Generalizations of Ostrowski type inequalities via $F$-convexity

1.   Introduction

Let $f :I \rightarrow \mathbb{R}$ be continuous on $\left( a ,b \right)$, whose derivative $f ^{\prime }:(a,b)\rightarrow \mathbb{R}$ is bounded on $(a,b)$, i.e.,

Then

for all $x \in \left[ a ,b \right] .$ The inequality can also be written in another form as follow:

where $\left\vert f^{\prime }\left( t\right)\right\vert\leq M.$ The constant $\frac{1}{4}$ is the best and cannot be replaced by a smaller one.

In 1938, Ukrainian mathematician Alexander Markowich Ostrowski established the above inequality which is called Ostrowski's inequality in the literature. It has attracted the attention of many researchers since the day it was introduced and one of its strengths is that it can be used to estimate the deviation of functional value from its integral mean. We would like to bring some papers to the attention of interested readers who want more information about Ostrowski-type inequalities. In ^[1], Anastassiou presented classical inequalities of this type for outside of the convexity concept. Besides, in ^[14], the authors have proved some Ostrowski's type inequalities for differentiable mappings that do not satisfy the properties of convexity. To provide more information related to Ostrowski type inequalities for convex and different kinds of convex functions, we suggest the papers ^[4,5,6]. Besides these references, for a collection of important results dealing with Ostrowski's inequality, we refer to the paper ^[9].

Convexity is one of the most frequently used concepts to obtain new Ostrowski type inequalities. Let us remind the definition of this class that we will use in this article. A function $f :I\rightarrow \mathbb{R}$ is convex on $I$ where $I\subseteq \mathbb{R}$ is an interval if the following inequality holds:

for all $x ,y \in I$ and $t\in \left[ 0,1\right] .$ If the inequality $\left( 1.2\right)$ changes the direction then $f$ is called concave function.

The concept of convexity has been perhaps the most popular subject of the inequality theory. Undoubtedly, the fact that this concept has many interesting applications by taking an active role. In recent years, many generalizations of convexity have been established by several researchers. We will mention some of them briefly as follow. In ^[15], the authors have given some general inequalities that involve convexity. In ^[7], the readers can find some integral inequalities that obtained by using the concept of concave functions. In ^[8,10], the authors have mentioned some different kinds of convex functions and associated inequalities.

The following class of functions that Poljak introduced is providing a stronger condition than convexity ^[16].

If the inequality

is valid for all $x ,y \in I$ and $t\in \left[ 0,1\right] ,$ the function $f :I\rightarrow \mathbb{R}$ is called strongly convex with modulus $c>0.$

After the emergence of this concept, many articles have been published on this subject. For example, we can see interesting basic results for this class in ^[2,12]. In ^[13], Nikodem and Pales presented inequalities for inner product spaces. Besides, we have Ostrowski type ^[17] and majorization type ^[11] results for this function class in the literature. Since strongly convex functions are important strengthening of the classical convex functions, we can expect better estimates related to deviation of functional value from its integral mean.

Take into consideration from this thought, Set et al. gave the following Ostrowski type inequalites.

Theorem 1.1. ^[17] Let $f :I\subset \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function on $I^{\circ }$ such that $f ^{\prime }\in L\left[ a ,b \right] ,$ where $a ,b \in I,$ with $a <b .$ If $\left\vert f ^{\prime }\right\vert$ is strongly convex on $\left[ a ,b \right]$ with respect to $c>0,$ $\left\vert f ^{\prime }\right\vert \leq M$ and $M\geq \max \left\{ \frac{c\left( x -a \right) ^{2}}{6},\frac{c\left( b -x \right) ^{2} }{6}\right\}$, then

for all $x ,y \in \left[ a ,b \right]$.

Theorem 1.2. ^[17] Let $f :I\subset \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function on $I^{\circ }$ such that $f ^{\prime }\in L\left[ a ,b \right]$, where $a ,b \in I,$ with $a <b .$ If $\left\vert f ^{\prime }\right\vert ^{q}$ is strongly convex on $\left[ a ,b \right]$ with respect to $c>0,$ $\left\vert f ^{\prime }\right\vert \leq M$ $\ $and $M\geq \max \left\{ \frac{ c\left( x -a \right) ^{2}}{6},\frac{c\left( b -x \right) ^{2}}{6}\right\} ,$ then

for all $x ,y \in \left[ a ,b \right] ,$ $q>1$ and $\frac{1}{p}+\frac{1}{q} = 1$.

In ^[2], Adamek defined a generalization of strong convexity which is called $F$-convex.

Definition 1.1. ^[2] Let $F: \mathbb{R} \rightarrow \mathbb{R}$ be a fixed function. A function $f:I\rightarrow \mathbb{R}$ is called $F$-convex if

for all $x ,y\in I$ and $t\in \left[ 0,1\right] .$

Readers can find other interesting results for $F$-convex functions in ^[3]. This class of functions is also a generalization of the approximate convex and semiconvex function classes.

2.   Lemmas

To prove our main results, we use the following two lemmas introduced in ^[5,6] respectively.

Lemma 2.1. ^[6] Let $f :I\subset \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function on $I^{\circ }$, where $a ,b \in I$, with $a <b$. If $f ^{\prime }\in L\left[ a ,b \right]$, then

for each $x \in \left[ a ,b \right]$.

Lemma 2.2. ^[5] Let $f :I\subset \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function on $I^{\circ }$, where $a ,b \in I,$ with $a <b .$ If $f ^{\prime }\in L\left[ a ,b \right] ,$ then

for each $t\in \left[ 0,1\right] ,$ where

for all $x \in \left( a ,b \right) .$

3.   Main results

In this section, we obtained new Ostrowski-type results for $F$-convex functions using Lemmas 2.1 and 2.2.

Theorem 3.1. Let $f :I\subset \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function on $I^{\circ }$, where $a ,b \in I,$ with $a <b$. If $\left\vert f ^{\prime }\right\vert$ is $F$ -convex on $\left[ a ,b \right]$ with $\left\vert f ^{\prime }\right\vert \leq M$ $\ $and $M\geq \max \left\{ \frac{F\left( x -a \right) }{6},\frac{F\left( b -x \right) }{6}\right\}$, where $F: \mathbb{R} \rightarrow \mathbb{R}$ continous on $\left[ a ,b \right] ,$ then

for all $x ,y \in \left[ a ,b \right]$ .

Proof. First, if we modify the right hand side of Lemma 2.1, we get

then we have

Since $\left\vert f^{\prime }\right\vert$ is $F$-convex on $\left[ a,b\right]$ and $\left\vert f^{\prime }\right\vert \leq M,$ we get

and

By using these results we can easily see that

which completes the proof.

Remark 3.1. Choosing $F\left( x \right) = 0$ in $\left( 3.1\right) ,$ then we have the inequality $\left( 1.1\right) .$

Remark 3.2. If we choose $F\left( x\right) = cx^{2}$ in $\left( 3.1\right)$, then we obtain $\left( 1.3\right) .$

Corollary 3.1. If we take $x = \frac{a+b}{2}$ in $\left( 3.1\right) ,$ then we have

Theorem 3.2. Let $f :I\subset \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function on $I^{\circ }$ such that $f ^{\prime }\in L\left[ a ,b \right]$, where $a ,b \in I,$ with $a <b .$ If $\left\vert f ^{\prime }\right\vert ^{q}$ is $F$-convex on $\left[ a ,b \right]$ with $\left\vert f ^{\prime }\right\vert \leq M$ $\ $and $M^{q}\geq \max \left\{ \frac{F\left( x -a \right) }{6},\frac{F\left( b -x \right) }{6}\right\}$, where $F: \mathbb{R} \rightarrow \mathbb{R}$ continous on $\left[ a ,b \right]$, then

for all $x ,y \in \left[ a ,b \right] ,$ $q>1$ and $\frac{1}{p}+\frac{1}{q} = 1$.

Proof. Using $\left( 3.2\right)$ and the well-known Hölder inequality for $q>1,$ we can write

Since $\left\vert f^{\prime }\right\vert ^{q}$ is $F$-convex on $\left[ a,b\right]$ and $\left\vert f^{\prime }\right\vert \leq M,$ we get

and

Also we have

Combining these results, the proof is completed.

Remark 3.3. Choosing $F\left( x \right) = cx ^{2}$ in $\left( 3.3\right)$ we obtain $\left( 1.4\right)$.

Corollary 3.2. Choosing $F\left( x \right) = 0$ in $\left( 3.3\right) ,$ then we have

Corollary 3.3. If we take $x = \frac{a+b}{2}$ in $\left( 3.3\right) ,$ then we have

Theorem 3.3. Let $f :I\subset \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function on $I^{\circ }$ such that $f ^{\prime }\in L\left[ a ,b \right]$, where $a ,b \in I,$ with $a <b .$ If $\left\vert f ^{\prime }\right\vert ^{q}$ is $F$-convex on $\left[ a ,b \right]$ with $\left\vert f ^{\prime }\right\vert \leq M$ $\ $and $M^{q}\geq \max \left\{ \frac{F\left( x -a \right) }{6},\frac{F\left( b -x \right) }{6}\right\}$, where $F: \mathbb{R} \rightarrow \mathbb{R}$ continous on $\left[ a ,b \right]$, then

for all $x ,y \in \left[ a ,b \right] ,$ $q>1$ and $\frac{1}{p}+\frac{1}{q} = 1$.

Proof. Using $\left( 3.2\right)$ and applying Young's inequality, we have

Since $\left\vert f ^{\prime }\right\vert ^{q}$ is $F$-convex on $\left[ a ,b \right]$ and $\left\vert f ^{\prime }\right\vert \leq M,$ we know from the proof of the last theorem that

and

If we write the last three results in $\left( 3.5\right)$, then we have

so that the desired inequality is achieved.

Corollary 3.4. Choosing $F\left( x \right) = 0$ in $\left( 3.4\right) ,$ then we have

Corollary 3.5. If we take $x = \frac{a+b}{2}$ in $\left( 3.4\right) ,$ then we have

Corollary 3.6. Choosing $F\left( x \right) = cx ^{2}$ in $\left( 3.4\right)$, then the following Ostrowski type inequality for strongly convex functions holds:

Theorem 3.4. Let $f :I\subset \left[ 0,\infty \right) \rightarrow \mathbb{R}$ be a differentiable function on $I^{\circ }$ such that $f ^{\prime }\in L\left[ a ,b \right] ,$ where $a ,b \in I,$ with $a <b .$ If $\left\vert f ^{\prime }\right\vert ^{q}$ is $F$-convex on $\left[ a ,b \right]$ with $\left\vert f ^{\prime }\right\vert \leq M$ and $M^{q}\geq \frac{1}{6}F\left( b -a \right)$, where $F: \mathbb{R} \rightarrow \mathbb{R}$ continous on $\left[ a ,b \right] ,$ then

for all $x ,y \in \left[ a ,b \right] ,$ $q>1$ and $\frac{1}{p}+\frac{1}{q} = 1.$

Proof. From Lemma 2.2, we have

where

By use of Hölder inequality for $q>1,$ in $\left( 3.7\right)$ we have

Since $\left\vert f ^{\prime }\right\vert ^{q}$ is $F$-convex on $\left[ a ,b \right]$ and $\left\vert f ^{\prime }\right\vert \leq M,$ we get

Also we have

Combining these, the proof is completed.

Corollary 3.7. If we choose $F\left( x\right) = 0$ in $\left( 3.6\right) ,$ then we have

Corollary 3.8. If we take $x = \frac{a+b}{2}$ in $\left( 3.6\right) ,$ then we have

Corollary 3.9. If we choose $F\left( x \right) = cx ^{2}$ in $\left( 3.6\right)$, then we obtain the following inequality for strongly convex functions:

Theorem 3.5. Let $f :I\subset \left[ 0,\infty \right) \rightarrow \mathbb{R}$ be a differentiable function on $I^{\circ }$ such that $f ^{\prime }\in L\left[ a ,b \right] ,$ where $a ,b \in I,$ with $a <b .$ If $\left\vert f ^{\prime }\right\vert ^{q}$ is $F$-convex on $\left[ a ,b \right]$ with $\left\vert f ^{\prime }\right\vert \leq M$ and $\ M^{q}\geq \frac{1}{6}F\left( b -a \right)$, where $F: \mathbb{R} \rightarrow \mathbb{R}$ continous on $\left[ a ,b \right]$, then

for all $x ,y \in \left[ a ,b \right] ,$ $q>1$ and $\frac{1}{p}+\frac{1}{q} = 1.$

Proof. Using $\left( 3.7\right)$ and applying Young's inequality, we obtain

Since $\left\vert f ^{\prime }\right\vert ^{q}$ is $F$-convex on $\left[ a ,b \right]$ and $\left\vert f ^{\prime }\right\vert \leq M,$ we have from the previous results:

also

Eventually we get

Corollary 3.10. If we take $x = \frac{a+b}{2}$ in $\left( 3.8\right) ,$ then we have

Corollary 3.11. If we choose $F\left( x \right) = cx^{2}$ in $\left( 3.8 \right)$, then we have the following inequality for strongly convex functions:

4.   Conclusions

In studies on Ostrowski type inequalities, the main purpose is to try to obtain the best possible upper limits. For this purpose, in this article, some ostrowski type inequalities were obtained with the help of a new generalization of strongly convex functions, which is an important strengthening of convexity. Some of the results obtained are generalizations of the existing inequalities in the literature, while others are the most general Ostrowski type inequalities obtained for the strongly convex functions class.

Acknowledgements

The third and the fourth authors would like to thank Prince Sultan University for paying the APC and for the support through the TAS research lab.

Conflict of interest

The authors declare that they have no competing interests.