$\left(p\in \mathbb{N}\right)$ and suitable constants ${\alpha }_i\in \mathbb{R}$ $\left(i=0,\dots ,n-1\right)$ and ${\beta }_j>0$$\left(j=1,\dots ,p\right)$.

It follows from the first part of the proof that $g\in F_C^n\left(\phi ,\psi \right)$.

By applying the monotone convergence theorem, we obtain that

$$\underset{X}{\int }{f}_l\circ \phi d\mu \to \underset{X}{\int }f\circ \phi d\mu \mathrm{and}\underset{Y}{\int }{f}_l\circ \psi d\nu \to \underset{Y}{\int }f\circ \psi d\nu .$$

It can be seen that it is sufficient to prove (14) for elementary n-convex functions like g, but it follows from the conditions.

(ii) Assume n is odd, and let $c\in C°$ be fixed.

In this case Theorem 4 implies that f is the pointwise limit of a sequence of elementary n-convex functions such that the sequence is decreasing on $C\cap \left]-\infty ,c\right]$ and increasing on $C\cap \left[c,\infty \right[$. Then the proof of part (i) can be copied with suitable modifications.

(b) The verification of (a) can be followed.

The proof is complete. □

Remark 3. 

The problem considered in the previous theorem has been studied by many authors for finite measures and compact intervals. A detailed analysis can be found in paper [13] for probability measures. Other approaches can be found in papers [5,21,22]. Theorem 6 is particularly interesting because C can be an interval of any type, and μ, ν can be signed measures.

The condition $\nu \left(Y\right)=0$ shows that part (b) of the statement is only interesting if the measure ν is a signed measure; it is meaningless in the case of nonnegative measures. If C is a compact interval, then the result can be found in [23] for convex functions and in [24] for n-convex functions.

The previous remark shows that the investigation of inequality (17) is a deeper problem if we only require its fulfillment for nonnegative n-convex functions. In the case of convex functions, this problem has been extensively studied, leading to the interesting and important topic of Steffensen–Popoviciu measures (see [17] and the references therein; for more recent results, see [15,19]). To the best of my knowledge, there is no similar result for n-convex functions.

The following theorem is an attempt in this direction for 3-convex functions.

Theorem 7. 

Let $\left(X,\mathcal{A},\mu \right)$ and $\left(Y,\mathcal{B},\nu \right)$ be measure spaces, where μ and ν are finite signed measures. Let $a,b\in \mathbb{R}$, $a<b$, and let $\phi :X\to \left[a,b\right]$, $\psi :Y\to \left[a,b\right]$ be functions such that φ is $\mathcal{A}$-measurable and ψ is $\mathcal{B}$-measurable. Then

(a) For every continuous and increasing 3-convex function $f:\left[a,b\right]\to \mathbb{R}$ inequality

$$\underset{X}{\int }f\circ \phi d\mu \le \underset{Y}{\int }f\circ \psi d\nu$$

(19)

holds if and only if

$$\mu \left(X\right)=\nu \left(Y\right),\underset{X}{\int }\phi d\mu \le \underset{Y}{\int }\psi d\nu ,$$

$$\underset{\left\{\phi \ge w\right\}}{\int }{\left(\phi -w\right)}^{2}d\mu \le \underset{\left\{\psi \ge w\right\}}{\int }{\left(\psi -w\right)}^{2}d\nu ,w\in \left]a,b\right[$$

(20)

and

$$\underset{\left\{\phi <w\right\}}{\int }{\left(\phi -w\right)}^{2}d\mu \ge \underset{\left\{\psi <w\right\}}{\int }{\left(\psi -w\right)}^{2}d\nu ,w\in \left]a,b\right[.$$

(b) For every continuous nonnegative and increasing 3-convex function $f:\left[a,b\right]\to \mathbb{R}$ inequality (19) holds if and only if

$$\mu \left(X\right)\le \nu \left(Y\right),\underset{X}{\int }\left(\phi -a\right)d\mu \le \underset{Y}{\int }\left(\psi -a\right)d\nu ,$$

(20) and

$$\underset{\left\{\phi <w\right\}}{\int }{\left(w-a\right)}^2-{\left(\phi -w\right)}^{2}d\mu \ge \underset{\left\{\psi <w\right\}}{\int }{\left(w-a\right)}^2-{\left(\psi -w\right)}^{2}d\nu ,w\in \left]a,b\right[$$

are satisfied.

Proof. 

The conditions guarantee that $f\circ \phi \in L\left(\mu \right)$ and $f\circ \psi \in L\left(\nu \right)$ for every continuous and increasing 3-convex function $f:\left[a,b\right]\to \mathbb{R}$.

We can proceed similarly to the proof of Theorem 6, using the fact that the functions

$$t\to \beta ,t\to t-a,t\to {\left(t-w\right)}_{+}^2,\beta \in \mathbb{R},w\in \left]a,b\right[,t\in \left[a,b\right]$$

(21)

and

$$t\to {\left(w-a\right)}^2-{\left(t-w\right)}_{-}^2,w\in \left]a,b\right[,t\in \left[a,b\right]$$

(22)

are continuous and increasing 3-convex functions, and applying Theorem 5 instead of Theorem 4.

Furthermore, they are all nonnegative if $\beta$ is nonnegative.

The proof is complete. □

Remark 4. 

The previous theorem generalizes parts ($a_1$) and ($a_2$) of Theorem 6 of [15] to 3-convex functions.

5. Applications

First we give the discrete version of Theorem 6.

Theorem 8. 

Let $X:=\left\{1,\dots ,l\right\}$ for some $l\ge 1$, and let $Y:=\left\{1,\dots ,m\right\}$ for some $m\ge 1$. Let $C\subset \mathbb{R}$ be an interval with nonempty interior. Assume ${\left(p_i\right)}_{i=1}^l$ and ${\left(q_j\right)}_{j=1}^m$ are real sequences, and ${\left(s_i\right)}_{i=1}^l$ and ${\left(t_j\right)}_{j=1}^m$ are sequences from C. Then for every continuous n-convex function $f:C\to \mathbb{R}$ inequality

$$\sum _{i=1}^l{p}_{i}f\left(s_i\right)\le \sum _{j=1}^m{q}_{j}f\left(t_j\right)$$

(23)

holds if and only if

$$\sum _{i=1}^l{p}_i{s}_i^k=\sum _{j=1}^m{q}_j{t}_j^k,k=0,1,\dots ,n-1$$

(24)

and

$$\sum _{\left\{i\in X\mid s_i\ge w\right\}}{p}_i{\left(s_i-w\right)}^{n-1}\le \sum _{\left\{j\in Y\mid t_j\ge w\right\}}{q}_j{\left(t_j-w\right)}^{n-1},w\in C°.$$

(25)

Proof. 

By introducing the measure spaces $\left(X,P\left(X\right),\mu \right)$ and $\left(Y,P\left(Y\right),\nu \right)$, where

$$\mu :=\sum _{i=1}^l{p}_i{\epsilon }_i,\nu :=\sum _{j=1}^m{q}_j{\epsilon }_j,$$

the result follows from Theorem 6.

The proof is complete. □

Before analyzing the statement, we will present two known results related to it.

Theorem 9 

(see Sherman [14]). Let l, $m\in {\mathbb{N}}_{+}$, and assume that ${\displaystyle \sum _{i=1}^l}{p}_i{\epsilon }_{s_i}$ and ${\displaystyle \sum _{j=1}^m}{q}_j{\epsilon }_{t_j}$ are probability measures on ${\mathcal{B}}_{\mathbb{R}}$. Then the following assertions are equivalent:

(a) $s_1,\dots ,s_l$ belong to the convex hull of $t_1,\dots ,t_m$ and for every continuous convex function f defined on the convex hull of $t_1,\dots ,t_m$ the inequality (23) holds.

(b) There exists an $l\times m$ matrix $A=\left(a_{ij}\right)$ such that

$$a_{ij}\ge 0,i=1,\dots ,l,j=1,\dots ,m,$$

$${\displaystyle \sum _{j=1}^m}{a}_{ij}=1,i=1,\dots ,l,$$

$$q_j=\sum _{i=1}^l{a}_{ij}{p}_i,j=1,\dots ,m$$

and

$$s_i={\displaystyle \sum _{j=1}^m}{a}_{ij}{t}_j,i=1,\dots ,l.$$

Theorem 10 

(see Horváth [15]). Let $X:=\left\{1,\dots ,l\right\}$ for some $l\ge 1$, let $Y:=\left\{1,\dots ,m\right\}$ for some $m\ge 1$, and let $C\subset \mathbb{R}$ be an interval with nonempty interior. Assume that ${\left(p_i\right)}_{i=1}^l$ and ${\left(q_j\right)}_{j=1}^m$ are real sequences, and ${\left(s_i\right)}_{i=1}^l$ and ${\left(t_j\right)}_{j=1}^m$ are sequences from C. Let $u_1>u_2>\dots >u_o$ be the different elements of ${\left(s_i\right)}_{i=1}^l$ and ${\left(t_j\right)}_{j=1}^m$ in decreasing order $\left(1\le o\le l+m\right)$. Then the inequality (23) holds for every continuous convex function $f:C\to \mathbb{R}$ if and only if

$$\sum _{i=1}^l{p}_i=\sum _{j=1}^m{q}_j,\sum _{i=1}^l{p}_i{s}_i=\sum _{j=1}^m{q}_j{t}_j$$

(26)

and

$$\sum _{\left\{i\in X\mid s_i\ge u_k\right\}}{p}_i{s}_i-\sum _{\left\{j\in Y\mid t_j\ge u_k\right\}}{q}_j{t}_j$$

$$\le u_k\left(\sum _{\left\{i\in X\mid s_i\ge u_k\right\}}{p}_i-\sum _{\left\{j\in Y\mid t_j\ge u_k\right\}}{q}_j\right),k=1,\dots ,o.$$

(27)

Remark 5. 

(a) The previous two results provide necessary and sufficient conditions for the validity of inequality (23) for certain convex functions. Theorem 9 can be applied for discrete measures and requires the existence of a suitable matrix, which must be prepared in order to apply the theorem. Theorem 10 can be applied not only to measures, but also to signed measures, and the fulfillment of the conditions can be verified in a finite number of steps using formal calculation.

(b) Inequality (23) is studied for n-convex functions with discrete signed measures in Theorem 8. The conditions in (24) can be verified by formal calculation, similar to those in (26). Let $u_1>u_2>\dots >u_o$ be the different elements of ${\left(s_i\right)}_{i=1}^l$ and ${\left(t_j\right)}_{j=1}^m$ in decreasing order $\left(1\le o\le l+m\right)$. Condition (25) is satisfied if the inequality

$$0\le \sum _{\left\{j\in Y\mid t_j\ge w\right\}}{q}_j{\left(t_j-w\right)}^{n-1}-\sum _{\left\{i\in X\mid s_i\ge w\right\}}{p}_i{\left(s_i-w\right)}^{n-1},w\in \left[u_{k+1},u_k\right[$$

(28)

holds for any $k=1,\dots ,o-1$. For fixed k, the right-hand side of (28) is a polynomial of degree at most $n-1$, whose nonnegativity can be verified in a finite number of steps.

It is an interesting question whether it would be possible to formulate a condition similar to condition (27).

Before moving on to the next result, we give the known characterization of Steffensen–Popoviciu measures.

Theorem 11. 

Let $a,b\in \mathbb{R}$, $a<b$, and let μ be a finite signed measure on the Borel subsets of $\left[a,b\right]$. Then

(a) (see [17]) μ is a Steffensen–Popoviciu measure, that is $\mu \left(\left[a,b\right]\right)\ge 0$ and

$${\displaystyle \underset{\left[a,b\right]}{\int }}fd\mu \ge 0$$

(29)

for every nonnegative, continuous, and convex function $f:\left[a,b\right]\to \mathbb{R}$ if and only if

$$\underset{\left[a,w\right]}{\int }\left(w-s\right)d\mu \left(s\right)\ge 0,\underset{\left[w,b\right]}{\int }\left(s-w\right)d\mu \left(s\right)\ge 0,w\in \left]a,b\right[.$$

(b) (see [15]) $\mu \left(\left[a,b\right]\right)\ge 0$ and the inequality (29) holds for every nonnegative, continuous, and increasing convex function $f:\left[a,b\right]\to \mathbb{R}$ if and only if

$$\underset{\left[w,b\right]}{\int }\left(s-w\right)d\mu \left(s\right)\ge 0,w\in \left]a,b\right[.$$

In the next theorem, we study the same problem as in the previous theorem, but instead of convex functions, we consider 3-convex functions.

Theorem 12. 

Let $a,b\in \mathbb{R}$, $a<b$, and let μ be a finite signed measure on the Borel subsets of $\left[a,b\right]$. Then

$${\displaystyle \underset{\left[a,b\right]}{\int }}fd\mu \ge 0$$

(30)

for every nonnegative, continuous, and increasing 3-convex function $f:\left[a,b\right]\to \mathbb{R}$ if and only if

$$\mu \left(\left[a,b\right]\right)\ge 0,{\displaystyle \underset{\left[a,b\right]}{\int }}\left(s-a\right)d\mu \left(s\right)\ge 0,$$

$$\underset{\left[a,w\right]}{\int }{\left(s-w\right)}^{2}d\mu \left(s\right)\le {\left(w-a\right)}^2\mu \left(\left[a,b\right]\right),w\in \left]a,b\right[$$

and

$${\displaystyle \underset{\left[w,b\right]}{\int }}{\left(s-w\right)}^{2}d\mu \left(s\right)\ge 0,w\in \left]a,b\right[.$$

Proof. 

We have seen in the proof of Theorem 7 that the functions (21) and (22) are continuous, nonnegative, and increasing 3-convex functions if $\beta \ge 0$, and therefore Theorem 5 (b) can be applied.

The proof is complete. □

Remark 6. 

It can be seen that a statement analogous to Theorem 11 (b) has been proven for 3-convex functions. The conditions are obviously necessary for the inequality (30) to hold for any nonnegative, continuous and 3-convex function $f:\left[a,b\right]\to \mathbb{R}$. Formulating and proving a statement analogous to Theorem 11 (a) for 3-convex functions remains an open problem.

The next result corresponds to the following problem.

Problem 1 

(see Problem 3 of [16]). Given a Borel probability measure μ on $\left[a,b\right]$, find numbers α, β, $\gamma \in \left[0,1\right]$ such that for every continuous 3-convex function $f:\left[a,b\right]\to \mathbb{R}$ we have

$$\gamma f\left(\left(1-\alpha \right)a+\alpha b\right)+\left(1-\gamma \right)f\left(\left(1-\beta \right)a+\beta b\right)\le {\displaystyle \frac{1}{b-a}}{\displaystyle \underset{\left[a,b\right]}{\int }}fd\mu .$$

The first solution to this problem was provided by Bessenyei and Páles [7] in the case where $\mu$ equals ${\displaystyle \frac{dx}{b-a}}$.

Theorem 13 

(see Corollary 2 of [7]). Let $a,b\in \mathbb{R}$, $a<b$. Then for every continuous 3-convex function $f:\left[a,b\right]\to \mathbb{R}$, inequalities

$${\displaystyle \frac{1}{4}}f\left(a\right)+{\displaystyle \frac{3}{4}}f\left({\displaystyle \frac{a+2b}{3}}\right)\le {\displaystyle \frac{1}{b-a}}{\displaystyle \underset{\left[a,b\right]}{\int }}f\left(s\right)ds\le {\displaystyle \frac{3}{4}}f\left({\displaystyle \frac{2a+b}{3}}\right)+{\displaystyle \frac{1}{4}}f\left(b\right)$$

hold.

Sosztok [8] generalized the previous result in the following way.

Theorem 14 

(see Theorem 3 of [8]). Let $a,b\in \mathbb{R}$, $a<b$. If μ is a probability measure on ${\mathcal{B}}_{\left[a,b\right]}$ such that

$${\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{i}d\mu \left(s\right)={\displaystyle \frac{1}{b-a}}{\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{i}ds,i=1,2,$$

then for every continuous 3-convex function $f:\left[a,b\right]\to \mathbb{R}$, inequalities

$${\displaystyle \frac{1}{4}}f\left(a\right)+{\displaystyle \frac{3}{4}}f\left({\displaystyle \frac{a+2b}{3}}\right)\le {\displaystyle \underset{\left[a,b\right]}{\int }}fd\mu \le {\displaystyle \frac{3}{4}}f\left({\displaystyle \frac{2a+b}{3}}\right)+{\displaystyle \frac{1}{4}}f\left(b\right)$$

hold.

It should be noted that in both papers, the authors examine and solve the analogous problem not only for 3-convex functions, but also for n-convex functions. The previous two results are Hermite–Hadamard-type inequalities in the context of 3-convex functions.

The following statement provides a more general answer to Problem 1 than the previous two results.

Theorem 15. 

Let $a,b\in \mathbb{R}$, $a<b$. Let μ be a probability measure on ${\mathcal{B}}_{\left[a,b\right]}$ for which $\mu \ne {\epsilon }_t$ for any $a<t<b$ and $\mu \ne \lambda {\epsilon }_a+\left(1-\lambda \right){\epsilon }_b$ for any $0\le \lambda \le 1$. Then there exist unique $a<c<b$ and $0<\alpha <1$, and there exist unique $a<d<b$ and $0<\beta <1$ such that for every continuous 3-convex function $f:\left[a,b\right]\to \mathbb{R}$, inequalities

$$\alpha f\left(a\right)+\left(1-\alpha \right)f\left(c\right)\le {\displaystyle \underset{\left[a,b\right]}{\int }}fd\mu \le \left(1-\beta \right)f\left(d\right)+\beta f\left(b\right)$$

(31)

hold.

Proof. 

For simplicity, we only examine the first inequality in (31); the second one can be treated similarly.

Let the probability measure ${\nu }_{c,\alpha }$ be defined on ${\mathcal{B}}_{\left[a,b\right]}$ by ${\nu }_{c,\alpha }=\alpha ·{\epsilon }_a+\left(1-\alpha \right)·{\epsilon }_c$$\left(a<c<b,0<\alpha <1\right)$. Then the examined inequality can be written in the following form:

$${\displaystyle \underset{\left[a,b\right]}{\int }}fd{\nu }_{c,\alpha }\le {\displaystyle \underset{\left[a,b\right]}{\int }}fd\mu .$$

By Theorem 6, this inequality holds for every continuous 3-convex function $f:\left[a,b\right]\to \mathbb{R}$ if and only if

$${\displaystyle \underset{\left[a,b\right]}{\int }}sd{\nu }_{c,\alpha }\left(s\right)={\displaystyle \underset{\left[a,b\right]}{\int }}sd\mu \left(s\right),{\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d{\nu }_{c,\alpha }\left(s\right)={\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu \left(s\right)$$

(32)

and

$$\underset{\left[w,b\right]}{\int }{\left(s-w\right)}^{2}d{\nu }_{c,\alpha }\left(s\right)\le \underset{\left[w,b\right]}{\int }{\left(s-w\right)}^{2}d\mu \left(s\right),w\in \left]a,b\right[.$$

(33)

Corollary 2 (a) shows that there is only one $a<c<b$ and $0<\alpha <1$ for which (32) holds. It follows that there is at most one suitable probability measure.

We still need to verify that condition (33) is also satisfied for ${\nu }_{c,\alpha }$.

We could refer to Theorem 4.3 of [13], but since we do not need the generality in [13], we provide a simple proof for the sake of completeness and clarity.

If $c\le w<b$, then

$$\underset{\left[w,b\right]}{\int }{\left(s-w\right)}^{2}d{\nu }_{c,\alpha }\left(s\right)=0\le \underset{\left[w,b\right]}{\int }{\left(s-w\right)}^{2}d\mu \left(s\right),$$

so we only need to prove that for all $a<w<c$

$$\underset{\left[w,b\right]}{\int }{\left(s-w\right)}^{2}d{\nu }_{c,\alpha }\left(s\right)=\left(1-\alpha \right){\left(c-w\right)}^2\le \underset{\left[w,b\right]}{\int }{\left(s-w\right)}^{2}d\mu \left(s\right).$$

(34)

Let $F_{\mu }:\mathbb{R}\to \mathbb{R}$, $F_{\mu }\left(t\right):=\mu \left(i{d}_{\left[a,b\right]}<t\right)$ be the distribution function of $\mu$, and let $F_{{\nu }_{c,\alpha }}:\mathbb{R}\to \mathbb{R}$, $F_{{\nu }_{c,\alpha }}\left(t\right):={\nu }_{c,\alpha }\left(i{d}_{\left[a,b\right]}<t\right)$ be the distribution function of ${\nu }_{c,\alpha }$, respectively.

Since the measures $\mu$ and ${\nu }_{c,\alpha }$ are concentrated on $\left[a,b\right]$, $F_{\mu }\left(t\right)=F_{{\nu }_{c,\alpha }}\left(t\right)=0$$\left(t\le a\right)$ and $F_{\mu }\left(t\right)=F_{{\nu }_{c,\alpha }}\left(t\right)=1$$\left(b<t\right)$.

If $F_{{\nu }_{c,\alpha }}\left(t\right)\le F_{\mu }\left(t\right)$ $\left(a<t<c\right)$, then (34) is obviously satisfied since

$$\begin{gathered} \underset{\left[w,b\right]}{\int }{\left(s-w\right)}^{2}d{\nu }_{c,\alpha }\left(s\right)=\underset{\left[w,c\right]}{\int }{\left(s-w\right)}^{2}d{\nu }_{c,\alpha }\left(s\right) \\ \le \underset{\left[w,c\right]}{\int }{\left(s-w\right)}^{2}d\mu \left(s\right)+\underset{\left]c,b\right]}{\int }{\left(s-w\right)}^{2}d\mu \left(s\right). \end{gathered}$$

If $F_{\mu }\left(t\right)\le F_{{\nu }_{c,\alpha }}\left(t\right)$ $\left(a<t\le c\right)$, then $F_{\mu }\left(t\right)\le F_{{\nu }_{c,\alpha }}\left(t\right)$ $\left(a\le t\le b\right)$, and from this it follows that $F_{\mu }\left(t\right)=F_{{\nu }_{c,\alpha }}\left(t\right)$ $\left(a\le t\le b\right)$, since $\mu$ and ${\nu }_{c,\alpha }$ are both probability measures.

Therefore, we only need to show (34) when there is a point $a<t_0<c$ such that $F_{\mu }\left(t\right)\le \alpha =F_{{\nu }_{c,\alpha }}\left(t\right)$$\left(a<t\le t_0\right)$ and $F_{{\nu }_{c,\alpha }}\left(t\right)=\alpha <F_{\mu }\left(t\right)$$\left(t_0<t\le c\right)$.

Let the function $g:\left[a,c\right]\to \mathbb{R}$ be defined by

$$g\left(w\right):={\displaystyle \frac{1}{2}}\underset{\left[w,b\right]}{\int }{\left(s-w\right)}^{2}d\mu \left(s\right)-{\displaystyle \frac{1}{2}}\left(1-\alpha \right){\left(c-w\right)}^2.$$

Then $g\left(a\right)=0$ and $g\left(c\right)\ge 0$.

By applying the theorem on the differentiability of parameter-dependent integrals, g is differentiable (and hence it is continuous) and

$$g^{\prime }\left(w\right)=-\underset{\left[w,b\right]}{\int }\left(s-w\right)d\mu \left(s\right)+\left(1-\alpha \right)\left(c-w\right),w\in \left[a,c\right].$$

Then $g^{\prime }$ is continuous and $g_{+}^{\prime }\left(a\right)=0$, $g_{-}^{\prime }\left(c\right)\le 0$.

By applying the theorem on the differentiability of parameter-dependent integrals again, we obtain that the right-hand derivative of $g^{\prime }$ exists on $\left]a,c\right[$ and since $F_{\mu }$ is left-continuous,

$$g_{+}^{″}\left(w\right)=\alpha -F_{\mu }\left(w\right),w\in \left]a,c\right[.$$

Since $g^{\prime }$ is continuous, $g_{+}^{″}\left(w\right)\ge 0$$\left(a<w\le t_0\right)$, and $g_{-}^{″}\left(w\right)<0$$\left(t_0<w<c\right)$, $g^{\prime }$ is increasing on $\left[a,t_0\right]$ and strictly decreasing on $\left[t_0,c\right]$. It follows from $g_{+}^{\prime }\left(a\right)=0$ and $g_{-}^{\prime }\left(c\right)\le 0$ that either $g^{\prime }\left(w\right)\ge 0$ $\left(a\le w\le c\right)$ or there is a point $t_1\in \left[t_0,c\right[$ such that $g^{\prime }\left(w\right)\ge 0$$\left(a\le w\le t_1\right)$ and $g^{\prime }\left(w\right)<0$$\left(t_1<w\le c\right)$.

In the first case g is increasing on $\left[a,c\right]$, and hence $g\left(a\right)=0$ implies $g\left(w\right)\ge 0$ $\left(a\le w\le c\right)$. In the second case g is increasing on $\left[a,t_1\right]$ and g is decreasing on $\left[t_1,c\right]$, and hence $g\left(a\right)=0$ and $g\left(c\right)\ge 0$ yield that $g\left(w\right)\ge 0$$\left(a\le w\le c\right)$.

The proof is complete. □

Remark 7. 

It follows from Lemma 3 that the specific forms of constants c, α and d, β in the previous theorem are

$$c={\displaystyle \frac{{\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu \left(s\right)-a{\displaystyle \underset{\left[a,b\right]}{\int }}sd\mu \left(s\right)}{{\displaystyle \underset{\left[a,b\right]}{\int }}sd\mu \left(s\right)-a}},\alpha ={\displaystyle \frac{{\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu \left(s\right)-{\left({\displaystyle \underset{\left[a,b\right]}{\int }}sd\mu \left(s\right)\right)}^2}{{\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu \left(s\right)-2a{\displaystyle \underset{\left[a,b\right]}{\int }}sd\mu \left(s\right)+a^2}}$$

and

$$d={\displaystyle \frac{{\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu \left(s\right)-b{\displaystyle \underset{\left[a,b\right]}{\int }}sd\mu \left(s\right)}{b-{\displaystyle \underset{\left[a,b\right]}{\int }}sd\mu \left(s\right)}},\beta ={\displaystyle \frac{{\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu \left(s\right)-{\left({\displaystyle \underset{\left[a,b\right]}{\int }}sd\mu \left(s\right)\right)}^2}{{\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu \left(s\right)-2b{\displaystyle \underset{\left[a,b\right]}{\int }}sd\mu \left(s\right)+b^2}}.$$

Remark 8. 

(a) Theorem 15 includes Theorems 13 and 14 as special cases.

(b) The proof of Theorem 13 uses the properties of Hermite interpolating polynomials and the approximability of n-convex functions by n-convex functions in $C^{\infty }$. The proof of Theorem 14 is based primarily on the results of paper [13]. As we have already mentioned, the proof of inequality (34) does not use the results of the paper [13]; it is a new and direct approach.

Finally, we give the form of inequality (31) analogous to Fejér’s inequality.

Corollary 3. 

Let $a,b\in \mathbb{R}$, $a<b$, and let $g:\left[a,b\right]\to \mathbb{R}$ be a nonnegative and Lebesgue integrable function for which

$${\displaystyle \underset{\left[a,b\right]}{\int }}g\left(s\right)ds=1\mathit{and}g\left(a+b-t\right)=g\left(t\right),t\in \left[a,b\right].$$

Then for every continuous 3-convex function $f:\left[a,b\right]\to \mathbb{R}$, inequalities

$$\alpha f\left(a\right)+\left(1-\alpha \right)f\left(c\right)\le {\displaystyle \underset{\left[a,b\right]}{\int }}f\left(s\right)g\left(s\right)ds\le \left(1-\beta \right)f\left(d\right)+\beta f\left(b\right)$$

hold, where the unique constants c, α and d, β are

$$c={\displaystyle \frac{2{\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu -a\left(a+b\right)}{b-a}},d={\displaystyle \frac{2{\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu -b\left(a+b\right)}{b-a}}$$

and

$$\alpha =\beta ={\displaystyle \frac{4{\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu -{\left(a+b\right)}^2}{4\left({\displaystyle \underset{\left[a,b\right]}{\int }}{s}^{2}d\mu -ab\right)}}.$$

Proof. 

Let the probability measure $\mu$ be defined on ${\mathcal{B}}_{\left[a,b\right]}$ by $\mu \left(A\right):={\displaystyle \underset{A}{\int }}g\left(s\right)ds$. Applying Theorem 15 to this probability measure, we obtain the result.

The proof is complete. □

6. Discussion

In this paper, we primarily study majorization-type inequalities related to n-convex (specifically 3-convex) functions. The obtained results, including the approximation theorems used in the proofs, are interesting and novel primarily because they hold not only for compact but also for arbitrary intervals. We also succeeded in proving majorization-type results for (nonnegative) increasing and 3-convex functions defined on compact intervals. Various problems arising from the field of convexity can be addressed with the results obtained, which illustrate their applicability.