Petrović’s inequality on coordinates and related results

Abstract

In this paper, the authors extend Petrović’s inequality to coordinates in the plane. The authors consider functionals due to Petrović’s inequality in plane and discuss its properties for certain class of coordinated log-convex functions. Also, the authors proved related mean value theorems.

Keywords:

• Petrović’s inequality
• log-convexity
• convex functions on coordinates

AMS Subject Classifications:

• Primary 26A51
• Secondary 26D15

Public Interest Statement

A real-valued function defined on an interval is called convex if the line segment between any two points on the graph of the function lies above or on the graph. Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. One of the important subclass of convex functions is log-convex functions. Apparently, it would seem that log-convex functions would be unremarkable because they are simply related to convex functions. But they have some surprising properties. Recently, the concept of convex functions has been generalized by many mathematicians and different functions related or close to convex functions are defined. In this work, the variant of Petrovic’s inequality for convex functions on coordinates is given. Few generalization of the results related to it are given.

1. Introduction

A function $f:[a,b]\to \mathbb{R}$ is called mid-convex or convex in Jensen sense if for all $x,y\in [a,b]$, the inequality$$f\left(\frac{x+y}{2}\right)\le \frac{f(x)+f(y)}{2}$$

is valid.

In 1905, J. Jensen was the first to define convex functions using above inequality (see, Jensen, 1905) and draw attention to their importance.

Definition 1

A function $f:[a,b]\to \mathbb{R}$ is said to be convex if(1.1) $$\begin{array}{c}\hfill f(tx+(1-t)y)\le tf(x)+(1-t)f(y)\end{array}$$(1.1)

holds, for all $x,y\in [a,b]$ and $t\in [0,1]$. A function f is said to be strictly convex if strict inequality holds in (1.1).

A mapping $f:\mathrm{\Delta }\to \mathbb{R}$ is said to be convex in $\mathrm{\Delta }$ if$$\begin{array}{c}\hfill f(tx+(1-t)z,ty+(1-t)w)\le tf(x,y)+(1-t)f(z,w)\end{array}$$

for all $(x,y),(z,w)\in \mathrm{\Delta }$, where $\mathrm{\Delta }:=[a,b]\times [c,d]\subset {\mathbb{R}}^2$ and $t\in$ [0,1].

In Dragomir (2001) gave the definition of convex functions on coordinates as follows.

Definition 2

Let $\mathrm{\Delta }=[a,b]\times [c,d]\subseteq {\mathbb{R}}^2$ and $f:\mathrm{\Delta }\to \mathbb{R}$ be a mapping. Define partial mappings(1.2) $$\begin{array}{c}\hfill f_y:[a,b]\to \mathbb{R}\text{by}{f}_y(u)=f(u,y)\end{array}$$(1.2)

and(1.3) $$\begin{array}{c}\hfill f_x:[c,d]\to \mathbb{R}\text{by}{f}_x(v)=f(x,v).\end{array}$$(1.3)

Then f is said to be convex on coordinates (or coordinated convex) in $\mathrm{\Delta }$ if $f_y$ and $f_x$ are convex on [a, b] and [c, d] respectively for all $x\in [a,b]$ and $y\in [c,d]$. A mapping f is said to be strictly convex on coordinates (or strictly coordinated convex) in $\mathrm{\Delta }$ if $f_y$ and $f_x$ are strictly convex on [a, b] and [c, d] respectively for all $x\in [a,b]$ and $y\in [c,d]$.

One of the important subclass of convex functions is log-convex functions. Apparently, it would seem that log-convex functions would be unremarkable because they are simply related to convex functions. But they have some surprising properties. The Laplace transform of a non-negative function is a log-convex. The product of log-convex functions is log-convex. Due to their interesting properties, the log-convex functions appear frequently in many problems of classical analysis and probability theory, e.g. see (Niculescu, 2012; Xi & Qi, 2015; Zhang & Jiang, 2012) and the references therein.

Definition 3

A function $f:I\to {\mathbb{R}}_{+}$ is called log-convex on I if$$f(\mathit{\alpha }x+\mathit{\beta }y)\le f^{\mathit{\alpha }}(x)f^{\mathit{\beta }}(y)$$

where $\mathit{\alpha }+\mathit{\beta }=1$$\mathit{\alpha },\mathit{\beta }\ge 0$ and $x,y\in I$.

Definition 4

A function $f:\mathrm{\Delta }\to {\mathbb{R}}_{+}$ is called log-convex on coordinates in $\mathrm{\Delta }$ if partial mappings defined in (1.2) and (1.3) are log-convex on [a, b] and [c, d], respectively, for all $x\in [a,b]$ and $y\in [c,d]$.

Remark 1

Every log-convex function is log convex on coordinates but the converse is not true in general. For example, $f:{[0,1]}^2\to [0,\infty )$ defined by $f(x,y)=e^{xy}$ is convex on coodinates but not convex.

In Pečarić, Proschan, and Tong (1992, p. 154), Petrović’s inequality for convex function is stated as follows.

Theorem 1

Let $[0,a)\subset \mathbb{R}$, $(x_1,\dots ,x_n)\in {(0,a]}^n$ and $(p_1,\dots ,p_n)$ be nonnegative n-tuples such that$$\begin{array}{c}\hfill \sum _{i=1}^n{p}_i{x}_i⩾x_j\text{for}j=1,2,3,\dots ,n\text{and}\sum _{i=1}^n{p}_i{x}_i\in [0,a).\end{array}$$

If f is a convex function on [0, a), then the inequality(1.4) $$\begin{array}{c}\hfill \sum _{i=1}^n{p}_{i}f(x_i)⩽f\left(\sum _{i=1}^n{p}_i{x}_i\right)+\left(\sum _{i=1}^n{p}_i-1\right)f(0)\end{array}$$(1.4)

is valid.

Remark 2

If f is strictly convex, then strict inequality holds in (1.4) unless $x_1=\cdots =x_n$ and ${\sum }_{i=1}^n{p}_i=1$.

Remark 3

For $p_i=1$$(i=1,\dots ,n)$, the above inequality becomes(1.5) $$\begin{array}{c}\hfill \sum _{i=1}^{n}f(x_i)⩽f\left(\sum _{i=1}^n{x}_i\right)+\left(n-1\right)f(0).\end{array}$$(1.5)

This was proved by Petrović in 1932 (see Petrović, 1932).

In this paper, we extend Petrović’s inequality to coordinates in the plane. We consider functionals due to Petrović’s inequality in plane and discuss its properties for certain class of coordinated log-convex functions. Also we proved related mean value theorems.

2. Main results

In the following theorem, we give our first result that is Petrović’s inequality for coordinated convex functions.

Theorem 2

Let $\mathrm{\Delta }=[0,a)\times [0,b)\subset {\mathbb{R}}^2$, $(x_1,\dots ,x_n)\in {(0,a]}^n$, $(y_1,\dots ,y_n)\in {(0,b]}^n$, $(p_1,\dots ,p_n)$ and $(q_1,\dots ,q_n)$ be non-negative n-tuples such that ${\sum }_{i=1}^n{p}_i{x}_i\ge x_j$ and ${\sum }_{i=1}^n{q}_i{y}_i\ge y_j$ for $j=1,\dots ,n$. Also let ${\sum }_{i=1}^n{p}_i{x}_i\in [0,a),{\sum }_{i=1}^n{p}_i\ge 1$ and ${\sum }_{i=1}^n{q}_i{y}_i\in [0,b)$. If $f:\mathrm{\Delta }\to \mathbb{R}$ is coordinated convex function, then(2.1) $$\begin{array}{cc}\hfill \sum _{i,j=1}^n{p}_i{q}_{j}f(x_i,y_j)⩽& f\left(\sum _{i=1}^n{p}_i{x}_i,\sum _{j=1}^n{q}_j{y}_j\right)+\left(\sum _{j=1}^n{q}_j-1\right)f\left(\sum _{i=1}^n{p}_i{x}_i,0\right)\hfill \\ \hfill +& \left(\sum _{i=1}^n{p}_i-1\right)\left[f\left(0,\sum _{j=1}^n{q}_j{y}_j\right)+\left(\sum _{j=1}^n{q}_j-1\right)f(0,0)\right].\hfill \end{array}$$(2.1)

Proof

Let $f_x:[0,b)\to \mathbb{R}$ and $f_y:[0,a)\to \mathbb{R}$ be mappings such that $f_x(v)=f(x,v)$ and $f_y(u)=f(u,y).$ Since f is coordinated convex on $\mathrm{\Delta }$, therefore $f_y$ is convex on [0, a). By Theorem 1, one has$$\begin{array}{c}\hfill \sum _{i=1}^n{p}_i{f}_y(x_i)⩽f_y\left(\sum _{i=1}^n{p}_i{x}_i\right)+\left(\sum _{i=1}^n{p}_i-1\right)f_y(0).\end{array}$$

By setting $y=y_j$, we have$$\begin{array}{c}\hfill \sum _{i=1}^n{p}_{i}f(x_i,y_j)⩽f\left(\sum _{i=1}^n{p}_i{x}_i,y_j\right)+\left(\sum _{i=1}^n{p}_i-1\right)f(0,y_j),\end{array}$$

this gives(2.2) $$\begin{array}{c}\hfill \sum _{i=1}^n\sum _{j=1}^n{p}_i{q}_{j}f(x_i,y_j)⩽\sum _{j=1}^n{q}_{j}f\left(\sum _{i=1}^n{p}_i{x}_i,y_j\right)+\left(\sum _{i=1}^n{p}_i-1\right)\sum _{j=1}^n{q}_{j}f(0,y_j).\end{array}$$(2.2)

Again, using Theorem 1 on terms of right-hand side for second coordinates, we have$$\begin{array}{c}\hfill \sum _{j=1}^n{q}_{j}f\left(\sum _{i=1}^n{p}_i{x}_i,y_j\right)⩽f\left(\sum _{i=1}^n{p}_i{x}_i,\sum _{j=1}^n{q}_j{y}_j\right)+\left(\sum _{j=1}^n{q}_j-1\right)f\left(\sum _{i=1}^n{p}_i{x}_i,0\right)\end{array}$$

and$$\begin{array}{c}\hfill \left(\sum _{i=1}^n{p}_i-1\right)\sum _{j=1}^n{q}_{j}f(0,y_j)⩽\left(\sum _{i=1}^n{p}_i-1\right)\left[f\left(0,\sum _{j=1}^n{q}_j{y}_j\right)+\left(\sum _{j=1}^n{q}_j-1\right)f(0,0)\right].\end{array}$$

Using above inequities in (2.2), we get inequality (2.1). $\square$

Remark 4

If f is strictly coordinated convex, then above inequality is strict unless all $x_i$’s and $y_i$’s are not equal or ${\sum }_{i=1}^n{p}_i\ne 1$ and ${\sum }_{j=1}^n{q}_j\ne 1$.

Remark 5

If we take $y_i=0$ and $q_j=1$, (i,j=1,...,n) with $f(x_i,0)\mapsto f(x_i)$, then we get inequality (1.4).

Let $I\subseteq \mathbb{R}$ be an interval and $f:I\to \mathbb{R}$ be a function. Then for distinct points $u_i\in I,i=0,1,2$. The divided differences of first and second order are defined as follows:(2.3) $$\begin{array}{cc}\hfill [u_i,u_{i+1},f]& =\frac{f(u_{i+1})-f(u_i)}{u_{i+1}-u_i},(i=0,1)\hfill \end{array}$$(2.3) (2.4) $$\begin{array}{cc}\hfill [u_0,u_1,u_2,f]& =\frac{[u_1,u_2,f]-[u_0,u_1,f]}{u_2-u_0}\hfill \end{array}$$(2.4)

the values of the divided differences are independent of the order of the points $u_0,u_1,u_2$ and may be extended to include the cases when some or all points are equal, that is(2.5) $$\begin{array}{c}\hfill [u_0,u_0,f]=\underset{u_1\to u_0}{lim}[u_0,u_1,f]=f^{\prime }(u_0)\end{array}$$(2.5)

provided that $f^{\prime }$ exists. Now passing the limit $u_1\to u_0$ and replacing $u_2$ by u in second-order divided difference, we have(2.6) $$\begin{array}{c}\hfill [u_0,u_0,u,f]=\underset{u_1\to u_0}{lim}[u_0,u_1,u,f]=\frac{f(u)-f(u_0)-(u-u_0)f^{\prime }(u_0)}{{(u-u_0)}^2},u\ne u_0\end{array}$$(2.6)

provided that $f^{\prime }$ exists. Also, passing to the limit $u_i\to u$$(i=0,1,2)$ in second-order divided difference, we have(2.7) $$\begin{array}{c}\hfill [u,u,u,f]=\underset{u_i\to u}{lim}[u_0,u_1,u_2,f]=\frac{f^{″}(u)}{2}\end{array}$$(2.7)

provided that $f^{″}$ exists.

One can note that, if for all $u_0,u_1\in I$, $[u_0,u_1,f]\ge 0,$ then f is increasing on I and if for all $u_0,u_1,u_2\in I$, $[u_0,u_1,u_2,f]\ge 0,$ then f is convex on I.

Now we define some families of parametric functions which we use in sequal.

Let $I=[0,a)$ and $J=[0,b)$ be intervals and let for $t\in (c,d)\subseteq \mathbb{R}$, $f_t:I\times J\to \mathbb{R}$ be a mapping. Then we define functions$$f_{t,y}:I\to \mathbb{R}\text{by}{f}_{t,y}(u)=f_t(u,y)$$

and$$f_{t,x}:J\to \mathbb{R}\text{by}{f}_{t,x}(v)=f_t(x,v),$$

where $x\in I$ and $y\in J$.

Suppose $\mathcal{K}$ denotes the class of functions $f_t:I\times J\to \mathbb{R}$ for $t\in (c,d)$ such that$$t\mapsto [u_0,u_1,u_2,f_{t,y}]\forall u_0,u_1,u_2\in I$$

and$$t\mapsto [v_0,v_1,v_2,f_{t,x}]\forall v_0,v_1,v_2\in J$$

are log-convex functions in Jensen sense on (c, d) for all $x\in I$ and $y\in J$.

We define linear functional $\mathit{{\rm Y}}(f)$ as a non-negative difference of inequality (2.1)(2.8) $$\begin{array}{cc}\hfill \mathit{{\rm Y}}(f)=& f\left(\sum _{i=1}^n{p}_i{x}_i,\sum _{j=1}^n{q}_j{y}_j\right)+\left(\sum _{j=1}^n{q}_j-1\right)f\left(\sum _{i=1}^n{p}_i{x}_i,0\right)\hfill \\ \hfill +& \left(\sum _{i=1}^n{p}_i-1\right)\left[f\left(0,\sum _{j=1}^n{q}_j{y}_j\right)+\left(\sum _{j=1}^n{q}_j-1\right)f(0,0)\right]-\sum _{i,j=1}^n{p}_i{q}_{j}f(x_i,y_j).\hfill \end{array}$$(2.8)

Remark 6

Under the assumptions of Theorem 2, if f is coordinated convex in $\mathrm{\Delta }$, then $\mathit{{\rm Y}}(f)\ge 0$.

The following lemmas are given in Pečarić and Rehman (2008).

Lemma 1

Let $I\subseteq \mathbb{R}$ be an interval. A function $f:I\to (0,\infty )$ is log-convex in Jensen sense on I, that is, for each $r,t\in I$$$f(r)f(t)\ge f^2\left(\frac{t+r}{2}\right)$$

if and only if the relation$$m^{2}f(t)+2mnf\left(\frac{t+r}{2}\right)+n^{2}f(r)\ge 0$$

holds for each $m,n\in \mathbb{R}$ and $r,t\in I.$

Lemma 2

If f is convex function on interval I then for all $x_1,x_2,x_3\in I$ for which $x_1<x_2<x_3$, the following inequality is valid:$$(x_3-x_2)f(x_1)+(x_1-x_3)f(x_2)+(x_2-x_1)f(x_3)\ge 0.$$

Our next result comprises properties of functional defined in (2.8).

Theorem 3

Let the functional $\mathit{{\rm Y}}$ defined in (2.8) and $f_t\in \mathcal{K}$. Then the following are valid:

(a)	The function $t\mapsto \mathit{{\rm Y}}(f_t)$ is log-convex in Jensen sense on (c, d).
(b)	If the function $t\mapsto \mathit{{\rm Y}}(f_t)$ is continuous on (c,d), then it is log-convex on (c, d).
(c)	If $\mathit{{\rm Y}}(f_t)$ is positive, then for some $r<s<t$, where r, s, t $\in (c,d)$, one has (2.9) $$\begin{array}{c}\hfill {\left[\mathit{{\rm Y}}(f_s)\right]}^{t-r}\le {\left[\mathit{{\rm Y}}(f_r)\right]}^{t-s}{\left[\mathit{{\rm Y}}(f_t)\right]}^{s-r}.\end{array}$$(2.9)

Proof

 

(a)	Let $$h(u,v)=m^2{f}_t(u,v)+2mn{f}_{\frac{t+r}{2}}\left(u,v\right)+n^2{f}_r(u,v)$$ where $m,n\in \mathbb{R}$ and $t,r\in (c,d)$. We can consider $$h_y(u)=m^2{f}_{t,y}(u)+2mn{f}_{\frac{t+r}{2},y}(u)+n^2{f}_{r,y}(u)$$ and $$h_x(v)=m^2{f}_{t,x}(v)+2mn{f}_{\frac{t+r}{2},x}(v)+n^2{f}_{r,x}(v).$$ Now we take $$[u_0,u_1,u_2,h_y]=m^2[u_0,u_1,u_2,f_{t,y}]+2mn[u_0,u_1,u_2,f_{\frac{t+r}{2},y}]+n^2[u_0,u_1,u_2,f_{r,y}].$$ As $[u_0,u_1,u_2,f_{t,y}]$ is log convex in Jensen sense, so using Lemma 1, the right-hand side of above expression is non-negative, so $h_y$ is convex on I. Similarly, one can show that $h_x$ is also convex on J, which concludes h is coordinated convex on $\mathrm{\Delta }$. By Remark 6, $\mathit{{\rm Y}}(h)\ge 0$, that is, $$m^2\mathit{{\rm Y}}(f_t)+2mn\mathit{{\rm Y}}(f_{\frac{t+r}{2}})+n^2\mathit{{\rm Y}}(f_r)\ge 0,$$ so $t\mapsto \mathit{{\rm Y}}(f_t)$ is log-convex in Jensen sense on (c,d).
(b)	Additionally, we have $t\mapsto \mathit{{\rm Y}}(f_t)$ is continuous on (c, d), hence we have $t\mapsto \mathit{{\rm Y}}(f_t)$ is log-convex on (c, d).
(c)	Since $t\mapsto \mathit{{\rm Y}}(f_t)$ is log-convex on (c, d), therefore for $r,s,t\in (c,d)$ with $r<s<t$ and $f(t)=log\mathit{{\rm Y}}(t)$ in Lemma 2, we have $$(t-s)log\mathit{{\rm Y}}(f_r)+(r-t)log\mathit{{\rm Y}}(f_s)+(s-r)log\mathit{{\rm Y}}(f_t)\ge 0,$$ which is equivalent to (2.9).

$\square$

Example 1

Let $t\in (0,\infty )$ and ${\mathit{\phi }}_t:{[0,\infty )}^2\to \mathbb{R}$ be a function defined as(2.10) $$\begin{array}{c}\hfill {\mathit{\phi }}_t(u,v)=\left\{\begin{array}{cc}\frac{u^t{v}^t}{t(t-1)},\hfill & t\ne 1,\hfill \\ uv(logu+logv),\hfill & t=1.\hfill \end{array}\right.\end{array}$$(2.10)

Define partial mappings$${\mathit{\phi }}_{t,v}:[0,\infty )\to \mathbb{R}\text{by}{\mathit{\phi }}_{t,v}(u)={\mathit{\phi }}_t(u,v)$$

and$${\mathit{\phi }}_{t,u}:[0,\infty )\to \mathbb{R}\text{by}{\mathit{\phi }}_{t,u}(v)={\mathit{\phi }}_t(u,v).$$

As we have$$[u,u,u,{\mathit{\phi }}_{t,v}]=\frac{{\mathit{\partial }}^2{\mathit{\phi }}_{t,v}}{\mathit{\partial }{u}^2}=u^{t-2}{v}^t\ge 0\forall t\in (0,\infty ).$$

This gives $t\mapsto [u_0,u_0,u_0,{\mathit{\phi }}_{t,v}]$ is log-convex in Jensen sense. Similarly, one can deduce that $t\mapsto [v_0,v_0,v_0,{\mathit{\phi }}_{t,u}]$ is also log-convex in Jensen sense. If we choose $f_t={\mathit{\phi }}_t$ in Theorem 3, we get log convexity of the functional $\mathit{{\rm Y}}({\mathit{\phi }}_t)$.

In special case, if we choose ${\mathit{\phi }}_t(u,v)={\mathit{\phi }}_t(u,1)$, then we get (Butt, Pečarić, & Rehman, A. U. 2011, Example 3).

Example 2

Let $t\in [0,\infty )$ and ${\mathit{\delta }}_t:{[0,\infty )}^2\to \mathbb{R}$ be a function defined as(2.11) $$\begin{array}{c}\hfill {\mathit{\delta }}_t(u,v)=\left\{\begin{array}{cc}\frac{uv{e}^{uvt}}{t},\hfill & t\ne 0,\hfill \\ u^2{v}^2,\hfill & t=0.\hfill \end{array}\right.\end{array}$$(2.11)

Define partial mappings$${\mathit{\delta }}_{t,v}:[0,\infty )\to \mathbb{R}\text{by}{\mathit{\delta }}_{t,v}(u)={\mathit{\delta }}_t(u,v)$$

and$${\mathit{\delta }}_{t,u}:[0,\infty )\to \mathbb{R}\text{by}{\mathit{\delta }}_{t,u}(v)={\mathit{\delta }}_t(u,v)$$

for all $u,v\in [0,\infty ).$

As we have$$[u,u,u,{\mathit{\delta }}_{t,v}]=\frac{{\mathit{\partial }}^2{\mathit{\delta }}_{t,v}}{\mathit{\delta }{u}^2}=e^{uvt}(2v^2+u{v}^2)\ge 0\forall t\in (0,\infty ).$$

This gives $t\mapsto [u_0,u_0,u_0,{\mathit{\delta }}_{t,v}]$ is log convex in Jensen sense. Similarly, one can deduce that $t\mapsto [v_0,v_0,v_0,{\mathit{\delta }}_{t,u}]$ is also log-convex in Jensen sense. If we choose $f_t={\mathit{\delta }}_t$ in Theorem 3, we get log convexity of the functional $\mathit{{\rm Y}}({\mathit{\delta }}_t)$.

In special case, if we choose ${\mathit{\delta }}_t(u,v)={\mathit{\delta }}_t(u,1)$, then we get (Butt et al., 2011, Example 8).

Example 3

Let $t\in [0,\infty )$ and ${\mathit{\gamma }}_t:{[0,\infty )}^2\to \mathbb{R}$ be a function defined as(2.12) $$\begin{array}{c}\hfill {\mathit{\gamma }}_t(u,v)=\left\{\begin{array}{cc}\frac{e^{uvt}}{t},\hfill & t\ne 0,\hfill \\ uv,\hfill & t=0.\hfill \end{array}\right.\end{array}$$(2.12)

Define partial mappings$${\mathit{\gamma }}_{t,v}:[0,\infty )\to \mathbb{R}\text{by}{\mathit{\gamma }}_{t,v}(u)={\mathit{\gamma }}_t(u,v)$$

and$${\mathit{\gamma }}_{t,u}:[0,\infty )\to \mathbb{R}\text{by}{\mathit{\gamma }}_{t,u}(v)={\mathit{\gamma }}_t(u,v).$$

As we have$$[u,u,u,{\mathit{\gamma }}_{t,v}]=\frac{{\mathit{\partial }}^2{\mathit{\gamma }}_{t,v}}{\mathit{\partial }{u}^2}=t{v}^2{e}^{uvt}\ge 0\forall t\in (0,\infty ).$$

This gives $t\mapsto [u_0,u_0,u_0,{\mathit{\gamma }}_{t,v}]$ is log-convex in Jensen sense. Similarly one can deduce that $t\mapsto [v_0,v_0,v_0,{\mathit{\gamma }}_{t,u}]$ is also log-convex in Jensen sense. If we choose $f_t={\mathit{\gamma }}_t$ in Theorem 3, we get log convexity of the functional $\mathit{{\rm Y}}({\mathit{\gamma }}_t)$.

In special case, if we choose ${\mathit{\gamma }}_t(u,v)={\mathit{\gamma }}_t(u,1)$, then we get (Butt et al., 2011, Example 9).

3. Mean value theorems

If a function is twice differentiable on an interval I, then it is convex on I if and only if its second order derivative is non-negative. If a function $f(X):=f(x,y)$ has continuous second-order partial derivatives on interior of $\mathrm{\Delta }$, then it is convex on $\mathrm{\Delta }$ if the Hessian matrix$$\begin{array}{c}\hfill H_f(X)=\left(\begin{array}{cc}\frac{{\mathit{\partial }}^{2}f(X)}{\mathit{\partial }{x}^2}\hfill & \frac{{\mathit{\partial }}^{2}f(X)}{\mathit{\partial }y\mathit{\partial }x}\hfill \\ \frac{{\mathit{\partial }}^{2}f(X)}{\mathit{\partial }x\mathit{\partial }y}\hfill & \frac{{\mathit{\partial }}^{2}f(X)}{\mathit{\partial }{y}^2}\hfill \end{array}\right)\end{array}$$

is non-negative definite, that is, $\mathit{v}{H}_f(X){\mathit{v}}^t$ is non-negative for all real non-negative vector v.

It is easy to see that $f:\mathrm{\Delta }\to \mathbb{R}$ is coordinated convex on $\mathrm{\Delta }$ iff$$\begin{array}{c}\hfill f_x^{″}(y)=\frac{{\mathit{\partial }}^{2}f(x,y)}{\mathit{\partial }{y}^2}\text{and}{f}_y^{″}(x)=\frac{{\mathit{\partial }}^{2}f(x,y)}{\mathit{\partial }{x}^2}\end{array}$$

are non-negative for all interior points (x, y) in ${\mathrm{\Delta }}^2$.

Lemma 3

Let $f:\mathrm{\Delta }\to \mathbb{R}$ be a function such that$$\begin{array}{c}\hfill m_1\le \frac{{\mathit{\partial }}^{2}f(x,y)}{\mathit{\partial }{x}^2}\le M_1\end{array}$$

and$$\begin{array}{c}\hfill m_2\le \frac{{\mathit{\partial }}^{2}f(x,y)}{\mathit{\partial }{y}^2}\le M_2\end{array}$$

for all interior points (x, y) in ${\mathrm{\Delta }}^2$. Consider the function ${\mathit{\psi }}_1,{\mathit{\psi }}_2:\mathrm{\Delta }\to \mathbb{R}$ defined as$$\begin{array}{c}\hfill {\mathit{\psi }}_1=\frac{1}{2}max\{M_1,M_2\}(x^2+y^2)-f(x,y)\end{array}$$$$\begin{array}{c}\hfill {\mathit{\psi }}_2=f(x,y)-\frac{1}{2}min\{m_1,m_2\}(x^2+y^2)\end{array}$$

then ${\mathit{\psi }}_1,{\mathit{\psi }}_2$ are convex on coordinates in $\mathrm{\Delta }$.

Proof

Since$$\begin{array}{c}\hfill \frac{{\mathit{\partial }}^2{\mathit{\psi }}_1(x,y)}{\mathit{\partial }{x}^2}=max\{M_1,M_2\}-\frac{{\mathit{\partial }}^{2}f(x,y)}{\mathit{\partial }{x}^2}\ge 0\end{array}$$

and$$\begin{array}{c}\hfill \frac{{\mathit{\partial }}^2{\mathit{\psi }}_1(x,y)}{\mathit{\partial }{y}^2}=max\{M_1,M_2\}-\frac{{\mathit{\partial }}^{2}f(x,y)}{\mathit{\partial }{y}^2}\ge 0\end{array}$$

for all interior points (x, y) in $\mathrm{\Delta }$, ${\mathit{\psi }}_1$ is convex on coordinates in $\mathrm{\Delta }$. Similarly, one can prove that ${\mathit{\psi }}_2$ is also convex on coordinates in $\mathrm{\Delta }$.

$\square$

Theorem 4

Let $f:\mathrm{\Delta }\to \mathbb{R}$ be a mapping which has continuous partial derivatives of second order in $\mathrm{\Delta }$ and $\mathit{\phi }(x,y):=x^2+y^2$. Then, there exist $({\mathit{\beta }}_1,{\mathit{\gamma }}_1)$ and $({\mathit{\beta }}_2,{\mathit{\gamma }}_2)$ in the interior of $\mathrm{\Delta }$ such that$$\mathit{{\rm Y}}(f)=\frac{1}{2}\frac{{\mathit{\partial }}^{2}f({\mathit{\beta }}_1,{\mathit{\gamma }}_1)}{\mathit{\partial }{x}^2}\mathit{{\rm Y}}(\mathit{\phi })$$

and$$\mathit{{\rm Y}}(f)=\frac{1}{2}\frac{{\mathit{\partial }}^{2}f({\mathit{\beta }}_2,{\mathit{\gamma }}_2)}{\mathit{\partial }{y}^2}\mathit{{\rm Y}}(\mathit{\phi })$$

provided that $\mathit{{\rm Y}}(\mathit{\phi })$ is non-zero.

Proof

Since f has continuous partial derivatives of second order in $\mathrm{\Delta }$, there exist real numbers $m_1,m_2,M_1$ and $M_2$ such that$$m_1\le \frac{{\mathit{\partial }}^{2}f(x,y)}{\mathit{\partial }{x}^2}\le M_1\text{and}{m}_2\le \frac{{\mathit{\partial }}^{2}f(x,y)}{\mathit{\partial }{y}^2}\le M_2,$$

for all $(x,y)\in \mathrm{\Delta }$.

Now consider functions ${\mathit{\psi }}_1$ and ${\mathit{\psi }}_2$ defined in Lemma 3. As ${\mathit{\psi }}_1$ is convex on coordinates in $\mathrm{\Delta }$,$$\mathit{{\rm Y}}({\mathit{\psi }}_1)\ge 0,$$

that is$$\begin{array}{c}\hfill \mathit{{\rm Y}}\left(\frac{1}{2}max\{M_1,M_2\}\mathit{\phi }(x,y)-f(x,y)\right)\ge 0,\end{array}$$

this leads us to(3.1) $$\begin{array}{c}\hfill 2\mathit{{\rm Y}}(f)\le max\{M_1,M_2\}\mathit{{\rm Y}}(\mathit{\phi }).\end{array}$$(3.1)

On the other hand, for function ${\mathit{\psi }}_2$, one has(3.2) $$\begin{array}{c}\hfill min\{m_1,m_2\}\mathit{{\rm Y}}(\mathit{\phi })\le 2\mathit{{\rm Y}}(f).\end{array}$$(3.2)

As $\mathit{{\rm Y}}(\mathit{\phi })\ne 0$, combining inequalities (3.1) and (3.2), we get$$min\{m_1,m_2\}\le \frac{2\mathit{{\rm Y}}(f)}{\mathit{{\rm Y}}(\mathit{\phi })}\le max\{M_1,M_2\}.$$

Then there exist $({\mathit{\beta }}_1,{\mathit{\gamma }}_1)$ and $({\mathit{\beta }}_2,{\mathit{\gamma }}_2)$ in the interior of $\mathrm{\Delta }$ such that$$\begin{array}{c}\hfill \frac{2\mathit{{\rm Y}}(f)}{\mathit{{\rm Y}}(\mathit{\phi })}=\frac{{\mathit{\partial }}^{2}f({\mathit{\beta }}_1,{\mathit{\gamma }}_1)}{\mathit{\partial }{x}^2}\end{array}$$

and$$\begin{array}{c}\hfill \frac{2\mathit{{\rm Y}}(f)}{\mathit{{\rm Y}}(\mathit{\phi })}=\frac{{\mathit{\partial }}^{2}f({\mathit{\beta }}_2,{\mathit{\gamma }}_2)}{\mathit{\partial }{y}^2},\end{array}$$

hence the required result follows.

$\square$

Theorem 5

Let ${\mathit{\psi }}_1,{\mathit{\psi }}_2:\mathrm{\Delta }\to \mathbb{R}$ be mappings which have continuous partial derivatives of second order in $\mathrm{\Delta }$. Then there exists $({\mathit{\eta }}_1,{\mathit{\xi }}_1)$ and $({\mathit{\eta }}_2,{\mathit{\xi }}_2)$ in $\mathrm{\Delta }$ such that(3.3) $$\begin{array}{c}\hfill \frac{\mathit{{\rm Y}}({\mathit{\psi }}_1)}{\mathit{{\rm Y}}({\mathit{\psi }}_2)}=\frac{\frac{{\mathit{\partial }}^2{\mathit{\psi }}_1({\mathit{\eta }}_1,{\mathit{\xi }}_1)}{\mathit{\partial }{x}^2}}{\frac{{\mathit{\partial }}^2{\mathit{\psi }}_2({\mathit{\eta }}_1,{\mathit{\xi }}_1)}{\mathit{\partial }{x}^2}}\end{array}$$(3.3)

and

Proof

We define the mapping $P:\mathrm{\Delta }\to \mathbb{R}$ such that$$\begin{array}{c}\hfill P=k_1{\mathit{\psi }}_1-k_2{\mathit{\psi }}_2,\end{array}$$

where $k_1=\mathit{{\rm Y}}({\mathit{\psi }}_2)$ and $k_2=\mathit{{\rm Y}}({\mathit{\psi }}_1)$.

Using Theorem 4 with $f=P$, we have$$\begin{array}{c}\hfill 2\mathit{{\rm Y}}(P)=0=\left\{k_1\frac{{\mathit{\partial }}^2{\mathit{\psi }}_1}{\mathit{\partial }{x}^2}-k_2\frac{{\mathit{\partial }}^2{\mathit{\psi }}_2}{\mathit{\partial }{x}^2}\right\}\mathit{{\rm Y}}(\mathit{\phi })\end{array}$$

and$$\begin{array}{c}\hfill 2\mathit{{\rm Y}}(P)=0=\left\{k_1\frac{{\mathit{\partial }}^2{\mathit{\psi }}_1}{\mathit{\partial }{y}^2}-k_2\frac{{\mathit{\partial }}^2{\mathit{\psi }}_2}{\mathit{\partial }{y}^2}\right\}\mathit{{\rm Y}}(\mathit{\phi }).\end{array}$$

Since $\mathit{{\rm Y}}(\mathit{\phi })\ne 0$, we have$$\begin{array}{c}\hfill \frac{k_2}{k_1}=\frac{\frac{{\mathit{\partial }}^2{\mathit{\psi }}_1({\mathit{\eta }}_1,{\mathit{\xi }}_1)}{\mathit{\partial }{x}^2}}{\frac{{\mathit{\partial }}^2{\mathit{\psi }}_2({\mathit{\eta }}_1,{\mathit{\xi }}_1)}{\mathit{\partial }{x}^2}}\end{array}$$

and$$\begin{array}{c}\hfill \frac{k_2}{k_1}=\frac{\frac{{\mathit{\partial }}^2{\mathit{\psi }}_1({\mathit{\eta }}_2,{\mathit{\xi }}_2)}{\mathit{\partial }{y}^2}}{\frac{{\mathit{\partial }}^2{\mathit{\psi }}_2({\mathit{\eta }}_2,{\mathit{\xi }}_2)}{\mathit{\partial }{y}^2}},\end{array}$$

which are equivalent to required results. $\square$

Additional information

Funding

The author received no direct funding for this research.

Notes on contributors

Atiq Ur Rehman

Atiq ur Rehman and Ghulam Farid are assistant professors in the Department of Mathematics at the COMSATS Institute of Information Technology (CIIT), Attock, Pakistan. Their primary research interests include real functions, mathematical inequalities, and difference equation.

Muhammad Mudessir

Muhammad Mudessir has successfully completed his MS degree in mathematics from CIIT in this year. He is a teacher in Government Pilot Secondary School, Attock, Pakistan. His area of research includes convex analysis and inequalities in mathematics.

Hafiza Tahira Fazal

Hafiza Tahira Fazal received her master of philosophy degree from National College of Business Administration and Economics, Lahore, Pakistan. She is working as a lecturer in the Department of Mathematics at the University of Lahore, Sargodha, Pakistan from last two years. Her area of research includes inequalities in mathematics.