On Ostrowski Type Inequalities via the Extended Version of Montgomery’s Identity

Abstract

In this paper, we obtain new Ostrowski type inequalities by using the extended version of Montgomery identity and Green’s functions. We also give estimations of the difference between two integral means.

1. Introduction

A renowned integral inequality involving mapping with bounded derivative known as Ostrowski’s inequality was presented by Alexander Markovich Ostrowski in 1938 [1], which may be produced in various ways by using distinct techniques: direct calculation, Lagrange mean value theorem, Montgomery’s identity, etc., and can be stated as:

Theorem 1.

Let ζ be a real-valued continuous mapping on $[a_1,b_1]$ and differentiable on $(a_1,b_1)$ such that ${\zeta }^{\prime }$ is bounded by some real constant K. Then

$$\begin{array}{ccc}\hfill \left|{\displaystyle \zeta \left(\upsilon \right)-\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt}\right|& \le & \left[\frac{1}{4}+\frac{{\left(\upsilon -\frac{a_1+b_1}{2}\right)}^2}{{\left(b_1-a_1\right)}^2}\right]K(b_1-a_1)\hfill \\ \hfill & =& \left[\frac{{(\upsilon -a_1)}^2+{(b_1-\upsilon )}^2}{2(b_1-a_1)}\right]K.\hfill \end{array}$$

(1)

Here, the constant $\frac{1}{4}$ is the best possible in the sense that it cannot be replaced by any smaller constant. In recent versions, K is usually replaced by ${\displaystyle \parallel {\zeta }^{\prime }{\parallel }_{\infty }=ess\underset{\upsilon \in (a_1,b_1)}{sup}\left|{\zeta }^{\prime }\left(\upsilon \right)\right|<\infty }$. This outcome is also valid for functions of bounded variation since ${\zeta }^{\prime }$ is bounded.

Before we further proceed, let us denote the class of absolutely continuous functions by $AC\left(I\right)$, which are defined on some real interval I.

Now we recall the celebrated Montgomery identity from “Inequalities for Functions and their Integrals and Derivatives” by Mitrinović et al. [2].

Theorem 2.

Let $\zeta :[a_1,b_1]\to \mathbb{R}$ be differentiable on $[a_1,b_1]$ and ${\zeta }^{\prime }:[a_1,b_1]\to \mathbb{R}$ integrable on $[a_1,b_1]$. Then,

$$\begin{array}{c}\hfill \zeta \left(\upsilon \right)=\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt+\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}P(\upsilon ,t){\zeta }^{\prime }\left(t\right)dt,\end{array}$$

(2)

where

$$\begin{array}{c}\hfill P(\upsilon ,t)=\left\{\begin{array}{cc}\hfill t-a_1,& \mathit{if}t\in [a_1,\upsilon ],\hfill \\ & \\ \hfill t-b_1,& \mathit{if}t\in (\upsilon ,b_1].\hfill \end{array}\right.\end{array}$$

(3)

The following extended Montgomery identity via Taylor’s formula was obtained in [3].

Theorem 3.

Let $\zeta :I\to \mathbb{R}$ be a function such that ${\zeta }^{\left(n-1\right)}\in AC\left(I\right)$ and $n\in \mathbb{N}$, $n\ge 2$, $I\subset \mathbb{R}$ an open interval, $a_1,b_1\in I$, $a_1<b_1$. Then, the following identity holds

$$\begin{array}{ccc}\hfill \zeta \left(\upsilon \right)& =& \frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt+\sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(a_1\right)}{k!\left(k+2\right)}\frac{{\left(\upsilon -a_1\right)}^{k+2}}{b_1-a_1}\hfill \\ & -& \sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(b_1\right)}{k!\left(k+2\right)}\frac{{\left(\upsilon -b_1\right)}^{k+2}}{b_1-a_1}+\frac{1}{\left(n-1\right)!}{\int }_{a_1}^{b_1}{\tau }_n^{[a_1,b_1]}\left(\upsilon ,s\right){\zeta }^{\left(n\right)}\left(s\right)ds\hfill \end{array}$$

(4)

where

$${\tau }_n^{[a_1,b_1]}\left(\upsilon ,s\right)=\left\{\begin{array}{cc}-\frac{{\left(\upsilon -s\right)}^n}{n\left(b_1-a_1\right)}+\frac{\upsilon -a_1}{b_1-a_1}{\left(\upsilon -s\right)}^{n-1},& a_1\le s\le \upsilon ,\\ & \\ -\frac{{\left(\upsilon -s\right)}^n}{n\left(b_1-a_1\right)}+\frac{\upsilon -b_1}{b_1-a_1}{\left(\upsilon -s\right)}^{n-1},& \upsilon <s\le b_1.\end{array}\right.$$

(5)

In [4], the following identities were obtained for ${\sum }_{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)$ and ${\int }_{{\alpha }_1}^{{\beta }_1}{p}_i\zeta \left(g\left(\upsilon \right)\right)d\upsilon$ by using the above-mentioned extension of Montgomery’s identity.

Theorem 4.

Let all the assumptions from Theorem 3 hold and let ${\upsilon }_i\in [a_1,b_1]$ and $p_i\in \mathbb{R}$ for $i\in \{1,2,\dots ,m\}$ ($m\in \mathbb{N}$) s.t. ${\sum }_{i=1}^m{p}_i=0$. Then,

$$\begin{array}{ccc}\hfill & & \sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)=\frac{1}{(b_1-a_1)}\left[\sum _{k=0}^{n-2}\frac{{\zeta }^{k+1}\left(a_1\right)}{k!(k+2)!}\sum _{i=1}^m{p}_i{({\upsilon }_i-a_1)}^{k+2}\right.\hfill \\ & & \left.-\sum _{k=0}^{n-2}\frac{{\zeta }^{k+1}\left(b_1\right)}{k!(k+2)!}\sum _{i=1}^m{p}_i{({\upsilon }_i-b_1)}^{k+2}\right]\hfill \\ & & +\frac{1}{(n-1)!}{\int }_{a_1}^{b_1}\left(\sum _{i=1}^m{p}_i{\tau }_n^{[a_1,b_1]}({\upsilon }_i,s)\right){\zeta }^{\left(n\right)}\left(s\right)ds,\hfill \end{array}$$

(6)

where ${\tau }_n^{[·,·]}(·,·)$ is defined in (5).

Theorem 5.

Let $\zeta :I\to \mathbb{R}$ be a function such that ${\zeta }^{\left(n-1\right)}\in AC\left(I\right)$ and $n\in \mathbb{N}$, $I\subset \mathbb{R}$ an open interval, $a_1,b_1\in I$, $a_1<b_1$. Let $g:[{\alpha }_1,{\beta }_1]\to [a_1,b_1]$ and $p:[{\alpha }_1,{\beta }_1]\to \mathbb{R}$ be integrable functions such that ${\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)d\upsilon =0$. Then,

$$\begin{array}{ccc}& & {\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)\zeta \left(g\left(\upsilon \right)\right)d\upsilon =\frac{1}{(b_1-a_1)}\left[\sum _{k=0}^{n-2}\frac{{\zeta }^{k+1}\left(a_1\right)}{k!(k+2)!}{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){(g\left(\upsilon \right)-a_1)}^{k+2}d\upsilon \right.\hfill \\ & & \left.-\sum _{k=0}^{n-2}\frac{{\zeta }^{k+1}\left(b_1\right)}{k!(k+2)!}{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){(g\left(\upsilon \right)-b_1)}^{k+2}d\upsilon \right]\hfill \\ & & +\frac{1}{(n-1)!}{\int }_{a_1}^{b_1}\left({\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){\tau }_n^{[{\alpha }_1,{\beta }_1]}(g\left(\upsilon \right),s)d\upsilon \right){\zeta }^{\left(n\right)}\left(s\right)ds,\hfill \end{array}$$

(7)

where ${\tau }_n^{[·,·]}(·,·)$ is defined in (5).

Definition 1.

We say $(p,q)$ is a pair of conjugate exponents if $1<p,q<\infty$ and $\frac{1}{p}+\frac{1}{q}=1,$ or if $p=1$ and $q=\infty ,$ or if $p=\infty$ and $q=1$.

Let us denote by $G:[a_1,b_1]\times [a_1,b_1]\to \mathbb{R}$ the Green’s function of the boundary value problem

$$z^{″}\left(\lambda \right)=0,z\left(a_1\right)=z\left(b_1\right)=0.$$

The function G is given by (see [5])

$$G(t,s)=\left\{\begin{array}{cc}\frac{(t-b_1)(s-a_1)}{b_1-a_1}\hfill & \mathrm{for}{a}_1\le s\le t,\hfill \\ \frac{(s-b_1)(t-a_1)}{b_1-a_1}\hfill & \mathrm{for}t\le s\le b_1\hfill \end{array}\right.$$

(8)

and integration by parts easily yields that for any function $\zeta \in C^2[a_1,b_1]$ the following identity holds

$$\zeta \left(\upsilon \right)=\frac{b_1-\upsilon }{b_1-a_1}\zeta \left(a_1\right)+\frac{\upsilon -a_1}{b_1-a_1}\zeta \left(b_1\right)+{\int }_{a_1}^{b_1}G(\upsilon ,s){\zeta }^{″}\left(s\right)ds.$$

(9)

The function G is continuous, symmetric, and convex with respect to both variables t and s.

As a special choice, an Abel–Gontscharoff polynomial for two-point right focal interpolating polynomials for $n=2$ can be given as (see [6]):

$$\zeta \left(\upsilon \right)=\zeta \left(a_1\right)+(\upsilon -a_1){\zeta }^{\prime }\left(b_1\right)+{\int }_{a_1}^{b_1}{G}_1(\upsilon ,t){\zeta }^{″}\left(t\right)dt,$$

(10)

where $G_1(s,t)$ is Green’s function for the two-point right focal problem defined as

$$G_1(s,t)=\left\{\begin{array}{cc}{a}_1-t\hfill & \mathrm{for}{a}_1\le t\le s,\hfill \\ a_1-s\hfill & \mathrm{for}s\le t\le b_1.\hfill \end{array}\right.$$

(11)

Motivated by the Abel–Gontscharoff identity (10) and related Green’s function (11), we would like to recall some new types of Green’s functions $G_l:[a_1,b_1]\times [a_1,b_1]⟶\mathbb{R}$ for $l\in \{2,3,4\}$, defined as in [7]:

$$G_2(s,t)=\left\{\begin{array}{cc}s-b_1\hfill & \mathrm{for}{a}_1\le t\le s,\hfill \\ t-b_1\hfill & \mathrm{for}s\le t\le b_1.\hfill \end{array}\right.$$

(12)

$$G_3(s,t)=\left\{\begin{array}{cc}s-a_1\hfill & \mathrm{for}{a}_1\le t\le s,\hfill \\ t-a_1\hfill & \mathrm{for}s\le t\le b_1.\hfill \end{array}\right.$$

(13)

$$G_4(s,t)=\left\{\begin{array}{cc}{b}_1-t\hfill & \mathrm{for}{a}_1\le t\le s,\hfill \\ b_1-s\hfill & \mathrm{for}s\le t\le b_1.\hfill \end{array}\right.$$

(14)

In [7], it is also shown that all four Green’s functions are symmetric and continuous. These new Green’s functions allow us to adopt some new identities, which are described as follows.

$$\zeta \left(\upsilon \right)=\zeta \left(b_1\right)+(b_1-\upsilon ){\zeta }^{\prime }\left(a_1\right)+{\int }_{a_1}^{b_1}{G}_2(\upsilon ,t){\zeta }^{″}\left(t\right)dt.$$

(15)

$$\zeta \left(\upsilon \right)=\zeta \left(b_1\right)-(b_1-a_1){\zeta }^{\prime }\left(b_1\right)+(\upsilon -a_1){\zeta }^{\prime }\left(a_1\right)+{\int }_{a_1}^{b_1}{G}_3(\upsilon ,t){\zeta }^{″}\left(t\right)dt.$$

(16)

$$\zeta \left(\upsilon \right)=\zeta \left(a_1\right)+(b_1-a_1){\zeta }^{\prime }\left(a_1\right)-(b_1-\upsilon ){\zeta }^{\prime }\left(b_1\right)+{\int }_{a_1}^{b_1}{G}_4(\upsilon ,s){\zeta }^{″}\left(t\right)dt.$$

(17)

The following result was obtained by using a Green’s function in [8].

Theorem 6.

Let $\zeta :I\to \mathbb{R}$ be a function such that ${\zeta }^{\left(n-1\right)}\in AC\left(I\right)$ and $n\in \mathbb{N}$, $n\ge 3$, $I\subset \mathbb{R}$ an open interval, $a_1,b_1\in I$, $a_1<b_1$. Moreover, let ${\upsilon }_i\in [a_1,b_1]$ and $p_i\in \mathbb{R}$ for $i\in \{1,2,\dots ,m\}$ ($m\in \mathbb{N}$) such that ${\sum }_{i=1}^m{p}_i=0$ and ${\sum }_{i=1}^m{p}_i{\upsilon }_i=0$ and let G be given by (8). Then

$$\begin{array}{c}\sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)=\frac{{\zeta }^{\prime }\left(a_1\right)-{\zeta }^{\prime }\left(b_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)ds\hfill \\ +\sum _{k=2}^{n-1}\frac{k}{(k-1)!}{\int }_{a_1}^{b_1}\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\\ \hfill +\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}{\zeta }^{\left(n\right)}\left(t\right)\left({\int }_{a_1}^{b_1}\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s){\tilde{\tau }}_{n-2}^{[a_1,b_1]}(s,t)ds\right)dt,\end{array}$$

(18)

where

$${\tilde{\tau }}_{n-2}^{[a_1,b_1]}\left(s,t\right)=\left\{\begin{array}{cc}\frac{1}{b_1-a_1}\left[\frac{{\left(s-t\right)}^{n-2}}{(n-2)}+\left(s-a_1\right){\left(s-t\right)}^{n-3}\right],& a_1\le t\le s,\\ & \\ \frac{1}{b_1-a_1}\left[\frac{{\left(s-t\right)}^{n-2}}{(n-2)}+\left(s-b_1\right){\left(s-t\right)}^{n-3}\right],& s<t\le b_1.\end{array}\right.$$

(19)

Furthermore, the following identity holds

$$\begin{array}{c}\sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)=\frac{{\zeta }^{\prime }\left(b_1\right)-{\zeta }^{\prime }\left(a_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)ds\hfill \\ +\sum _{k=3}^{n-1}\frac{k-2}{(k-1)!}{\int }_{a_1}^{b_1}\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\\ \hfill +\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}{\zeta }^{\left(n\right)}\left(t\right)\left({\int }_{a_1}^{b_1}\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s){\tau }_{n-2}^{[a_1,b_1]}(s,t)ds\right)dt,\end{array}$$

(20)

where ${\tau }_n^{[·,·]}(·,·)$ is as given in (5).

The integral version of Theorem 6 is given in the following theorem.

Theorem 7.

Let $\zeta :I\to \mathbb{R}$ be a function such that ${\zeta }^{\left(n-1\right)}\in AC\left(I\right)$ and $n\in \mathbb{N}$, $n\ge 3$, $I\subset \mathbb{R}$ an open interval, $a_1,b_1\in I$, $a_1<b_1$. Furthermore, let $g:[{\alpha }_1,{\beta }_1]\to [a_1,b_1]$ and $p:[{\alpha }_1,{\beta }_1]\to \mathbb{R}$ satisfy ${\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)d\upsilon =0$ and ${\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)g\left(\upsilon \right)d\upsilon =0$, and let ${\tau }_n^{[·,·]}(·,·)$, G and ${\tilde{\tau }}_n^{[·,·]}(·,·)$ be given by (5), (8) and (19). Then, the following two identities hold:

$$\begin{array}{c}{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)\zeta \left(g\left(\upsilon \right)\right)d\upsilon =\frac{{\zeta }^{\prime }\left(a_1\right)-{\zeta }^{\prime }\left(b_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G(g\left(\upsilon \right),s)d\upsilon ds\hfill \\ +\sum _{k=2}^{n-1}\frac{k}{(k-1)!}{\int }_{a_1}^{b_1}\left({\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G(g\left(\upsilon \right),s)d\upsilon \right)\\ \times \frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\\ \hfill +\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}{\zeta }^{\left(n\right)}\left(t\right)\left({\int }_{a_1}^{b_1}\left({\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G(g\left(\upsilon \right),s)d\upsilon \right){\tilde{\tau }}_{n-2}^{[a_1,b_1]}(s,t)ds\right)dt\end{array}$$

(21)

and

$$\begin{array}{c}{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)\zeta \left(g\left(\upsilon \right)\right)d\upsilon =\frac{{\zeta }^{\prime }\left(b_1\right)-{\zeta }^{\prime }\left(a_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G(g\left(\upsilon \right),s)d\upsilon ds\hfill \\ +\sum _{k=3}^{n-1}\frac{k-2}{(k-1)!}{\int }_{a_1}^{b_1}\left({\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G(g\left(\upsilon \right),s)d\upsilon \right)\\ \times \frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\\ \hfill +\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}{\zeta }^{\left(n\right)}\left(t\right)\left({\int }_{a_1}^{b_1}\left({\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G(g\left(\upsilon \right),s)d\upsilon \right){\tau }_{n-2}^{[a_1,b_1]}(s,t)ds\right)dt.\end{array}$$

(22)

In [9], the authors proved the following results using new Green type functions as mentioned above.

Theorem 8.

Let $\zeta :I\to \mathbb{R}$ be a function such that ${\zeta }^{\left(n-1\right)}\in AC\left(I\right)$ and $n\in \mathbb{N}$, $n\ge 3$, $I\subset \mathbb{R}$ an open interval, $a_1,b_1\in I$, $a_1<b_1$. Moreover, let ${\upsilon }_i\in [a_1,b_1]$ and $p_i\in \mathbb{R}$ for $i\in \{1,2,\dots ,m\}$ ($m\in \mathbb{N}$) such that ${\sum }_{i=1}^m{p}_i=0$ and ${\sum }_{i=1}^m{p}_i{\upsilon }_i=0$ and let $G_l$ be given by (11), (12), (13) and (14) by fixing $l\in \{1,2,3,4\}$. Then,

$$\begin{array}{c}\sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)=\frac{{\zeta }^{\prime }\left(a_1\right)-{\zeta }^{\prime }\left(b_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}\sum _{i=1}^m{p}_i{G}_l({\upsilon }_i,s)ds\hfill \\ +\sum _{k=2}^{n-1}\frac{k}{(k-1)!}{\int }_{a_1}^{b_1}\sum _{i=1}^m{p}_i{G}_l({\upsilon }_i,s)\frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\\ \hfill +\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}{\zeta }^{\left(n\right)}\left(t\right)\left({\int }_{a_1}^{b_1}\sum _{i=1}^m{p}_i{G}_l({\upsilon }_i,s){\tilde{\tau }}_{n-2}^{[a_1,b_1]}(s,t)ds\right)dt,\end{array}$$

(23)

where ${\tilde{\tau }}_{n-2}^{[·,·]}(·,·)$ is as given in (19).

Furthermore, the following identity holds

$$\begin{array}{c}\sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)=\frac{{\zeta }^{\prime }\left(b_1\right)-{\zeta }^{\prime }\left(a_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}\sum _{i=1}^m{p}_i{G}_l({\upsilon }_i,s)ds\hfill \\ +\sum _{k=3}^{n-1}\frac{k-2}{(k-1)!}{\int }_{a_1}^{b_1}\sum _{i=1}^m{p}_i{G}_l({\upsilon }_i,s)\frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\\ \hfill +\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}{\zeta }^{\left(n\right)}\left(t\right)\left({\int }_{a_1}^{b_1}\sum _{i=1}^m{p}_i{G}_l({\upsilon }_i,s){\tau }_{n-2}^{[a_1,b_1]}(s,t)ds\right)dt,\end{array}$$

(24)

where ${\tau }_n^{[·,·]}(·,·)$ is as given in (5).

The integral version of Theorem 8 is given in the following theorem.

Theorem 9.

Let $\zeta :I\to \mathbb{R}$ be a function such that ${\zeta }^{\left(n-1\right)}\in AC\left(I\right)$ and $n\in \mathbb{N}$, $n\ge 3$, $I\subset \mathbb{R}$ an open interval, $a_1,b_1\in I$, $a_1<b_1$. Furthermore, let $g:[{\alpha }_1,{\beta }_1]\to [a_1,b_1]$ and $p:[{\alpha }_1,{\beta }_1]\to \mathbb{R}$ satisfy ${\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)d\upsilon =0$ and ${\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)g\left(\upsilon \right)d\upsilon =0$, and let $G_l$ (fix $l\in \{1,2,3,4\}$), ${\tilde{\tau }}_n^{[·,·]}(·,·)$ and ${\tau }_n^{[·,·]}(·,·)$ be given by (11), (12), (13), (14), (19) and (5), respectively. Then, the following two identities hold:

$$\begin{array}{c}{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)\zeta \left(g\left(\upsilon \right)\right)d\upsilon =\frac{{\zeta }^{\prime }\left(a_1\right)-{\zeta }^{\prime }\left(b_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G_l(g\left(\upsilon \right),s)d\upsilon ds\hfill \\ +\sum _{k=2}^{n-1}\frac{k}{(k-1)!}{\int }_{a_1}^{b_1}\left({\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G_l(g\left(\upsilon \right),s)d\upsilon \right)\\ \times \frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\\ \hfill +\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}{\zeta }^{\left(n\right)}\left(t\right)\left({\int }_{a_1}^{b_1}\left({\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G_l(g\left(\upsilon \right),s)d\upsilon \right){\tilde{\tau }}_{n-2}^{[a_1,b_1]}(s,t)ds\right)dt\end{array}$$

(25)

and

$$\begin{array}{c}{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)\zeta \left(g\left(\upsilon \right)\right)d\upsilon =\frac{{\zeta }^{\prime }\left(b_1\right)-{\zeta }^{\prime }\left(a_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G_l(g\left(\upsilon \right),s)d\upsilon ds\hfill \\ +\sum _{k=3}^{n-1}\frac{k-2}{(k-1)!}{\int }_{a_1}^{b_1}\left({\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G_l(g\left(\upsilon \right),s)d\upsilon \right)\\ \times \frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\\ \hfill +\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}{\zeta }^{\left(n\right)}\left(t\right)\left({\int }_{a_1}^{b_1}\left({\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G_l(g\left(\upsilon \right),s)d\upsilon \right){\tau }_{n-2}^{[a_1,b_1]}(s,t)ds\right)dt.\end{array}$$

(26)

In the literature, Ostrowski type inequalities have been examined broadly, since they have widespread applications in numerical analysis, special mean(s), and probability theory, among many others. In [10,11], Ostrowski type inequalities for continuous functions and their application in numerical integration are conferred. Some other Ostrowski type inequalities can be spotted in [5,12,13,14]. Some recently obtained generalizations of Ostrowski type inequalities via weighted generalizations of the Montgomery identity established by using the Hermite interpolating polynomial and Green’s functions can be found in [15].

This paper aims to provide some Ostrowski type inequalities via the extended version of Montgomery’s identity and using some of Green’s functions. For this purpose, the paper is organized in the following manner. After this introduction, in Section 2, we obtain new discrete and integral nature Ostrowski type inequalities using the extended version of Montgomery’s identity. In Section 3, we establish Ostrowski type inequalities using different stated Green’s functions involving the extended Montgomery identity. Finally, in Section 4, we obtain some results which are the generalizations of the estimation of the difference of two integral means (generalized results from [16,17,18,19]).

2. Ostrwoski Type Inequalities via the Extended Montgomery Identity

Theorem 10.

Let all the assumptions of Theorem 4 be valid. Moreover, assume that $(p,q)$ is a pair of conjugate exponents. Let $|{\zeta }^{\left(n\right)}{|}^p:[a_1,b_1]\to \mathbb{R}$ be an R-integrable function for some $n\ge 2.$ Then we have

$$\begin{array}{c}\hfill \left|\zeta \left(\upsilon \right)-\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt+D_n^1[a_1,b_1]\left(\upsilon \right)\right|\le {\left({\int }_{a_1}^{b_1}{\left|{\lambda }_1\left(s\right)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\parallel }_p,\end{array}$$

(27)

where

$$\begin{array}{ccc}& & D_n^1[a_1,b_1]\left(\upsilon \right)=-\sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)\hfill \\ & & -\frac{1}{b_1-a_1}\left[\sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(a_1\right)}{k!\left(k+2\right)}\left[{\left(\upsilon -a_1\right)}^{k+2}-\sum _{i=1}^m{p}_i{({\upsilon }_i-a_1)}^{k+2}\right]\right.\hfill \\ & & \left.-\sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(b_1\right)}{k!\left(k+2\right)}\left[{\left(\upsilon -b_1\right)}^{k+2}-\sum _{i=1}^m{p}_i{({\upsilon }_i-b_1)}^{k+2}\right]\right]\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\lambda }_1\left(s\right)=\frac{1}{(n-1)!}\left[{\tau }_n^{[a_1,b_1]}(\upsilon ,s)-\sum _{i=1}^m{p}_i{\tau }_n^{[a_1,b_1]}({\upsilon }_i,s)\right].\end{array}$$

The constant on the right-hand side of (27) is the best possible for $p=1$ and sharp for $1<p\le \infty$.

Proof. 

By taking the difference of identities (4) and (6) we get

$$\begin{array}{ccc}& & \zeta \left(\upsilon \right)-\sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)-\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt\hfill \\ & & =\frac{1}{b_1-a_1}\left[\sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(a_1\right)}{k!\left(k+2\right)}\left[{\left(\upsilon -a_1\right)}^{k+2}-\sum _{i=1}^m{p}_i{({\upsilon }_i-a_1)}^{k+2}\right]\right.\hfill \\ & & \left.-\sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(b_1\right)}{k!\left(k+2\right)}\left[{\left(\upsilon -b_1\right)}^{k+2}-\sum _{i=1}^m{p}_i{({\upsilon }_i-b_1)}^{k+2}\right]\right]\hfill \\ & & +\frac{1}{(n-1)!}{\int }_{a_1}^{b_1}\left[{\tau }_n^{[a_1,b_1]}(\upsilon ,s)-\sum _{i=1}^m{p}_i{\tau }_n^{[a_1,b_1]}({\upsilon }_i,s)\right]{\zeta }^{\left(n\right)}\left(s\right)ds,\hfill \end{array}$$

where ${\tau }_n^{[·,·]}(·,·)$ is as defined in (5). After some rearrangements we get

$$\begin{array}{ccc}\hfill & & \zeta \left(\upsilon \right)-\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt-\sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)\hfill \\ & & -\frac{1}{b_1-a_1}\left[\sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(a_1\right)}{k!\left(k+2\right)}\left[{\left(\upsilon -a_1\right)}^{k+2}-\sum _{i=1}^m{p}_i{({\upsilon }_i-a_1)}^{k+2}\right]\right.\hfill \\ & & \left.-\sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(b_1\right)}{k!\left(k+2\right)}\left[{\left(\upsilon -b_1\right)}^{k+2}-\sum _{i=1}^m{p}_i{({\upsilon }_i-b_1)}^{k+2}\right]\right]\hfill \\ & & =\frac{1}{(n-1)!}{\int }_{a_1}^{b_1}\left[{\tau }_n^{[a_1,b_1]}(\upsilon ,s)-\sum _{i=1}^m{p}_i{\tau }_n^{[a_1,b_1]}({\upsilon }_i,s)\right]{\zeta }^{\left(n\right)}\left(s\right)ds.\hfill \end{array}$$

Now, in new notations as defined in Theorem 27, we have

$$\begin{array}{c}\hfill \zeta \left(\upsilon \right)-\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt+D_n^1[a_1,b_1]\left(\upsilon \right)={\int }_{a_1}^{b_1}{\lambda }_1\left(s\right){\zeta }^{\left(n\right)}\left(s\right)ds\end{array}$$

(28)

After applying Hölder’s inequality, we obtain

$$\begin{array}{ccc}\hfill & & \left|\zeta \left(\upsilon \right)-\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt+D_n^1[a_1,b_1]\left(\upsilon \right)\right|=\left|{\int }_{a_1}^{b_1}{\lambda }_1\left(s\right){\zeta }^{\left(n\right)}\left(s\right)ds\right|\hfill \end{array}$$

$$\begin{array}{ccc}& & \le {\int }_{a_1}^{b_1}\left|{\lambda }_1\left(s\right){\zeta }^{\left(n\right)}\left(s\right)\right|ds\le \parallel {\lambda }_1{\left(s\right)\parallel }_q{\parallel {\zeta }^{\left(n\right)}\parallel }_p\hfill \end{array}$$

(29)

$$\begin{array}{ccc}& & ={\left({\int }_{a_1}^{b_1}{\left|{\lambda }_1\left(s\right)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\parallel }_p.\hfill \end{array}$$

(30)

For the proof of the sharpness of the constant ${\left({\int }_{a_1}^{b_1}{\left|{\lambda }_1\left(s\right)\right|}^{q}ds\right)}^{\frac{1}{q}}$, let us define a function $\zeta$ for which the equality in (29) is obtained.

For $1<p<\infty$, take $\zeta$ to be such that

$$\begin{array}{c}\hfill {\zeta }^{\left(n\right)}\left(s\right)=sgn{\lambda }_1\left(s\right).{\left|{\lambda }_1\left(s\right)\right|}^{\frac{1}{p-1}}.\end{array}$$

For $p=\infty$, take

$$\begin{array}{c}\hfill {\zeta }^{\left(n\right)}\left(s\right)=sgn{\lambda }_1\left(s\right).\end{array}$$

For $p=1$, we shall prove that

$$\begin{array}{c}\hfill \left|{\int }_{a_1}^{b_1}{\lambda }_1\left(s\right){\zeta }^{\left(n\right)}\left(s\right)ds\right|\le \underset{s\in [a_1,b_1]}{max}\left|{\lambda }_1\left(s\right)\right|{\int }_{a_1}^{b_1}\left|{\zeta }^{\left(n\right)}\left(s\right)\right|ds\end{array}$$

(31)

is the best possible inequality. Since $|{\lambda }_1\left(s\right)|$ is continuous for $n\ge 2$ we suppose that $|{\lambda }_1\left(s\right)|$ attains its maximum at $s_0\in [a_1,b_1].$ First, we assume that ${\lambda }_1\left(s_0\right)>0.$ For $ϵ>0$, ${\zeta }_{ϵ}\left(s\right)$ is defined by

$$\begin{array}{c}\hfill {\zeta }_{ϵ}\left(s\right)=\left\{\begin{array}{cc}0,& a_1\le s\le s_0,\\ \\ {\displaystyle \frac{1}{ϵn!}{(s-s_0)}^n,}& s_0<s\le s_0+ϵ,\\ \\ {\displaystyle \frac{1}{(n-1)!}{(s-s_0)}^{n-1},}& s_0+ϵ<s\le b_1.\end{array}\right.\end{array}$$

We have

$$\begin{array}{c}\hfill \left|{\int }_{a_1}^{b_1}{\lambda }_1\left(s\right){\zeta }_{ϵ}^{\left(n\right)}\left(s\right)ds\right|=\left|{\int }_{s_0}^{s_0+ϵ}{\lambda }_1\left(s\right)\frac{1}{ϵ}ds\right|=\frac{1}{ϵ}{\int }_{s_0}^{s_0+ϵ}{\lambda }_1\left(s\right)ds.\end{array}$$

Now, from inequality (31), we have

$$\begin{array}{c}\hfill \frac{1}{ϵ}{\int }_{s_0}^{s_0+ϵ}{\lambda }_1\left(s\right)ds\le {\lambda }_1\left(s_0\right){\int }_{s_0}^{s_0+ϵ}ds={\lambda }_1\left(s_0\right).\end{array}$$

Since

$$\begin{array}{c}\hfill \underset{ϵ\to 0}{lim}\frac{1}{ϵ}{\int }_{s_0}^{s_0+ϵ}{\lambda }_1\left(s\right)ds={\lambda }_1\left(s_0\right),\end{array}$$

the statement follows. In the case ${\lambda }_1\left(s_0\right)<0,$ we define ${\zeta }_{ϵ}\left(s\right)$ by

$$\begin{array}{c}\hfill {\zeta }_{ϵ}\left(s\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{(n-1)!}{(s-s_0-ϵ)}^{n-1},}& a_1\le s\le s_0,\\ \\ {\displaystyle -\frac{1}{ϵn!}{(s-s_0-ϵ)}^n,}& s_0<s\le s_0+ϵ,\\ \\ 0,& s_0+ϵ<s\le b_1,\end{array}\right.\end{array}$$

and the rest of the proof is the same as above. □

Now we provide the integral version of Theorem 10.

Theorem 11.

Suppose all the assumptions of Theorem 5 hold and further assume that $(p,q)$ is a pair of conjugate exponents. Let $|{\zeta }^{\left(n\right)}{|}^p:[a_1,b_1]\to \mathbb{R}$ be an R-integrable function for some $n\ge 2.$ Then, we have

$$\begin{array}{c}\hfill \left|\zeta \left(\upsilon \right)-\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}f\left(t\right)dt+D_n^2[a_1,b_1]\left(\upsilon \right)\right|\le {\left({\int }_{a_1}^{b_1}{\left|{\lambda }_2\left(s\right)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\parallel }_p,\end{array}$$

(32)

where

$$\begin{array}{ccc}& & D_n^2[a_1,b_1]\left(\upsilon \right)=-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)\zeta \left(g\left(\upsilon \right)\right)d\upsilon \hfill \\ & & -\frac{1}{b_1-a_1}\left[\sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(a_1\right)}{k!\left(k+2\right)}\left[{\left(\upsilon -a_1\right)}^{k+2}-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){(g\left(\upsilon \right)-a_1)}^{k+2}d\upsilon \right]\right.\hfill \\ & & \left.-\sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(b_1\right)}{k!\left(k+2\right)}\left[{\left(\upsilon -b_1\right)}^{k+2}-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){(g\left(\upsilon \right)-b_1)}^{k+2}d\upsilon \right]\right]\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\lambda }_2\left(s\right)=\frac{1}{(n-1)!}\left[{\tau }_n^{[a_1,b_1]}(\upsilon ,s)-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){\tau }_n^{[a_1,b_1]}(g\left(\upsilon \right),s)d\upsilon \right].\end{array}$$

The constant on the right-hand side of (32) is the best possible for $p=1$ and sharp for $1<p\le \infty$.

Proof. 

By taking the difference of identities (4) and (7)

$$\begin{array}{ccc}& & \zeta \left(\upsilon \right)-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)\zeta \left(g\left(\upsilon \right)\right)d\upsilon -\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt\hfill \\ & & =\frac{1}{b_1-a_1}\left[\sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(a_1\right)}{k!\left(k+2\right)}\left[{\left(\upsilon -a_1\right)}^{k+2}-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){(g\left(\upsilon \right)-a_1)}^{k+2}d\upsilon \right]\right.\hfill \\ & & \left.-\sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(b_1\right)}{k!\left(k+2\right)}\left[{\left(\upsilon -b_1\right)}^{k+2}-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){(g\left(\upsilon \right)-b_1)}^{k+2}d\upsilon \right]\right]\hfill \\ & & +\frac{1}{(n-1)!}{\int }_{a_1}^{b_1}\left[T_n^{[a_1,b_1]}(\upsilon ,s)-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){\tau }_n^{[a_1,b_1]}(g\left(\upsilon \right),s)d\upsilon \right]{\zeta }^{\left(n\right)}\left(s\right)ds,\hfill \end{array}$$

where ${\tau }_n^{[·,·]}(·,\cdots )$ is as defined in (5). After some rearrangements, we get

$$\begin{array}{ccc}\hfill & & \zeta \left(\upsilon \right)-\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)\zeta \left(g\left(\upsilon \right)\right)d\upsilon \hfill \\ & & -\frac{1}{b_1-a_1}\left[\sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(a_1\right)}{k!\left(k+2\right)}\left[{\left(\upsilon -a_1\right)}^{k+2}-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){(g\left(\upsilon \right)-a_1)}^{k+2}d\upsilon \right]\right.\hfill \\ & & \left.-\sum _{k=0}^{n-2}\frac{{\zeta }^{\left(k+1\right)}\left(b_1\right)}{k!\left(k+2\right)}\left[{\left(\upsilon -b_1\right)}^{k+2}-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){(g\left(\upsilon \right)-b_1)}^{k+2}d\upsilon \right]\right]\hfill \\ & & =\frac{1}{(n-1)!}{\int }_{a_1}^{b_1}\left[T_n^{[a_1,b_1]}(\upsilon ,s)-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){\tau }_n^{[a_1,b_1]}(g\left(\upsilon \right),s)d\upsilon \right]{\zeta }^{\left(n\right)}\left(s\right)ds.\hfill \end{array}$$

Now, in new notations as defined in Theorem 32, we have

$$\begin{array}{c}\hfill \zeta \left(\upsilon \right)-\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt+D_n^2[a_1,b_1]\left(\upsilon \right)={\int }_{a_1}^{b_1}{\lambda }_2\left(s\right){\zeta }^{\left(n\right)}\left(s\right)ds.\end{array}$$

(33)

After applying Hölder’s inequality, we obtain

$$\begin{array}{ccc}\hfill & & \left|\zeta \left(\upsilon \right)-\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt+D_n^2[a_1,b_1]\left(\upsilon \right)\right|=\left|{\int }_{a_1}^{b_1}{\lambda }_2\left(s\right){\zeta }^{\left(n\right)}\left(s\right)ds\right|\hfill \end{array}$$

$$\begin{array}{ccc}& & \le {\int }_{a_1}^{b_1}\left|{\lambda }_2\left(s\right){\zeta }^{\left(n\right)}\left(s\right)\right|ds\le \parallel {\lambda }_2{\left(s\right)\parallel }_q{\parallel {\zeta }^{\left(n\right)}\parallel }_p\hfill \end{array}$$

(34)

$$\begin{array}{ccc}& & ={\left({\int }_{a_1}^{b_1}{\left|{\lambda }_2\left(s\right)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\parallel }_p,\hfill \end{array}$$

(35)

which proves the inequality. The proof for the best possibility and sharpness are similar to that in Theorem 10. □

3. Ostrwoski Type Inequalities via the Extended Montgomery Identity Using Greens’ Functions

Theorem 12.

If all the assumptions of Theorem 6 hold and G and ${\tilde{\tau }}_{n-2}^{[·,·]}(·,·)$ are as given in (8) and (19), then we have

$$\begin{array}{ccc}\hfill & & \left|\zeta \left(\upsilon \right)-\sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)-\frac{(b_1-\upsilon )\zeta \left(a_1\right)+(\upsilon -a_1)\zeta \left(b_1\right)}{b_1-a_1}\right.\hfill \\ & & -\frac{{\zeta }^{\prime }\left(a_1\right)-{\zeta }^{\prime }\left(b_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\right)ds\hfill \\ & & -\sum _{k=2}^{n-1}\frac{k}{(k-1)!}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\right)\hfill \\ & & \left.\times \frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\right|\hfill \\ & & \le {\left({\int }_{a_1}^{b_1}{\left|{\lambda }_3\left(s\right)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\parallel }_p,\hfill \end{array}$$

(36)

where

$$\begin{array}{c}\hfill {\lambda }_3\left(s\right)=\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\right){\tilde{\tau }}_{n-2}^{[a_1,b_1]}(s,t)ds.\end{array}$$

Moreover, the following inequality holds

$$\begin{array}{ccc}\hfill & & \left|\zeta \left(\upsilon \right)-\sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)-\frac{(b_1-\upsilon )\zeta \left(a_1\right)+(\upsilon -a_1)\zeta \left(b_1\right)}{b_1-a_1}\right.\hfill \\ & & -\frac{{\zeta }^{\prime }\left(b_1\right)-{\zeta }^{\prime }\left(a_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\right)ds\hfill \\ & & -\sum _{k=3}^{n-1}\frac{k-2}{(k-1)!}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\right)\hfill \\ & & \left.\times \frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\right|\hfill \\ & & \le {\left({\int }_{a_1}^{b_1}{\left|{\lambda }_4\left(s\right)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\parallel }_p,\hfill \end{array}$$

(37)

where

$$\begin{array}{c}\hfill {\lambda }_4\left(s\right)=\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\right){\tau }_{n-2}^{[a_1,b_1]}(s,t)ds,\end{array}$$

where ${\tau }_n^{[·,·]}(·,·)$ is as defined in (5) .

Proof. 

Differentiating the function f in (4) twice gives

$$\begin{array}{c}{\zeta }^{\prime \prime }\left(s\right)=\frac{{\zeta }^{\prime }\left(a_1\right)-{\zeta }^{\prime }\left(b_1\right)}{b_1-a_1}+\sum _{k=2}^{n-1}\frac{k}{(k-1)!}\frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}\hfill \\ \hfill +\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}{\tilde{\tau }}_{n-2}^{[a_1,b_1]}(s,t){\zeta }^{\left(n\right)}\left(t\right)dt.\end{array}$$

(38)

Inserting (38) in (9) we get

$$\begin{array}{c}\zeta \left(\upsilon \right)=\frac{b_1-\upsilon }{b_1-a_1}\zeta \left(a_1\right)+\frac{\upsilon -a_1}{b_1-a_1}\zeta \left(b_1\right)+\frac{{\zeta }^{\prime }\left(a_1\right)-{\zeta }^{\prime }\left(b_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}G(\upsilon ,s)ds\hfill \\ +\sum _{k=2}^{n-1}\frac{k}{(k-1)!}{\int }_{a_1}^{b_1}G(\upsilon ,s)\frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\\ \hfill +\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}{\zeta }^{\left(n\right)}\left(t\right)\left({\int }_{a_1}^{b_1}G(\upsilon ,s){\tilde{\tau }}_{n-2}^{[a_1,b_1]}(s,t)ds\right)dt.\end{array}$$

(39)

By taking the difference of (39) and (18), after some rearrangements we get

$$\begin{array}{c}\zeta \left(\upsilon \right)-\sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)-\frac{(b_1-\upsilon )\zeta \left(a_1\right)+(\upsilon -a_1)\zeta \left(b_1\right)}{b_1-a_1}\hfill \\ -\frac{{\zeta }^{\prime }\left(a_1\right)-{\zeta }^{\prime }\left(b_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\right)ds\\ -\sum _{k=2}^{n-1}\frac{k}{(k-1)!}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\right)\\ \times \frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\\ \hfill =\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}{\zeta }^{\left(n\right)}\left(t\right)\left[{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\right){\tilde{\tau }}_{n-2}^{[a_1,b_1]}(s,t)ds\right]dt.\end{array}$$

(40)

Let

$$\begin{array}{c}\hfill {\lambda }_3\left(s\right)=\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\right){\tilde{\tau }}_{n-2}^{[a_1,b_1]}(s,t)ds.\end{array}$$

After applying Hölder’s inequality we obtain

$$\begin{array}{ccc}& & \left|\zeta \left(\upsilon \right)-\sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)-\frac{(b_1-\upsilon )\zeta \left(a_1\right)+(\upsilon -a_1)\zeta \left(b_1\right)}{b_1-a_1}\right.\hfill \\ & & -\frac{{\zeta }^{\prime }\left(a_1\right)-{\zeta }^{\prime }\left(b_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\right)ds\hfill \\ & & -\sum _{k=2}^{n-1}\frac{k}{(k-1)!}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-\sum _{i=1}^m{p}_{i}G({\upsilon }_i,s)\right)\hfill \\ & & \left.\times \frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\right|\hfill \\ & & =\left|{\int }_{a_1}^{b_1}{\lambda }_3\left(s\right){\zeta }^{\left(n\right)}\left(t\right)dt\right|\le {\int }_{a_1}^{b_1}\left|{\lambda }_3\left(s\right){\zeta }^{\left(n\right)}\left(t\right)\right|dt\le \parallel {\lambda }_3{\left(s\right)\parallel }_q{\parallel {\zeta }^{\left(n\right)}\parallel }_p\hfill \\ & & ={\left({\int }_{a_1}^{b_1}{\left|{\lambda }_3\left(s\right)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\parallel }_p,\hfill \end{array}$$

which proves the inequality. The proof for the best possibility and sharpness are similar to Theorem 10.

Furthermore, by using Formula (4) on the function ${\zeta }^{″}$, replacing n by $n-2(n\ge 3)$ and rearranging the indices we get

$$\begin{array}{c}{\zeta }^{″}\left(s\right)=\frac{{\zeta }^{\prime }\left(b_1\right)-{\zeta }^{\prime }\left(a_1\right)}{b_1-a_1}+\sum _{k=3}^{n-1}\frac{k-2}{(k-1)!}\frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}\hfill \\ \hfill +\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}{\tau }_{n-2}^{[a_1,b_1]}(s,t){\zeta }^{\left(n\right)}\left(t\right)dt.\end{array}$$

(41)

Inserting (41) in (9), we obtain

$$\begin{array}{c}\zeta \left(\upsilon \right)=\frac{b_1-\upsilon }{b_1-a_1}\zeta \left(a_1\right)+\frac{\upsilon -a_1}{b_1-a_1}\zeta \left(b_1\right)+\frac{{\zeta }^{\prime }\left(b_1\right)-{\zeta }^{\prime }\left(a_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}G(\upsilon ,s)ds\hfill \\ +\sum _{k=3}^{n-1}\frac{k-2}{(k-1)!}{\int }_{a_1}^{b_1}G(\upsilon ,s)\frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\\ \hfill +\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}{\zeta }^{\left(n\right)}\left(t\right)\left({\int }_{a_1}^{b_1}G(\upsilon ,s){\tau }_{n-2}^{[a_1,b_1]}(s,t)ds\right)dt.\end{array}$$

(42)

By taking the difference of (42) and (20) and after some rearrangements by applying Hölder inequality, we get (37). □

Theorem 13.

If all the assumptions of Theorem 6 hold and $G_1$ and ${\tilde{\tau }}_{n-2}^{[·,·]}(·,·)$ are as given in (11) and (19), then we have

$$\begin{array}{ccc}\hfill & & \left|\zeta \left(\upsilon \right)-\sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)-\zeta \left(a_1\right)-(\upsilon -a_1){\zeta }^{\prime }\left(b_1\right)\right.\hfill \\ & & -\frac{{\zeta }^{\prime }\left(a_1\right)-{\zeta }^{\prime }\left(b_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}\left(G_1(\upsilon ,s)-\sum _{i=1}^m{p}_i{G}_1({\upsilon }_i,s)\right)ds\hfill \\ & & -\sum _{k=2}^{n-1}\frac{k}{(k-1)!}{\int }_{a_1}^{b_1}\left(G_1(\upsilon ,s)-\sum _{i=1}^m{p}_i{G}_1({\upsilon }_i,s)\right)\hfill \\ & & \left.\times \frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\right|\hfill \\ & & \le {\left({\int }_{a_1}^{b_1}{\left|{\lambda }_5\left(s\right)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\parallel }_p,\hfill \end{array}$$

(43)

where

$$\begin{array}{c}\hfill {\lambda }_5\left(s\right)=\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}\left(G_1(\upsilon ,s)-\sum _{i=1}^m{p}_i{G}_1({\upsilon }_i,s)\right){\tilde{\tau }}_{n-2}^{[a_1,b_1]}(s,t)ds.\end{array}$$

Moreover, the following inequality holds

$$\begin{array}{ccc}\hfill & & \left|\zeta \left(\upsilon \right)-\sum _{i=1}^m{p}_i\zeta \left({\upsilon }_i\right)-\zeta \left(a_1\right)-(\upsilon -a_1){\zeta }^{\prime }\left(b_1\right)\right.\hfill \\ & & -\frac{{\zeta }^{\prime }\left(b_1\right)-{\zeta }^{\prime }\left(a_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}\left(G_1(\upsilon ,s)-\sum _{i=1}^m{p}_i{G}_1({\upsilon }_i,s)\right)ds\hfill \\ & & -\sum _{k=3}^{n-1}\frac{k-2}{(k-1)!}{\int }_{a_1}^{b_1}\left(G_1(\upsilon ,s)-\sum _{i=1}^m{p}_i{G}_1({\upsilon }_i,s)\right)\hfill \\ & & \left.\times \frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\right|\hfill \\ & & \le {\left({\int }_{a_1}^{b_1}{\left|{\lambda }_6\left(s\right)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\parallel }_p,\hfill \end{array}$$

(44)

where

$$\begin{array}{c}\hfill {\lambda }_6\left(s\right)=\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}\left(G_1(\upsilon ,s)-\sum _{i=1}^m{p}_i{G}_1({\upsilon }_i,s)\right){\tau }_{n-2}^{[a_1,b_1]}(s,t)ds,\end{array}$$

for ${\tau }_n^{[·,·]}(·,·)$ as defined in (5).

Remark 1.

The proof of Theorem 13 is similar to the proof of Theorem 12 except for the use of Identity (10) in place of (9), so we omitted the details. In a similar way, we can state and prove results for $G_2,G_3$ and $G_4$ by using Identities (15), (16) and (17), respectively.

Now we give the integral version of the Theorem 12 only. Analogous results for Theorem 13 and Remark 1 are also valid. As the proofs are similar, the details are omitted.

Theorem 14.

If all the assumptions of Theorem 7 hold and G and ${\tilde{\tau }}_{n-2}^{[·,·]}(·,·)$ are as given in (8) and (19), then we have

$$\begin{array}{ccc}\hfill & & \left|\zeta \left(\upsilon \right)-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)\zeta \left(g\left(\upsilon \right)\right)d\upsilon -\frac{(b_1-\upsilon )\zeta \left(a_1\right)+(\upsilon -a_1)\zeta \left(b_1\right)}{b_1-a_1}\right.\hfill \\ & & -\frac{{\zeta }^{\prime }\left(a_1\right)-{\zeta }^{\prime }\left(b_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G(g\left(\upsilon \right),s)d\upsilon \right)ds\hfill \\ & & -\sum _{k=2}^{n-1}\frac{k}{(k-1)!}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G(g\left(\upsilon \right),s)d\upsilon \right)\hfill \\ & & \left.\times \frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\right|\hfill \\ & & \le {\left({\int }_{a_1}^{b_1}{\left|{\lambda }_7\left(s\right)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\parallel }_p,\hfill \end{array}$$

(45)

where

$$\begin{array}{c}\hfill {\lambda }_7\left(s\right)=\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G(g\left(\upsilon \right),s)d\upsilon \right){\tilde{\tau }}_{n-2}^{[a_1,b_1]}(s,t)ds.\end{array}$$

Moreover, the following inequality holds

$$\begin{array}{ccc}\hfill & & \left|\zeta \left(\upsilon \right)-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)\zeta \left(g\left(\upsilon \right)\right)d\upsilon -\frac{(b_1-\upsilon )\zeta \left(a_1\right)+(\upsilon -a_1)\zeta \left(b_1\right)}{b_1-a_1}\right.\hfill \\ & & -\frac{{\zeta }^{\prime }\left(b_1\right)-{\zeta }^{\prime }\left(a_1\right)}{b_1-a_1}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G(g\left(\upsilon \right),s)d\upsilon \right)ds\hfill \\ & & -\sum _{k=3}^{n-1}\frac{k-2}{(k-1)!}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G(g\left(\upsilon \right),s)d\upsilon \right)\hfill \\ & & \left.\times \frac{{\zeta }^{\left(k\right)}\left(a_1\right){(s-a_1)}^{k-1}-{\zeta }^{\left(k\right)}\left(b_1\right){(s-b_1)}^{k-1}}{b_1-a_1}ds\right|\hfill \\ & & \le {\left({\int }_{a_1}^{b_1}{\left|{\lambda }_8\left(s\right)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\parallel }_p,\hfill \end{array}$$

(46)

where

$$\begin{array}{c}\hfill {\lambda }_8\left(s\right)=\frac{1}{(n-3)!}{\int }_{a_1}^{b_1}\left(G(\upsilon ,s)-{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)G(g\left(\upsilon \right),s)d\upsilon \right){\tau }_{n-2}^{[a_1,b_1]}(s,t)ds.\end{array}$$

for ${\tau }_n^{[·,·]}(·,·)$ as defined in (5).

4. Estimations of the Difference of Two Integral Means

In this section, we generalize the results from [17,18].

We have four possible cases for the two intervals $[a_1,b_1]$ and $[c_1,d_1]$ if $[a_1,b_1]\cup [c_1,d_1]\ne \varnothing$. The first case is $[c_1,d_1]\subset [a_1,b_1]$ and the second $[a_1,b_1]\cap [c_1,d_1]=[c_1,b_1].$ We simply get the other two possible cases by interchanging $a_1\leftrightarrow c_1,b_1\leftrightarrow d_1.$

Theorem 15.

Let $\zeta :[a_1,b_1]\cup [c_1,d_1]\to \mathbb{R}$ be a function such that ${\zeta }^{(n-1)}\in AC\left(I\right)$ function for some $n\ge 2.$ If $\upsilon \in [a_1,b_1]\cup [c_1,d_1]\ne \varnothing ,$ and $p_i\in \mathbb{R}$ for $i\in \{1,2,\dots ,m\}$ ($m\in \mathbb{N}$) such that ${\sum }_{i=1}^m{p}_i=0$, then we have

$$\begin{array}{c}\hfill \frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt-\frac{1}{d_1-c_1}{\int }_{c_1}^{d_1}\zeta \left(t\right)dt-D_n^1[a_1,b_1]\left(\upsilon \right)+D_n^1[c_1,d_1]\left(\upsilon \right)\\ \hfill =\frac{1}{(n-1)!}{\int }_{min\{a_1,c_1\}}^{max\{b_1,d_1\}}{K}_n(\upsilon ,s){\zeta }^{\left(n\right)}\left(s\right)ds,\end{array}$$

(47)

where in the case $[c_1,d_1]\subset [a_1,b_1]$

$$\begin{array}{ccc}& & K_n(\upsilon ,s)\hfill \\ & & =\left\{\begin{array}{cc}{\displaystyle \sum _{i=1}^m{p}_i{\tau }_n^{[a_1,b_1]}({\upsilon }_i,s)-{\tau }_n^{[a_1,b_1]}(\upsilon ,s),}& s\in [a_1,c_1\rangle ,\\ {\displaystyle \sum _{i=1}^m{p}_i\left[{\tau }_n^{[a_1,b_1]}({\upsilon }_i,s)-{\tau }_n^{[c_1,d_1]}({\upsilon }_i,s)\right]}\\ {\displaystyle -{\tau }_n^{[a_1,b_1]}(\upsilon ,s)+{\tau }_n^{[c_1,d_1]}(\upsilon ,s),}& s\in \langle c_1,d_1],\\ {\displaystyle \sum _{i=1}^m{p}_i{\tau }_n^{[a_1,b_1]}({\upsilon }_i,s)-{\tau }_n^{[a_1,b_1]}(\upsilon ,s).}& s\in \langle d_1,b_1],\end{array}\right.\hfill \end{array}$$

and in the case $[a_1,b_1]\cap [c_1,d_1]=[c_1,b_1]$

$$\begin{array}{ccc}& & K_n(\upsilon ,s)\hfill \\ & & =\left\{\begin{array}{cc}{\displaystyle \sum _{i=1}^m{p}_i{\tau }_n^{[a_1,b_1]}({\upsilon }_i,s)-{\tau }_n^{[a_1,b_1]}(\upsilon ,s),}& s\in [a_1,c_1\rangle ,\\ {\displaystyle \sum _{i=1}^m{p}_i\left[{\tau }_n^{[a_1,b_1]}({\upsilon }_i,s)-{\tau }_n^{[c_1,d_1]}({\upsilon }_i,s)\right]}\\ {\displaystyle -{\tau }_n^{[a_1,b_1]}(\upsilon ,s)+-{\tau }_n^{[c_1,d_1]}(\upsilon ,s),}& s\in \langle c_1,b_1],\\ {\displaystyle -\sum _{i=1}^m{p}_i{\tau }_n^{[c_1,d_1]}({\upsilon }_i,s)-{\tau }_n^{[c_1,d_1]}(\upsilon ,s).}& s\in \langle b_1,d_1].\end{array}\right.\hfill \end{array}$$

Proof. 

By taking the difference of the identities in (28) for intervals $[a_1,b_1]$ and $[c_1,d_1]$, we get Formula (47). □

Theorem 16.

Assume $(p,q)$ is a pair of conjugate exponents, that is $1<p,q<\infty$ and $\frac{1}{p}+\frac{1}{q}=1.$ Let $|{\zeta }^{\left(n\right)}{|}^p:[a_1,b_1]\to \mathbb{R}$ be an R-integrable function for some $n\ge 2.$ Then, we have

$$\begin{array}{c}\hfill \left|\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt-\frac{1}{d_1-c_1}{\int }_{c_1}^{d_1}\zeta \left(t\right)dt-D_n^1[a_1,b_1]\left(\upsilon \right)+D_n^1[c_1,d_1]\left(\upsilon \right)\right|\\ \hfill \le \frac{1}{(n-1)!}{\left({\int }_{min\{a_1,c_1\}}^{max\{b_1,d_1\}}{\left|K_n(\upsilon ,s)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\left(s\right)\parallel }_p,\end{array}$$

(48)

for every $\upsilon \in [a_1,b_1]\cap [c_1,d_1].$ The constant ${\left({\int }_{min\{a_1,c_1\}}^{max\{b_1,d_1\}}{\left|K_n(\upsilon ,s)\right|}^{q}ds\right)}^{\frac{1}{q}}$ is sharp for $1<p\le \infty$ and the best possible for $p=1.$

Proof. 

Using the identity (47) and applying the Hölder inequality, we obtain

$$\begin{array}{ccc}& & \left|\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt-\frac{1}{d_1-c_1}{\int }_{c_1}^{d_1}\zeta \left(t\right)dt-D_n^1[a_1,b_1]\left(\upsilon \right)+D_n^1[c_1,d_1]\left(\upsilon \right)\right|\hfill \\ & & \le \frac{1}{(n-1)!}{\int }_{min\{a_1,c_1\}}^{max\{b_1,d_1\}}\left|K_n(\upsilon ,s)\right|\left|{\zeta }^{\left(n\right)}\left(s\right)\right|ds\hfill \\ & & \le \frac{1}{(n-1)!}{\left({\int }_{min\{a_1,c_1\}}^{max\{b_1,d_1\}}{\left|K_n(\upsilon ,s)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\left(s\right)\parallel }_p,\hfill \end{array}$$

which proves the inequality. The proofs for the best possibility and the sharpness are similar to Theorem 10. □

Theorem 17.

Let $\zeta :[a_1,b_1]\cup [c_1,d_1]\to \mathbb{R}$ be a function such that ${\zeta }^{(n-1)}\in AC\left(I\right)$ function for some $n\ge 2$ and $g:[{\alpha }_1,{\beta }_1]\to [a_1,b_1]$ and $p:[{\alpha }_1,{\beta }_1]\to \mathbb{R}$ be integrable functions such that ${\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)d\upsilon =0.$ If $\upsilon \in [a_1,b_1]\cup [c_1,d_1]\ne \varnothing ,$ then we have

$$\begin{array}{c}\hfill \frac{1}{b_1-a_1}{\int }_{a_1}^b\zeta \left(t\right)dt-\frac{1}{d_1-c_1}{\int }_{c_1}^{d_1}\zeta \left(t\right)dt-D_n^2[a_1,b_1]\left(\upsilon \right)+D_n^2[c_1,d_1]\left(\upsilon \right)\\ \hfill =\frac{1}{(n-1)!}{\int }_{min\{a_1,c_1\}}^{max\{b_1,d_1\}}{M}_n(\upsilon ,s){\zeta }^{\left(n\right)}\left(s\right)ds,\end{array}$$

(49)

where in the case $[c_1,d_1]\subset [a_1,b_1]$

$$\begin{array}{ccc}& & M_n(\upsilon ,s)\hfill \\ & & =\left\{\begin{array}{cc}{\displaystyle {\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){\tau }_n^{[a_1,b_1]}(g\left(\upsilon \right),s)d\upsilon -{\tau }_n^{[a_1,b_1]}(\upsilon ,s),}& s\in [a_1,c_1\rangle ,\\ {\displaystyle {\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)\left[{\tau }_n^{[a_1,b_1]}(g\left(\upsilon \right),s)-{\tau }_n^{[c_1,d_1]}(g\left(\upsilon \right),s)\right]d\upsilon }\\ {\displaystyle -{\tau }_n^{[a_1,b_1]}(\upsilon ,s)+{\tau }_n^{[c_1,d_1]}(\upsilon ,s),}& s\in \langle c_1,d_1],\\ {\displaystyle {\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){\tau }_n^{[a_1,b_1]}(g\left(\upsilon \right),s)d\upsilon -{\tau }_n^{[a_1,b_1]}(\upsilon ,s).}& s\in \langle d_1,b_1],\end{array}\right.\hfill \end{array}$$

and in the case $[a_1,b_1]\cap [c_1,d_1]=[c_1,b_1]$

$$\begin{array}{ccc}& & M_n(\upsilon ,s)\hfill \\ & & =\left\{\begin{array}{cc}{\displaystyle {\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){\tau }_n^{[a_1,b_1]}(g\left(\upsilon \right),s)d\upsilon -{\tau }_n^{[a_1,b_1]}(\upsilon ,s),}& s\in [a_1,c_1\rangle ,\\ {\displaystyle {\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right)\left[{\tau }_n^{[a_1,b_1]}(g\left(\upsilon \right),s)-{\tau }_n^{[c_1,d_1]}(g\left(\upsilon \right),s)\right]d\upsilon }\\ {\displaystyle -{\tau }_n^{[a_1,b_1]}(\upsilon ,s)+{\tau }_n^{[c_1,d_1]}(\upsilon ,s),}& s\in \langle c_1,b_1],\\ {\displaystyle -{\int }_{{\alpha }_1}^{{\beta }_1}p\left(\upsilon \right){\tau }_n^{[c_1,d_1]}(g\left(\upsilon \right),s)d\upsilon -{\tau }_n^{[c_1,d_1]}(\upsilon ,s).}& s\in \langle b_1,d_1].\end{array}\right.\hfill \end{array}$$

Proof. 

By taking the difference of the identities in (33) for interval $[a_1,b_1]$ and $[c_1,d_1]$ we get the Formula (49). □

Theorem 18.

Assume $(p,q)$ is a pair of conjugate exponents, that is $1<p,q<\infty$ and $\frac{1}{p}+\frac{1}{q}=1.$ Let $|{\zeta }^{\left(n\right)}{|}^p:[a_1,b_1]\to \mathbb{R}$ be an R-integrable function for some $n\ge 2.$ Then, we have

$$\begin{array}{c}\hfill \left|\frac{1}{b-a}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt-\frac{1}{d_1-c_1}{\int }_{c_1}^{d_1}\zeta \left(t\right)dt-D_n^2[a_1,b_1]\left(\upsilon \right)+D_n^2[c_1,d_1]\left(\upsilon \right)\right|\\ \hfill \le \frac{1}{(n-1)!}{\left({\int }_{min\{a_1,c_1\}}^{max\{b_1,d_1\}}{\left|M_n(\upsilon ,s)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\left(s\right)\parallel }_p,\end{array}$$

(50)

$\forall \upsilon \in [a_1,b_1]\cap [c_1,d_1].$ The constant ${\left({\int }_{min\{a_1,c_1\}}^{max\{b_1,d_1\}}{\left|M_n(\upsilon ,s)\right|}^{q}ds\right)}^{\frac{1}{q}}$ is the best possible for $p=1$ and sharp for $1<p\le \infty .$

Proof. 

Using the identity (49) and applying the Hölder inequality, we obtain

$$\begin{array}{ccc}& & \left|\frac{1}{b_1-a_1}{\int }_{a_1}^{b_1}\zeta \left(t\right)dt-\frac{1}{d_1-c_1}{\int }_{c_1}^{d_1}\zeta \left(t\right)dt-D_n^2[a_1,b_1]\left(\upsilon \right)+D_n^2[c_1,d_1]\left(\upsilon \right)\right|\hfill \\ & & \le \frac{1}{(n-1)!}{\int }_{min\{a_1,c_1\}}^{max\{b_1,d_1\}}\left|M_n(\upsilon ,s)\right|\left|{\zeta }^{\left(n\right)}\left(s\right)\right|ds\hfill \\ & & \le \frac{1}{(n-1)!}{\left({\int }_{min\{a_1,c_1\}}^{max\{b_1,d_1\}}{\left|M_n(\upsilon ,s)\right|}^{q}ds\right)}^{\frac{1}{q}}{\parallel {\zeta }^{\left(n\right)}\left(s\right)\parallel }_p,\hfill \end{array}$$

which proves the inequality. The proofs for the best possibility and sharpness are similar to Theorem 10. □

5. Conclusions

In this paper, we obtained some new Ostrowski type inequalities via the extended version of Montgomery’s identity and new Green’s functions. The results we acquired contain the identities for the sum and the integral. We also estimated the difference between two integral means. These results were obtained by taking the difference of the extended Montgomery identity for its sum and integral identities. It would be interesting to explore whether our method can be used to find such results for the weighted version of the extended Montgomery identity. Thus, the study of the estimation of the difference between two weighted integral means is a suggested future work.