Inequality on Three Intervals for Higher-Order Convex Functions

Abstract

In this article, an inequality on three intervals for convex functions is extended to inequalities on three intervals for higher-order convex functions. Some corollaries and applications are mentioned.

1. Introduction

Recall the definition of higher-order convex functions [1,2,3].

Definition 1. 

Suppose f is a real-valued function on $[a,b]$. An n-th order divided difference of f at distinct points $x_0,\dots ,x_n$ in $[a,b]$ may be defined by

$$\left[x_i\right]f=f\left(x_i\right),(i=0,\dots ,n)$$

and

$$[x_0,\dots ,x_n]f=([x_1,\dots ,x_n]f-[x_0,\dots ,x_{n-1}]f)/(x_n-x_0).$$

A function $f:[a,b]\to \mathbb{R}$ is called n-convex on $[a,b]$, iff the following inequality holds for all choices of $(n+1)$ distinct points in $[a,b]$,

$$[x_0,\dots ,x_n]f\ge 0.$$

This unifies non-negative functions (0-convex), non-decreasing functions (1-convex), the convex function in the usual sense (2-convex) and higher-order convex functions. For n-times differentiable functions, n-convex is $f^{\left(n\right)}\ge 0$.

Inequalities for higher-order convex functions are usually analogs of the Jensen-type inequality for 2-convex (convex) functions or Levinson-type inequality [4,5,6,7,8,9] for 3-convex functions. The inequalities of n-convex functions for general $n(n\ge 2)$ are truly extensions for these specific inequalities, so it is natural for researchers to consider extending the inequalities (Jensen type, Hermite–Hadamard type [10], majorization type…) for convex functions to higher-order convex versions.

In this article, we will focus on several points. Can we establish inequalities for higher-order convex functions with some simple conditions (without assuming too many terms to be non-negative)? Can we establish inequalities for higher-order convex functions under different conditions than those in the existing references? What difference can we make in terms of inequalities in information theory from the perspective of higher-order convexity than convexity in the usual sense?

In [11], we establish the following inequality for convex functions on three intervals, which will also be useful to higher-order convex functions in this paper.

Theorem 1. 

Let $a_0<a_1<a_2<a_3$  and  $x:[a_0,a_3]\to \mathbb{R}$. Let  $p:[a_0,a_1]\cup [a_2,a_3]\to {\mathbb{R}}_{+}$  and  $q:[a_1,a_2]\to {\mathbb{R}}_{+}$  be non-negative. If  $x\left(t\right)$  is increasing for  $t\in [a_0,a_3]$  and

$$\begin{array}{cc}\hfill & {\int }_{a_1}^{a_2}q\left(t\right)x\left(t\right)dt={\int }_{a_0}^{a_1}p\left(t\right)x\left(t\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)x\left(t\right)dt,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & {\int }_{a_1}^{a_2}q\left(t\right)dt={\int }_{a_0}^{a_1}p\left(t\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)dt,\hfill \end{array}$$

(2)

then

$${\int }_{a_1}^{a_2}q\left(t\right)\varphi \left(x\left(t\right)\right)dt\le {\int }_{a_0}^{a_1}p\left(t\right)\varphi \left(x\left(t\right)\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi \left(x\left(t\right)\right)dt$$

(3)

holds for every convex function ϕ such that the integrals above exist.

In [11], this is used to prove some generalized Levinson inequality and Ky Fan-type inequalities. Some equivalence of the Hermite–Hadamard-type inequality and connections to the Lah–Ribarič inequality are also pointed out. In [12] this inequality is extended and proves more inequalities.

In [3] (p. 143), the following types of identities in Theorem 2 and Theorem 3 are proven and used to establish some inequalities for higher-order convex functions. The Green function is defined by

$$\begin{array}{c}\hfill G(t,s)=\left\{\begin{array}{cc}{\displaystyle \frac{(t-b)(s-a)}{b-a}},\hfill & a\le s\le t,\hfill \\ {\displaystyle \frac{(s-b)(t-a)}{b-a}},\hfill & t\le s\le b.\hfill \end{array}\right.\end{array}$$

The function G is convex with respect to both s and t.

Theorem 2. 

Let integrable functions  $g:[\alpha ,\beta ]\to [a,b]$,  $p:[\alpha ,\beta ]\to \mathbb{R}$  be such that

$$\begin{array}{c}\hfill {\int }_{\alpha }^{\beta }p\left(x\right)dx=0, {\int }_{\alpha }^{\beta }p\left(x\right)g\left(x\right)dx=0.\end{array}$$

(4)

Let  $n\ge 3$  and  $f:I\to \mathbb{R},[a,b]\subset I$  be a function with  $f^{(n-1)}$  absolutely continuous. Then we have

$$\begin{array}{cc}\hfill & {\int }_{\alpha }^{\beta }p\left(x\right)f\left(g\left(x\right)\right)dx\hfill \\ \hfill =& \sum _{k=0}^{n-3}{\displaystyle \frac{f^{(k+2)}\left(a\right)}{k!}}{\int }_a^b\left({\int }_{\alpha }^{\beta }p\left(x\right)G(g\left(x\right),t)dx\right){(t-a)}^{k}dt\hfill \\ \hfill +& {\displaystyle \frac{1}{(n-3)!}}{\int }_a^b{f}^{\left(n\right)}\left(s\right)\left({\int }_s^b\left({\int }_{\alpha }^{\beta }p\left(x\right)G(g\left(x\right),t)dx\right){(t-s)}^{n-3}dt\right)ds.\hfill \end{array}$$

Theorem 3. 

Let  $n\in \mathbb{N},(n\ge 3)$,  $f:I\to \mathbb{R}$  be a function with  $f^{(n-1)}$  absolutely continuous and  $I\subset \mathbb{R}$  be an open interval with  $a,b\in I,a<b$. Also, let  $g:[\alpha ,\beta ]\to [a,b]$,  $p:[\alpha ,\beta ]\to \mathbb{R}$  be two functions with

$$\begin{array}{c}\hfill {\int }_{\alpha }^{\beta }p\left(x\right)dx=0, {\int }_{\alpha }^{\beta }p\left(x\right)g\left(x\right)dx=0.\end{array}$$

(5)

Then

$$\begin{array}{cc}\hfill & {\int }_{\alpha }^{\beta }p\left(x\right)f\left(g\left(x\right)\right)dx={\displaystyle \frac{f^{\prime }\left(a\right)-f^{\prime }\left(b\right)}{b-a}}{\int }_a^b{\int }_{\alpha }^{\beta }p\left(x\right)G(g\left(x\right),s)dxds\hfill \\ \hfill +& \sum _{k=2}^{n-1}{\displaystyle \frac{k}{(k-1)!}}{\int }_a^b\left({\int }_{\alpha }^{\beta }p\left(x\right)G(g\left(x\right),s)dx\right){\displaystyle \frac{f^{\left(k\right)}\left(a\right){(s-a)}^{k-1}-f^{\left(k\right)}\left(b\right){(s-b)}^{k-1}}{b-a}}ds\hfill \\ \hfill +& {\displaystyle \frac{1}{(n-3)!}}{\int }_a^b{f}^{\left(n\right)}\left(t\right)\left({\int }_a^b\left({\int }_{\alpha }^{\beta }p\left(x\right)G(g\left(x\right),s)dx\right){\tilde{T}}_{n-2}(s,t)ds\right)dt,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\tilde{T}}_{n-2}(s,t)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{b-a}}\left[{\displaystyle \frac{{(s-t)}^{n-2}}{n-2}}+(s-a){(s-t)}^{n-3}\right],\hfill & a\le t\le s\le b,\hfill \\ \hfill & \\ {\displaystyle \frac{1}{b-a}}\left[{\displaystyle \frac{{(s-t)}^{n-2}}{n-2}}+(s-b){(s-t)}^{n-3}\right],\hfill & a\le s\le t\le b.\hfill \end{array}\right.\end{array}$$

The assumptions (1) and (2) are similar to (4) and (5).

These results inspire us to consider higher-order convex functions for Theorem 1. The difficulty is that “inequality” in Theorem 1 can be expressed as an integral of $f^{″}$ with a non-negative weight function but may not be expressed by some higher-order derivative $f^{\left(n\right)}$; identities in Theorem 2 and Theorem 3 are expressed by an integral of higher-order derivative $f^{\left(n\right)}$, but the weight function may not be non-negative.

In this paper, we will establish inequality for some higher-order convex functions without assuming the weight function to be non-negative because the inequality on three intervals naturally possesses this property. The second section is particularly devoted to inequalities for 3-convex functions, and some necessary lemmas and corollaries are also mentioned. The third section is devoted to inequalities for general $(n+2)$-convex functions, and we will show that these kinds of inequalities hold under different assumptions. The fourth section consists of an application to Csiszár $\varphi$-divergence in information theory.

2. Three Interval Inequality for 3-Convex Functions

In this section, we establish inequality on three intervals for 3-convex functions. The following lemma [12] is essential in our proof. Here

$$\begin{array}{c}\hfill {(x-s)}_{+}=\left\{\begin{array}{cc}0,\hfill & x<s,\hfill \\ x-s,\hfill & x\ge s.\hfill \end{array}\right.\end{array}$$

Lemma 1. 

Let  $a_0<a_1<a_2<a_3$  and  $x:[a_0,a_3]\to \mathbb{R}$. Let  $p:[a_0,a_1]\cup [a_2,a_3]\to {\mathbb{R}}_{+}$  and  $q:[a_1,a_2]\to {\mathbb{R}}_{+}$  be non-negative. If  $x\left(t\right)$  is an increasing function for  $t\in [a_0,a_3]$  and (1), (2) hold, then for  $\varphi :I\to \mathbb{R}$  with  ${\varphi }^{\prime }$  absolutely continuous, we get

$$\begin{array}{c}\hfill {\int }_{a_0}^{a_1}p\left(t\right)\varphi \left(x\left(t\right)\right)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi \left(x\left(t\right)\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi \left(x\left(t\right)\right)dt={\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}\left(s\right)k\left(s\right)ds,\end{array}$$

where

$$\begin{array}{c}\hfill k\left(s\right)={\int }_{a_0}^{a_1}p\left(t\right){(x\left(t\right)-s)}_{+}dt-{\int }_{a_1}^{a_2}q\left(t\right){(x\left(t\right)-s)}_{+}dt+{\int }_{a_2}^{a_3}p\left(t\right){(x\left(t\right)-s)}_{+}dt.\end{array}$$

Furthermore, we have  $k\left(s\right)\ge 0$.

This lemma points out the non-negativity of the weight function $k\left(s\right)$.

Remark 1. 

This lemma also holds if  $x\left(t\right)$ is decreasing, only changing

$$\begin{array}{c}\hfill {\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}\left(s\right)k\left(s\right)ds\end{array}$$

with

$$\begin{array}{c}\hfill {\int }_{x\left(a_3\right)}^{x\left(a_0\right)}{\varphi }^{″}\left(s\right)k\left(s\right)ds.\end{array}$$

It can be seen from a substitution that  $\widehat{x}\left(t\right)=x(a_0+a_3-t),\widehat{q}\left(t\right)=q(a_0+a_3-t),\widehat{p}\left(t\right)=p(a_0+a_3-t)$, and we can use a similar conclusion for  $\widehat{x},\widehat{p},\widehat{q}$.

The first main result below is for 3-convex functions.

Theorem 4. 

Let  $a_0<a_1<a_2<a_3$  and  $x:[a_0,a_3]\to \mathbb{R}$. Let two functions  $p:[a_0,a_1]\cup [a_2,a_3]\to {\mathbb{R}}_{+}$  and  $q:[a_1,a_2]\to {\mathbb{R}}_{+}$  be non-negative. If  $x\left(t\right)$  is an increasing function for  $t\in [a_0,a_3]$  and  (1), (2)  hold, then

$$\begin{array}{cc}\hfill & {\int }_{a_0}^{a_1}p\left(t\right)\varphi \left(x\left(t\right)\right)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi \left(x\left(t\right)\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi \left(x\left(t\right)\right)dt\hfill \\ \hfill & +{\int }_{x\left(a_0\right)}^{x\left(a_3\right)+1}{\varphi }^{‴}\left(u\right)K\left(u\right)du\hfill \\ \hfill =& {\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)+1)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)+1)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)+1)dt\hfill \end{array}$$

holds for every 3-differentiable ϕ such that the above integrals exist, where  $K\left(u\right)\ge 0$  is defined by

$$\begin{array}{c}\hfill K\left(u\right)={\int }_{u-1}^{u}k\left(s\right)ds,\end{array}$$

and

$$\begin{array}{c}\hfill k\left(s\right)={\int }_{a_0}^{a_1}p\left(t\right){(x\left(t\right)-s)}_{+}dt-{\int }_{a_1}^{a_2}q\left(t\right){(x\left(t\right)-s)}_{+}dt+{\int }_{a_2}^{a_3}p\left(t\right){(x\left(t\right)-s)}_{+}dt\end{array}$$

for  $x\left(a_0\right)\le s\le x\left(a_3\right)$;  $k\left(s\right)=0$  for  $s>x\left(a_3\right)$  or  $s<x\left(a_0\right)$.

Proof. 

Utilize Lemma 1 to get

$$\begin{array}{cc}\hfill & {\int }_{a_0}^{a_1}p\left(t\right)\varphi \left(x\left(t\right)\right)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi \left(x\left(t\right)\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi \left(x\left(t\right)\right)dt={\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}\left(s\right)k_1\left(s\right)ds,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill k_1\left(s\right)={\int }_{a_0}^{a_1}p\left(t\right){(x\left(t\right)-s)}_{+}dt-{\int }_{a_1}^{a_2}q\left(t\right){(x\left(t\right)-s)}_{+}dt+{\int }_{a_2}^{a_3}p\left(t\right){(x\left(t\right)-s)}_{+}dt.\end{array}$$

And also regard $x\left(t\right)+1$ as a new function $\widehat{x}\left(t\right)$; similar identity conditions (1), (2) also hold; thus we can apply Lemma 1

$$\begin{array}{cc}\hfill & {\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)+1)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)+1)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)+1)dt\hfill \\ \hfill =& {\int }_{x\left(a_0\right)+1}^{x\left(a_3\right)+1}{\varphi }^{″}\left(s\right)k_2\left(s\right)ds,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & k_2\left(s\right)=\hfill \\ \hfill {\int }_{a_0}^{a_1}& p\left(t\right){(x\left(t\right)+1-s)}_{+}dt-{\int }_{a_1}^{a_2}q\left(t\right){(x\left(t\right)+1-s)}_{+}dt+{\int }_{a_2}^{a_3}p\left(t\right){(x\left(t\right)+1-s)}_{+}dt.\hfill \end{array}$$

Then we observe that

$$\begin{array}{c}\hfill {\int }_{x\left(a_0\right)+1}^{x\left(a_3\right)+1}{\varphi }^{″}\left(s\right)k_2\left(s\right)ds={\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s+1)k_2(s+1)ds={\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s+1)k_1\left(s\right)ds;\end{array}$$

thus the identity in the theorem is equivalent to

$$\begin{array}{cc}\hfill & {\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s+1)k_1\left(s\right)ds-{\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}\left(s\right)k_1\left(s\right)ds\hfill \\ \hfill =& {\int }_{x\left(a_0\right)}^{x\left(a_3\right)}({\varphi }^{″}(s+1)-{\varphi }^{″}\left(s\right))k_1\left(s\right)ds\hfill \\ \hfill =& {\int }_{x\left(a_0\right)}^{x\left(a_3\right)}\left({\int }_s^{s+1}{\varphi }^{‴}\left(u\right)du\right)k_1\left(s\right)ds\hfill \\ \hfill =& {\int }_{x\left(a_0\right)}^{x\left(a_3\right)+1}{\varphi }^{‴}\left(u\right){\int }_{u-1}^{u}k\left(s\right)dsdu,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill k\left(s\right)=\left\{\begin{array}{cc}{k}_1\left(s\right),\hfill & x\left(a_0\right)\le s\le x\left(a_3\right),\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.\end{array}$$

Then

$$K\left(u\right)={\int }_{u-1}^{u}k\left(s\right)ds\ge 0.$$

□

The following inequality is a direct corollary.

Corollary 1. 

Let  $a_0<a_1<a_2<a_3$  and  $x:[a_0,a_3]\to \mathbb{R}$. Let two functions  $p:[a_0,a_1]\cup [a_2,a_3]\to {\mathbb{R}}_{+}$  and  $q:[a_1,a_2]\to {\mathbb{R}}_{+}$  be non-negative. If  $x\left(t\right)$  is an increasing function for  $t\in [a_0,a_3]$  and  (1), (2) hold, then

$$\begin{array}{cc}\hfill & {\int }_{a_0}^{a_1}p\left(t\right)\varphi \left(x\left(t\right)\right)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi \left(x\left(t\right)\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi \left(x\left(t\right)\right)dt\hfill \\ \hfill \le & {\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)+1)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)+1)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)+1)dt\hfill \end{array}$$

holds for every 3-convex function ϕ such that the integrals above exist.

The following corollary is an $L_p$ estimation on the difference in the inequality above. Note that $p\left(t\right),q\left(t\right)$ are functions but $p,q$ are real numbers.

Corollary 2. 

Let  $a_0<a_1<a_2<a_3$  and  $x:[a_0,a_3]\to \mathbb{R}$. Let two functions  $p\left(t\right):[a_0,a_1]\cup [a_2,a_3]\to {\mathbb{R}}_{+}$  and  $q\left(t\right):[a_1,a_2]\to {\mathbb{R}}_{+}$  be non-negative. If  $x\left(t\right)$  is an increasing function for  $t\in [a_0,a_3]$  and (1), (2) hold, then for  $p>1,{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, inequality

$$\begin{array}{cc}\hfill & {\int }_{a_0}^{a_1}p\left(t\right)\varphi \left(x\left(t\right)\right)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi \left(x\left(t\right)\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi \left(x\left(t\right)\right)dt\hfill \\ \hfill & +{\left({\int }_{x\left(a_0\right)}^{x\left(a_3\right)+1}{\left({\varphi }^{‴}\left(u\right)\right)}^{p}du\right)}^{1/p}{\left({\int }_{x\left(a_0\right)}^{x\left(a_3\right)+1}{\left(K\left(u\right)\right)}^{q}du\right)}^{1/q}\hfill \\ \hfill \ge & {\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)+1)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)+1)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)+1)dt\hfill \end{array}$$

holds for every ϕ with  ${\varphi }^{‴}\ge 0$ such that the integrals exist, where

$$\begin{array}{c}\hfill K\left(u\right)={\int }_{u-1}^{u}k\left(s\right)ds,\end{array}$$

and

$$\begin{array}{c}\hfill k\left(s\right)={\int }_{a_0}^{a_1}p\left(t\right){(x\left(t\right)-s)}_{+}dt-{\int }_{a_1}^{a_2}q\left(t\right){(x\left(t\right)-s)}_{+}dt+{\int }_{a_2}^{a_3}p\left(t\right){(x\left(t\right)-s)}_{+}dt\end{array}$$

for  $x\left(a_0\right)\le s\le x\left(a_3\right)$;  $k\left(s\right)=0$  for  $s>x\left(a_3\right)$  or  $s<x\left(a_0\right)$.

Let $p\to +\infty$. We then have the following corollary, where the error term is more clear.

Corollary 3. 

Let  $a_0<a_1<a_2<a_3$  and  $x:[a_0,a_3]\to \mathbb{R}$. Let two functions  $p\left(t\right):[a_0,a_1]\cup [a_2,a_3]\to {\mathbb{R}}_{+}$  and  $q\left(t\right):[a_1,a_2]\to {\mathbb{R}}_{+}$  be non-negative. If  $x\left(t\right)$  is an increasing function for  $t\in [a_0,a_3]$  and (1), (2) hold, then

$$\begin{array}{cc}\hfill & {\int }_{a_0}^{a_1}p\left(t\right)\varphi \left(x\left(t\right)\right)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi \left(x\left(t\right)\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi \left(x\left(t\right)\right)dt\hfill \\ \hfill & +{\displaystyle \frac{{\parallel {\varphi }^{‴}\parallel }_{\infty }}{2}}\left({\int }_{a_0}^{a_1}p\left(t\right)x^2\left(t\right)dt-{\int }_{a_1}^{a_2}q\left(t\right)x^2\left(t\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)x^2\left(t\right)dt\right)\hfill \\ \hfill \ge & {\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)+1)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)+1)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)+1)dt\hfill \end{array}$$

holds for every ϕ with  ${\varphi }^{‴}\ge 0$  such that the integrals exist.

Proof. 

In the corollary above, as $q=1$, we have

$$\begin{array}{cc}\hfill & {\left({\int }_{x\left(a_0\right)}^{x\left(a_3\right)+1}{\left(K\left(u\right)\right)}^{q}du\right)}^{1/q}\hfill \\ \hfill =& {\int }_{x\left(a_0\right)}^{x\left(a_3\right)+1}{\int }_{u-1}^{u}k\left(s\right)dsdu\hfill \\ \hfill =& {\int }_{x\left(a_0\right)}^{x\left(a_3\right)}\left({\int }_{a_0}^{a_1}p\left(t\right){(x\left(t\right)-s)}_{+}dt-{\int }_{a_1}^{a_2}q\left(t\right){(x\left(t\right)-s)}_{+}dt+{\int }_{a_2}^{a_3}p\left(t\right){(x\left(t\right)-s)}_{+}dt\right)\hfill \\ \hfill =& {\int }_{a_0}^{a_1}p\left(t\right){\displaystyle \frac{{(x\left(t\right)-x\left(a_0\right))}^2}{2}}dt-{\int }_{a_1}^{a_2}q\left(t\right){\displaystyle \frac{{(x\left(t\right)-x\left(a_0\right))}^2}{2}}dt+{\int }_{a_2}^{a_3}p\left(t\right){\displaystyle \frac{{(x\left(t\right)-x\left(a_0\right))}^2}{2}}dt\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}\left({\int }_{a_0}^{a_1}p\left(t\right)x^2\left(t\right)dt-{\int }_{a_1}^{a_2}q\left(t\right)x^2\left(t\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)x^2\left(t\right)dt\right),\hfill \end{array}$$

in which the second equation was obtained due to the definition of $k\left(s\right)$, the third equation was obtained due to Fubini’s theorem and the fourth equation was obtained due to (1), (2). □

The second main theorem is similar to the above and is an analog of the Levinson-type inequality.

Theorem 5. 

Let  $a_0<a_1<a_2<a_3$  and  $x:[a_0,a_3]\to \mathbb{R}$. Let two functions  $p:[a_0,a_1]\cup [a_2,a_3]\to {\mathbb{R}}_{+}$  and  $q:[a_1,a_2]\to {\mathbb{R}}_{+}$  be non-negative. If  $x\left(t\right)$  is an increasing function for  $t\in [a_0,a_3]$  with  $0\le x\left(t\right)\le {\displaystyle \frac{1}{2}}$  and (1), (2) hold, then

$$\begin{array}{cc}\hfill & {\int }_{a_0}^{a_1}p\left(t\right)\varphi \left(x\left(t\right)\right)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi \left(x\left(t\right)\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi \left(x\left(t\right)\right)dt\hfill \\ \hfill \le & {\int }_{a_0}^{a_1}p\left(t\right)\varphi (1-x\left(t\right))dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (1-x\left(t\right))dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (1-x\left(t\right))dt\hfill \end{array}$$

holds for every 3-convex function ϕ such that the integrals above exist.

Proof. 

We just consider a 3-convex function with ${\varphi }^{″}$. Utilize Lemma 1 to get

$$\begin{array}{cc}\hfill & {\int }_{a_0}^{a_1}p\left(t\right)\varphi \left(x\left(t\right)\right)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi \left(x\left(t\right)\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi \left(x\left(t\right)\right)dt={\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}\left(s\right)k_1\left(s\right)ds,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill k_1\left(s\right)={\int }_{a_0}^{a_1}p\left(t\right){(x\left(t\right)-s)}_{+}dt-{\int }_{a_1}^{a_2}q\left(t\right){(x\left(t\right)-s)}_{+}dt+{\int }_{a_2}^{a_3}p\left(t\right){(x\left(t\right)-s)}_{+}dt.\end{array}$$

And also regard $1-x\left(t\right)$ as a new function $\widehat{x}\left(t\right)$; similar identity conditions also hold; thus we can apply Lemma 1 and Remark 1 to get

$$\begin{array}{cc}\hfill & {\int }_{a_0}^{a_1}p\left(t\right)\varphi (1-x\left(t\right))dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (1-x\left(t\right))dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (1-x\left(t\right))dt\hfill \\ \hfill =& {\int }_{1-x\left(a_3\right)}^{1-x\left(a_0\right)}{\varphi }^{″}\left(s\right)k_2\left(s\right)ds,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & k_2\left(s\right)=\hfill \\ \hfill {\int }_{a_0}^{a_1}& p\left(t\right){(1-x\left(t\right)-s)}_{+}dt-{\int }_{a_1}^{a_2}q\left(t\right){(1-x\left(t\right)-s)}_{+}dt+{\int }_{a_2}^{a_3}p\left(t\right){(1-x\left(t\right)-s)}_{+}dt.\hfill \end{array}$$

As $k_1\left(s\right),k_2\left(s\right)\ge 0$ and $\varphi$ is 3-convex, we have

$$\begin{array}{cc}\hfill & {\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}\left(s\right)k_1\left(s\right)ds\le {\varphi }^{″}\left(x\left(a_3\right)\right){\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{k}_1\left(s\right)ds,\hfill \\ \hfill & {\int }_{1-x\left(a_3\right)}^{1-x\left(a_0\right)}{\varphi }^{″}\left(s\right)k_2\left(s\right)ds\ge {\varphi }^{″}(1-x\left(a_3\right)){\int }_{1-x\left(a_3\right)}^{1-x\left(a_0\right)}{k}_2\left(s\right)ds.\hfill \end{array}$$

(6)

It is easy to observe that

$$\begin{array}{c}\hfill {\int }_{1-x\left(a_3\right)}^{1-x\left(a_0\right)}{k}_2\left(s\right)ds={\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{k}_2(1-s)ds,\end{array}$$

where

$$\begin{array}{c}\hfill k_2(1-s)={\int }_{a_0}^{a_1}p\left(t\right){(s-x\left(t\right))}_{+}dt-{\int }_{a_1}^{a_2}q\left(t\right){(s-x\left(t\right))}_{+}dt+{\int }_{a_2}^{a_3}p\left(t\right){(s-x\left(t\right))}_{+}dt.\end{array}$$

Then we consider

$$\begin{array}{cc}\hfill & k_1\left(s\right)-k_2(1-s)\hfill \\ \hfill =& {\int }_{a_0}^{a_1}p\left(t\right){(x\left(t\right)-s)}_{+}dt-{\int }_{a_1}^{a_2}q\left(t\right){(x\left(t\right)-s)}_{+}dt+{\int }_{a_2}^{a_3}p\left(t\right){(x\left(t\right)-s)}_{+}dt\hfill \\ \hfill & -{\int }_{a_0}^{a_1}p\left(t\right){(s-x\left(t\right))}_{+}dt+{\int }_{a_1}^{a_2}q\left(t\right){(s-x\left(t\right))}_{+}dt-{\int }_{a_2}^{a_3}p\left(t\right){(s-x\left(t\right))}_{+}dt\hfill \\ \hfill =& {\int }_{a_0}^{a_1}p\left(t\right)(x\left(t\right)-s)dt-{\int }_{a_1}^{a_2}q\left(t\right)(x\left(t\right)-s)dt+{\int }_{a_2}^{a_3}p\left(t\right)(x\left(t\right)-s)dt=0,\hfill \end{array}$$

in which the last step was obtained due to (1), (2).

Thus, we can apply the identities above to (6) with the facts that $\varphi$ is 3-convex and that $0\le x\left(t\right)\le {\displaystyle \frac{1}{2}}$,

$$\begin{array}{cc}\hfill & {\varphi }^{″}\left(x\left(a_3\right)\right){\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{k}_1\left(s\right)ds\hfill \\ \hfill \le & {\varphi }^{″}(1-x\left(a_3\right)){\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{k}_1\left(s\right)ds\hfill \\ \hfill =& {\varphi }^{″}(1-x\left(a_3\right)){\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{k}_2(1-s)ds\hfill \\ \hfill =& {\varphi }^{″}(1-x\left(a_3\right)){\int }_{1-x\left(a_3\right)}^{1-x\left(a_0\right)}{k}_2\left(s\right)ds.\hfill \end{array}$$

Combining (6), we obtain the desired inequality. □

3. Three Interval Inequality for ($\mathit{n}$ + 2)-Convex Functions

In this section, we consider inequalities on three intervals for general higher-order convex functions.

First we utilize Theorem 1 to prove the inequality of Theorem 2 type.

Theorem 6. 

Let two functions  $g:[\alpha ,\beta ]\to [a,b]$,  $p:[\alpha ,\beta ]\to \mathbb{R}$ be integrable with

$$\begin{array}{c}\hfill {\int }_{\alpha }^{\beta }p\left(x\right)dx=0,{\int }_{\alpha }^{\beta }p\left(x\right)g\left(x\right)dx=0\end{array}$$

(7)

with  $g\left(x\right)$ non-decreasing and

$$\begin{array}{c}\hfill p\left(x\right)\left\{\begin{array}{cc}\ge 0,\hfill & \alpha \le x\le a_1,\hfill \\ \le 0,\hfill & a_1\le x\le a_2,\hfill \\ \ge 0,\hfill & a_2\le x\le \beta .\hfill \end{array}\right.\end{array}$$

For $n\ge 3$, let  $f:I\to \mathbb{R},[a,b]\subset I$ be an n-convex function. Then we get the following inequality

$$\begin{array}{cc}\hfill & {\int }_{\alpha }^{\beta }p\left(x\right)f\left(g\left(x\right)\right)dx\ge \sum _{k=0}^{n-3}{\displaystyle \frac{f^{(k+2)}\left(a\right)}{k!}}{\int }_a^b\left({\int }_{\alpha }^{\beta }p\left(x\right)G(g\left(x\right),t)dx\right){(t-a)}^{k}dt,\hfill \end{array}$$

in which we further have

$$\begin{array}{c}\hfill {\int }_{\alpha }^{\beta }p\left(x\right)G(g\left(x\right),t)dx\ge 0.\end{array}$$

Proof. 

First, we can regard $[\alpha ,\beta ]$ as the whole three interval $[a_0,a_3]$ in Theorem 1 and $p\left(x\right)$ as $p,-q,p$. As G is convex, we can apply Theorem 1 to

$$\begin{array}{c}\hfill {\int }_{\alpha }^{\beta }p\left(x\right)G(g\left(x\right),t)dx\end{array}$$

in the last term in Theorem 2; with $f^{\left(n\right)}$ and ${(t-s)}^{n-3}$ also being non-negative, we get the inequality from Theorem 2. And

$$\begin{array}{c}\hfill {\int }_{\alpha }^{\beta }p\left(x\right)G(g\left(x\right),t)dx\ge 0\end{array}$$

also holds in the first $(n-2)$ terms. □

Then we utilize Theorem 1 to prove the inequality similar to Theorem 3. Note that in [3], the same inequality also holds under several different assumptions; they are all deduced from Theorem 3.

Theorem 7. 

Let  $n\ge 3$  be an even number and  $f:I\to \mathbb{R}$  be an n-convex function, where  $I\subset \mathbb{R}$  is an open interval with $a,b\in I,a<b$. Further, let two functions  $g:[\alpha ,\beta ]\to [a,b]$,  $p:[\alpha ,\beta ]\to \mathbb{R}$  be such that

$$\begin{array}{c}\hfill {\int }_{\alpha }^{\beta }p\left(x\right)dx=0,{\int }_{\alpha }^{\beta }p\left(x\right)g\left(x\right)dx=0.\end{array}$$

(8)

with  $g\left(x\right)$  non-decreasing and

$$\begin{array}{c}\hfill p\left(x\right)\left\{\begin{array}{cc}\ge 0,\hfill & \alpha \le x\le a_1,\hfill \\ \le 0,\hfill & a_1\le x\le a_2,\hfill \\ \ge 0,\hfill & a_2\le x\le \beta .\hfill \end{array}\right.\end{array}$$

Then we get

$$\begin{array}{cc}\hfill & {\int }_{\alpha }^{\beta }p\left(x\right)f\left(g\left(x\right)\right)dx\ge {\displaystyle \frac{f^{\prime }\left(a\right)-f^{\prime }\left(b\right)}{b-a}}{\int }_a^b{\int }_{\alpha }^{\beta }p\left(x\right)G(g\left(x\right),s)dxds\hfill \\ \hfill +& \sum _{k=2}^{n-1}{\displaystyle \frac{k}{(k-1)!}}{\int }_a^b\left({\int }_{\alpha }^{\beta }p\left(x\right)G(g\left(x\right),s)dx\right){\displaystyle \frac{f^{\left(k\right)}\left(a\right){(s-a)}^{k-1}-f^{\left(k\right)}\left(b\right){(s-b)}^{k-1}}{b-a}}ds,\hfill \end{array}$$

in which we further have

$$\begin{array}{c}\hfill {\int }_{\alpha }^{\beta }p\left(x\right)G(g\left(x\right),s)dx\ge 0.\end{array}$$

Proof. 

The first few steps of the proof is similar to the above theorem. Notice, for even n, ${\tilde{T}}_{n-2}(s,t)\ge 0$ in the last term in Theorem 3. So with Theorem 1, we can use the identity in Theorem 3 to prove the inequality. □

Another type of higher-order situation can be established as the following.

Theorem 8. 

Let  $a_0<a_1<a_2<a_3$  and  $x:[a_0,a_3]\to \mathbb{R}$. Let two functions  $p:[a_0,a_1]\cup [a_2,a_3]\to {\mathbb{R}}_{+}$  and  $q:[a_1,a_2]\to {\mathbb{R}}_{+}$  be non-negative. If  $x\left(t\right)$  is an increasing function for  $t\in [a_0,a_3]$ and

$$\begin{array}{cc}\hfill & {\int }_{a_1}^{a_2}q\left(t\right)x\left(t\right)dt={\int }_{a_0}^{a_1}p\left(t\right)x\left(t\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)x\left(t\right)dt,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\int }_{a_1}^{a_2}q\left(t\right)dt={\int }_{a_0}^{a_1}p\left(t\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)dt,\hfill \end{array}$$

(10)

then for  $n\in \mathbb{N}$, inequality

$$\begin{array}{cc}\hfill \sum _{i=0}^n& {(-1)}^i\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\left[{\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)-i)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)-i)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)-i)dt\right]\hfill \\ \hfill & \ge 0\hfill \end{array}$$

holds for every $(n+2)$-convex function ϕ such that the integrals above exist.

Proof. 

For each i, we can apply Lemma 1 on ${\widehat{x}}_i\left(t\right)=x\left(t\right)-i$ to obtain (as the condition of identities is satisfied according to (9) and (10))

$$\begin{array}{cc}\hfill & {\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)-i)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)-i)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)-i)dt\hfill \\ \hfill =& {\int }_{x\left(a_0\right)-i}^{x\left(a_3\right)-i}{\varphi }^{″}\left(s\right)k_i\left(s\right)ds,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & k_i\left(s\right)=\hfill \\ \hfill {\int }_{a_0}^{a_1}& p\left(t\right){(x\left(t\right)-i-s)}_{+}dt-{\int }_{a_1}^{a_2}q\left(t\right){(x\left(t\right)-i-s)}_{+}dt+{\int }_{a_2}^{a_3}p\left(t\right){(x\left(t\right)-i-s)}_{+}dt.\hfill \end{array}$$

Then it is easy to observe that

$$\begin{array}{c}\hfill {\int }_{x\left(a_0\right)-i}^{x\left(a_3\right)-i}{\varphi }^{″}\left(s\right)k_i\left(s\right)ds={\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s-i)k_i(s-i)ds={\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s-i)k_0\left(s\right)ds;\end{array}$$

thus the inequality in the theorem is equivalent to

$$\begin{array}{cc}\hfill & \sum _{i=0}^n{(-1)}^i\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\left[{\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s-i)k_0\left(s\right)ds\right]\hfill \\ \hfill =& {\int }_{x\left(a_0\right)}^{x\left(a_3\right)}\left[\sum _{i=0}^n{(-1)}^i\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){\varphi }^{″}(s-i)\right]k_0\left(s\right)ds.\hfill \end{array}$$

As $\varphi$ is an $(n+2)$-convex function, we predict that

$$\begin{array}{c}\hfill \sum _{i=0}^n{(-1)}^i\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){\varphi }^{″}(s-i)\ge 0,\end{array}$$

with $k_0\left(s\right)$ being non-negative, the inequality is proven. □

Remark 2. 

For higher-order extensions of Theorem 1, Corollary 1, the above Theorem 8 using ${\sum }_{i=0}^n{(-1)}^i\left(\genfrac{}{}{0pt}{}{n}{i}\right)$ of a divided difference is just one case of obtaining such an inequality for general higher-order convex functions; in reality, there are some other types.

One example is Corollary 1, which is a special case, but Theorem 5 is not a special case of Theorem 8 for $n=1$.

The second example is that for $n=2$, we have the following inequality for 4-convex functions according to Theorem 8:

$$\begin{array}{cc}\hfill \sum _{i=0}^2& {(-1)}^i\left({\displaystyle \genfrac{}{}{0pt}{}{2}{i}}\right)\left[{\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)-i)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)-i)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)-i)dt\right]\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

However, the conclusion below will show a more general inequality for 4-convex functions.

Theorem 9. 

Let $a_0<a_1<a_2<a_3$  and  $x:[a_0,a_3]\to \mathbb{R}$. Let two functions  $p:[a_0,a_1]\cup [a_2,a_3]\to {\mathbb{R}}_{+}$  and  $q:[a_1,a_2]\to {\mathbb{R}}_{+}$  be non-negative. If  $x\left(t\right)$  is an increasing function for $t\in [a_0,a_3]$  and

$$\begin{array}{cc}\hfill & {\int }_{a_1}^{a_2}q\left(t\right)x\left(t\right)dt={\int }_{a_0}^{a_1}p\left(t\right)x\left(t\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)x\left(t\right)dt,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & {\int }_{a_1}^{a_2}q\left(t\right)dt={\int }_{a_0}^{a_1}p\left(t\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)dt,\hfill \end{array}$$

(12)

and the sequence $b:=(b_1,\dots ,b_m)$  majorizes  $c:=(c_1,\dots ,c_m)$$(b\succ c)$, then

$$\begin{array}{cc}\hfill & \sum _{j=1}^m\left[{\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)+b_j)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)+b_j)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)+b_j)dt\right]\hfill \\ \hfill \ge & \sum _{j=1}^m\left[{\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)+c_j)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)+c_j)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)+c_j)dt\right]\hfill \end{array}$$

holds for every 4-convex function ϕ such that the integrals above exist.

Proof. 

For each $b_j$ (or $c_j$, similarly), we can apply Lemma 1 on ${\widehat{x}}_{b,j}\left(t\right)=x\left(t\right)+b_j$ to obtain (as the condition of identities is satisfied according to (11) and (12))

$$\begin{array}{cc}\hfill & {\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)+b_j)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)+b_j)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)+b_j)dt\hfill \\ \hfill =& {\int }_{x\left(a_0\right)+b_j}^{x\left(a_3\right)+b_j}{\varphi }^{″}\left(s\right)k_{b,j}\left(s\right)ds,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & k_{b,j}\left(s\right)=\hfill \\ \hfill {\int }_{a_0}^{a_1}& p\left(t\right){(x\left(t\right)+b_j-s)}_{+}dt-{\int }_{a_1}^{a_2}q\left(t\right){(x\left(t\right)+b_j-s)}_{+}dt+{\int }_{a_2}^{a_3}p\left(t\right){(x\left(t\right)+b_j-s)}_{+}dt.\hfill \end{array}$$

Then we observe that

$$\begin{array}{cc}\hfill {\int }_{x\left(a_0\right)+b_j}^{x\left(a_3\right)+b_j}{\varphi }^{″}\left(s\right)k_{b,j}\left(s\right)ds& ={\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s+b_j)k_{b,j}(s+b_j)ds\hfill \\ \hfill & ={\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s+b_j)k_0\left(s\right)ds;\hfill \end{array}$$

thus the inequality in the theorem is equivalent to

$$\begin{array}{cc}\hfill & \sum _{j=1}^m\left[{\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s+b_j)k_0\left(s\right)ds\right]-\sum _{j=1}^m\left[{\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s+c_j)k_0\left(s\right)ds\right]\hfill \\ \hfill =& {\int }_{x\left(a_0\right)}^{x\left(a_3\right)}\left[\sum _{j=1}^m{\varphi }^{″}(s+b_j)-\sum _{j=1}^m{\varphi }^{″}(s+c_j)\right]k_0\left(s\right)ds,\hfill \end{array}$$

and as $\varphi$ is 4-convex and $b\succ c$ (so $(b+s)\succ (c+s)$), we affirm that

$$\sum _{j=1}^m{\varphi }^{″}(s+b_j)-\sum _{j=1}^m{\varphi }^{″}(s+c_j)\ge 0;$$

meanwhile, considering $k_0\left(s\right)\ge 0$, the inequality is proven. □

If we want to establish a similar inequality involving $2n$ nodes like Theorem 9 for a general $(n+2)$-convex function, the following essential conclusion [3,13] (p. 57) is needed.

Lemma 2. 

Given real numbers $b_1,\dots ,b_n\in I$  and  $c_1,\dots ,c_n\in I$ with

$$\sum _{i=1}^n{b}_i^j=\sum _{i=1}^n{c}_i^j,j=1,\dots ,n-1,$$

the statements below are equivalent:

$$\begin{array}{cc}\hfill \left(i\right)& \sum _{i=1}^n{b}_i^n\ge \sum _{i=1}^n{c}_i^n;\hfill \\ \hfill \left(ii\right)& max\{b_i:i=1,\dots ,n\}\ge max\{c_i:i=1,\dots ,n\};\hfill \\ \hfill \left(iii\right)& {(-1)}^n\prod _{i=1}^n{b}_i\le {(-1)}^n\prod _{i=1}^n{c}_i;\hfill \\ \hfill \left(iv\right)& \sum _{i=1}^{n}f\left(b_i\right)\ge \sum _{i=1}^{n}f\left(c_i\right)forallfunctionsf:I\to \mathbb{R}provided{f}^{\left(n\right)}\ge 0.\hfill \end{array}$$

Theorem 10. 

Let $a_0<a_1<a_2<a_3$  and  $x:[a_0,a_3]\to \mathbb{R}$. Let two functions  $p:[a_0,a_1]\cup [a_2,a_3]\to {\mathbb{R}}_{+}$  and  $q:[a_1,a_2]\to {\mathbb{R}}_{+}$  be non-negative. Consider the case where  $x\left(t\right)$  is an increasing function for  $t\in [a_0,a_3]$;  $b_1,\dots ,b_n$  and  $c_1,\dots ,c_n$  are real numbers;

$$\begin{array}{cc}\hfill & {\int }_{a_1}^{a_2}q\left(t\right)x\left(t\right)dt={\int }_{a_0}^{a_1}p\left(t\right)x\left(t\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)x\left(t\right)dt,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & {\int }_{a_1}^{a_2}q\left(t\right)dt={\int }_{a_0}^{a_1}p\left(t\right)dt+{\int }_{a_2}^{a_3}p\left(t\right)dt,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & \sum _{i=1}^n{b}_i^j=\sum _{i=1}^n{c}_i^j,j=1,\dots ,n-1;\hfill \end{array}$$

(15)

and one of the statements below is satisfied:

$$\begin{array}{cc}\hfill \left(i\right)& \sum _{i=1}^n{b}_i^n\ge \sum _{i=1}^n{c}_i^n;\hfill \\ \hfill \left(ii\right)& max\{b_i:i=1,\dots ,n\}\ge max\{c_i:i=1,\dots ,n\};\hfill \\ \hfill \left(iii\right)& {(-1)}^n\prod _{i=1}^n{b}_i\le {(-1)}^n\prod _{i=1}^n{c}_i.\hfill \end{array}$$

Then

$$\begin{array}{cc}\hfill & \sum _{i=1}^n\left[{\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)+b_i)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)+b_i)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)+b_i)dt\right]\hfill \\ \hfill \ge & \sum _{i=1}^n\left[{\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)+c_i)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)+c_i)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)+c_i)dt\right]\hfill \end{array}$$

holds for every $(n+2)$-convex function ϕ such that the integrals above exist.

Proof. 

For each $b_i$ (or $c_i$, similarly), we can apply Lemma 1 on ${\widehat{x}}_{b,i}\left(t\right)=x\left(t\right)+b_i$ to obtain (as the condition of identities is satisfied according to (13) and (14))

$$\begin{array}{cc}\hfill & {\int }_{a_0}^{a_1}p\left(t\right)\varphi (x\left(t\right)+b_i)dt-{\int }_{a_1}^{a_2}q\left(t\right)\varphi (x\left(t\right)+b_i)dt+{\int }_{a_2}^{a_3}p\left(t\right)\varphi (x\left(t\right)+b_i)dt\hfill \\ \hfill =& {\int }_{x\left(a_0\right)+b_i}^{x\left(a_3\right)+b_i}{\varphi }^{″}\left(s\right)k_{b,i}\left(s\right)ds,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & k_{b,i}\left(s\right)=\hfill \\ \hfill {\int }_{a_0}^{a_1}& p\left(t\right){(x\left(t\right)+b_i-s)}_{+}dt-{\int }_{a_1}^{a_2}q\left(t\right){(x\left(t\right)+b_i-s)}_{+}dt+{\int }_{a_2}^{a_3}p\left(t\right){(x\left(t\right)+b_i-s)}_{+}dt.\hfill \end{array}$$

Then it is easy to observe that

$$\begin{array}{cc}\hfill {\int }_{x\left(a_0\right)+b_i}^{x\left(a_3\right)+b_i}{\varphi }^{″}\left(s\right)k_{b,i}\left(s\right)ds& ={\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s+b_i)k_{b,i}(s+b_i)ds\hfill \\ \hfill & ={\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s+b_i)k_0\left(s\right)ds;\hfill \end{array}$$

thus the inequality in the theorem is equivalent to

$$\begin{array}{cc}\hfill & \sum _{i=1}^n\left[{\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s+b_i)k_0\left(s\right)ds\right]-\sum _{i=1}^n\left[{\int }_{x\left(a_0\right)}^{x\left(a_3\right)}{\varphi }^{″}(s+c_i)k_0\left(s\right)ds\right]\hfill \\ \hfill =& {\int }_{x\left(a_0\right)}^{x\left(a_3\right)}\left[\sum _{i=1}^n{\varphi }^{″}(s+b_i)-\sum _{i=1}^n{\varphi }^{″}(s+c_i)\right]k_0\left(s\right)ds.\hfill \end{array}$$

As $\varphi$ is $(n+2)$-convex (${\varphi }^{″}$ is n-convex) and (due to (15))

$$\sum _{i=1}^n{(s+b_i)}^j=\sum _{i=1}^n{(s+c_i)}^j,j=1,\dots ,n-1,$$

we can use Lemma 2 to get

$$\sum _{i=1}^n{\varphi }^{″}(s+b_i)-\sum _{i=1}^n{\varphi }^{″}(s+c_i)\ge 0;$$

meanwhile, considering $k_0\left(s\right)\ge 0$, the inequality is proven. □

4. Application to Csiszár $\varphi$-Divergence

Recall some concepts of Csiszár $\varphi$-divergence [14,15,16,17,18,19,20,21,22,23,24,25]. For a convex function $\varphi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, the $\varphi$-divergence functional

$$I_{\varphi }(p,q):=\sum _{i=1}^n{q}_i\varphi \left({\displaystyle \frac{p_i}{q_i}}\right)$$

is a generalized definition for a measure of information, a “distance function” defined on the set of the probability distributions ${\mathbb{P}}_n$. By selecting this convex function $\varphi$ to be some special function, various divergences can be derived; refer to Chapter 1 in [15] and Chapter 9.2 in [16] as well as other mentioned references.

In [12], we compared two different pairs of $I_{\varphi }(p_1,q_1)$ and $I_{\varphi }(p_2,q_2)$

$$I_{\varphi }(p_1,q_1)\le I_{\varphi }(p_2,q_2)$$

for convex function $\varphi$, if

$$\begin{array}{c}\hfill {\displaystyle \frac{p_{1,i}}{q_{1,i}}}\in [m,M],i=1,\dots ,n;\\ \hfill {\displaystyle \frac{p_{2,i}}{q_{2,i}}}\in [0,m]\cup [M,\infty ),i=1,\dots ,n\end{array}$$

(16)

for some $m\le 1\le M$. In this section, we will further estimate the error between these two divergence functionals for 3-convex $\varphi$. The following discrete case of Corollary 1 is needed.

Corollary 4. 

Let $1\le r<s<m$  and consider the m-tuple x with  $x_1\le \dots \le x_m$. Let two sequences  $p_i,(i=1,\dots ,r,s+1,\dots ,m)$  and  $q_i,(i=r+1,\dots ,s)$ be non-negative. If

$$\begin{array}{cc}\hfill & \sum _{i=r+1}^s{q}_i{x}_i=\sum _{i=1}^r{p}_i{x}_i+\sum _{i=s+1}^m{p}_i{x}_i,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & \sum _{i=r+1}^s{q}_i=\sum _{i=1}^r{p}_i+\sum _{i=s+1}^m{p}_i,\hfill \end{array}$$

(18)

then for $\epsilon >0$, inequality

$$\begin{array}{cc}\hfill & \sum _{i=1}^r{p}_i\varphi \left(x_i\right)-\sum _{i=r+1}^s{q}_i\varphi \left(x_i\right)+\sum _{i=s+1}^m{p}_i\varphi \left(x_i\right)\hfill \\ \hfill \le & \sum _{i=1}^r{p}_i\varphi (x_i+\epsilon )-\sum _{i=r+1}^s{q}_i\varphi (x_i+\epsilon )+\sum _{i=s+1}^m{p}_i\varphi (x_i+\epsilon )\hfill \end{array}$$

(19)

holds for every 3-convex function ϕ.

Theorem 11. 

Let $\varphi :[0,\infty )\to \mathbb{R}$  be 3-convex. If  $p_1,q_1,p_2,q_2\in {\mathbb{P}}_n$  and (16) is satisfied, then for $\epsilon >0$, we have

$$\begin{array}{cc}\hfill & \sum _{i=1}^n{q}_{2,i}\varphi \left({\displaystyle \frac{p_{2,i}}{q_{2,i}}}-\epsilon \right)-\sum _{i=1}^n{q}_{1,i}\varphi \left({\displaystyle \frac{p_{1,i}}{q_{1,i}}}-\epsilon \right)\hfill \\ \hfill \le & I_{\varphi }(p_2,q_2)-I_{\varphi }(p_1,q_1)\hfill \\ \hfill \le & \sum _{i=1}^n{q}_{2,i}\varphi \left({\displaystyle \frac{p_{2,i}}{q_{2,i}}}+\epsilon \right)-\sum _{i=1}^n{q}_{1,i}\varphi \left({\displaystyle \frac{p_{1,i}}{q_{1,i}}}+\epsilon \right),\hfill \end{array}$$

(20)

if all $({\displaystyle \frac{p_{j,i}}{q_{j,i}}}-\epsilon )$ are in the domain.

Proof. 

Due to (16), we may set $x_1\le \dots \le x_m$ in Corollary 4 as ${\displaystyle \frac{p_{2,\left[1\right]}}{q_{2,\left[1\right]}}}\le \dots \le {\displaystyle \frac{p_{1,\left[1\right]}}{q_{1,\left[1\right]}}}\le \dots \le {\displaystyle \frac{p_{1,\left[n\right]}}{q_{1,\left[n\right]}}}\le \dots \le {\displaystyle \frac{p_{2,\left[n\right]}}{q_{2,\left[n\right]}}}$, where ${\displaystyle \frac{p_{2,\left[i\right]}}{q_{2,\left[i\right]}}},{\displaystyle \frac{p_{1,\left[i\right]}}{q_{1,\left[i\right]}}}$ are an increasing rearrangement of ${\displaystyle \frac{p_{2,i}}{q_{2,i}}},{\displaystyle \frac{p_{1,i}}{q_{1,i}}}$.

Furthermore, set $q_i$ in Corollary 4 as $q_{1,i}$, set $p_i$ in Corollary 4 as $q_{2,i}$ and set $s-r=n$ and $m=2n$. As $p_1,q_1,p_2,q_2\in {\mathbb{P}}_n$, conditions (17) and (18) are naturally satisfied; using (19) we obtain (20). □

Then we illustrate how to apply this theorem to some special examples, where $\varphi$ is 3-convex or 3-concave.

Definition 2. 

For

$$\varphi \left(t\right)=tlnt,t>0$$

the ϕ-divergence is

$$I_{\varphi }(p,q):=\sum _{i=1}^n{p}_{i}ln\left({\displaystyle \frac{p_i}{q_i}}\right),$$

the Kullback–Leibler distance.

Notice that $\varphi \left(t\right)=tlnt$ is 3-concave.

Definition 3. 

Let

$$\varphi \left(t\right)={\displaystyle \frac{1}{2}}{(1-\sqrt{t})}^2,t>0$$

The corresponding ϕ-divergence is

$$I_{\varphi }(p,q):={\displaystyle \frac{1}{2}}\sum _{i=1}^n{(\sqrt{q_i}-\sqrt{p_i})}^2,$$

the Hellinger distance.

Notice that $\varphi \left(t\right)={\displaystyle \frac{1}{2}}{(1-\sqrt{t})}^2$ is 3-concave.

Definition 4. 

For $\alpha >1$, let

$$\varphi \left(t\right)=t^{\alpha },t>0$$

The ϕ-divergence is

$$I_{\varphi }(p,q):=\sum _{i=1}^n{p}_i^{\alpha }{q}_i^{1-\alpha },$$

the α-order entropy. Furthermore, the Rényi divergence of order α is defined by

$$D_{\alpha }(p,q):={\displaystyle \frac{1}{\alpha -1}}ln\left(\sum _{i=1}^n{p}_i^{\alpha }{q}_i^{1-\alpha }\right).$$

Notice that $\varphi \left(t\right)=t^{\alpha }$ is 3-convex, if $\alpha (\alpha -1)(\alpha -2)>0$.

Thus, apart from the previous perspective of convexity (2-convexity) on these distance functions in the References, our 3-convexity perspective may give further estimations.

Another application of Corollary 4 is the comparison between two investment strategies over the same period of years (${\sum }_{i=r+1}^s{q}_i$ and ${\sum }_{i=1}^r{p}_i+{\sum }_{i=s+1}^m{p}_i$) with the same weighted arithmetic average annual yields. First recall the original 2-convex version of Corollary 4 under similar assumptions.

$$\begin{array}{c}\hfill \sum _{i=1}^r{p}_i\varphi \left(x_i\right)-\sum _{i=r+1}^s{q}_i\varphi \left(x_i\right)+\sum _{i=s+1}^m{p}_i\varphi \left(x_i\right)\ge 0\end{array}$$

holds for every 2-convex function $\varphi$. Taking $\varphi \left(x\right)=-lnx(x_i>0)$, we have

$$\begin{array}{c}\hfill \prod _{i=1}^r{x}_i^{p_i}\prod _{i=s+1}^m{x}_i^{p_i}\le \prod _{i=r+1}^s{x}_i^{q_i},\end{array}$$

(21)

in which the annual yield among each $p_i$ (or $q_i$) period of years is $(x_i-1)·100\%$. This inequality means that the overall income for a more stable investment strategy ${\prod }_{i=r+1}^s{x}_i^{q_i}$ is better than that for the strategy with more fluctuation (sometimes large $x_i$, sometimes small).

Now let us turn to Corollary 4 itself. If inequality (21) above is only a common sense investment, then the following inequality would be more interesting. Taking 3-convex function $\varphi \left(x\right)=lnx(x_i>0)$, we have

$$\begin{array}{c}\hfill \prod _{i=1}^r{x}_i^{p_i}\prod _{i=s+1}^m{x}_i^{p_i}÷\prod _{i=r+1}^s{x}_i^{q_i}\le \prod _{i=1}^r{(x_i+\epsilon )}^{p_i}\prod _{i=s+1}^m{(x_i+\epsilon )}^{p_i}÷\prod _{i=r+1}^s{(x_i+\epsilon )}^{q_i}.\end{array}$$

(22)

Indeed, no matter whether the annual yield is $(x_i-1)·100\%$ or $(x_i+\epsilon -1)·100\%$, the basic rule (21) will not change, which means the following:

$$\begin{array}{c}\hfill \prod _{i=1}^r{x}_i^{p_i}\prod _{i=s+1}^m{x}_i^{p_i}\le \prod _{i=r+1}^s{x}_i^{q_i},\\ \hfill \prod _{i=1}^r{(x_i+\epsilon )}^{p_i}\prod _{i=s+1}^m{(x_i+\epsilon )}^{p_i}\le \prod _{i=r+1}^s{(x_i+\epsilon )}^{q_i}.\end{array}$$

But (22) reveals that when the annual yields are all larger ($(x_i+\epsilon -1)·100\%$), there is less of a difference between two investment strategies than in the situation when the annual yields are all smaller ($(x_i-1)·100\%$). Especially when some $x_i$ are close to 0, a more fluctuating strategy would be much more devastating. When the market is strong, every strategy can make money, but the bad ones may only be exposed when the market turns weak.

5. Conclusions

In this paper, by utilizing some identities involving higher-order derivatives of functions, we obtain several inequalities for 3-convex functions as well as general $(n+2)$-convex functions, under various assumptions. Some are independent new inequalities; some are extensions of 2,3-convex function cases. From the perspective of 3-convex functions, we also obtain estimates for Csiszár $\varphi$-divergence.