Generalized n-Polynomial p-Convexity and Related Inequalities

Abstract

In this paper, we construct a new class of convex functions, so-called generalized n-polynomial p-convex functions. We investigate their algebraic properties and provide some relationships between these functions and other types of convex functions. We establish Hermite–Hadamard (H–H) inequality for the newly defined class of functions. Additionally, we derive refinements of H–H inequality for functions whose first derivatives in absolute value at certain power are generalized n-polynomial p-convex. When $p=-1$, our definition evolves into a new definition for the class of convex functions so-called generalized n-polynomial harmonically convex functions. The results obtained in this study generalize regarding those found in the existing literature. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields.

1. Introduction

Let $\aleph \subset$ $\mathbb{R}$ be an interval. Then, a function $\mathsf{\Lambda }:\aleph \to \mathbb{R}$ is said to be convex on ℵ if

$$\mathsf{\Lambda }\left(\mu \sigma +(1-\mu )\xi \right)\le \mu \mathsf{\Lambda }\left(\sigma \right)+\left(1-\mu \right)\mathsf{\Lambda }\left(\xi \right)$$

(1)

holds for all $\sigma ,\xi \in \aleph$ and $\mu \in [0,1]$.

Convex functions play a fundamental role in the realm of mathematical analysis, offering valuable insights into the principles of optimization, approximation and variational calculus. Their significance extends across a multitude of scientific and engineering disciplines, where they serve as indispensable tools for modeling complex phenomena and solving challenging problems.

The study of convex functions has garnered significant attention in recent decades, with researchers exploring their theoretical foundations, analytical properties and practical implications across various domains. For instance, in the realm of economics, convex functions play a pivotal role in modeling utility functions, production functions and cost functions, providing insights into consumer behavior, market equilibrium and resource allocation [1]. Moreover, convex optimization techniques have been extensively utilized in finance for portfolio optimization, risk management and option pricing, where convexity assumptions underpin many quantitative models [2].

In addition to these traditional applications, convexity concepts have found diverse and innovative applications in emerging fields such as quantum convexity and multiplicative convexity. In quantum information theory, for example, quantum convexity theory provides a framework for understanding the geometry of quantum states and operations, with applications in quantum communication, cryptography and computing [3,4]. Similarly, in the realm of multiplicative calculus, convexity theory offers novel perspectives on the behavior of multiplicative convex functions and their derivatives, leading to applications in various fields of pure and applied sciences [5,6,7,8,9]. In optimization theory, for instance, quasi-convex functions arise naturally in problems involving constraints that are not necessarily convex but possess certain desirable properties, leading to efficient algorithms for solving non-convex optimization problems [10]. Moreover, in image processing and computer vision, pseudo-convex optimization techniques are employed for image denoising, segmentation and reconstruction, leveraging the robustness and stability properties of pseudo-convex functions [11].

Among the pivotal discoveries in convexity theory is H–H inequality, unveiled by C. Hermite and J. Hadamard [12,13]. This inequality stands as one of the cornerstones in the broader framework of convexity and inequalities, contributing significantly to various scientific disciplines. It is stated as follows:

Let $\mathsf{\Lambda }:\left[\hslash ,\ell \right]\to \mathbb{R}$ be a convex function in ℵ for $\hslash ,\ell \in \aleph$ and $\hslash <\ell$. If $\mathsf{\Lambda }\in L\left[\hslash ,\ell \right]$, then

$$\mathsf{\Lambda }\left(\frac{\hslash +\ell }{2}\right)\le \frac{1}{\ell -\hslash }{\int }_{\hslash }^{\ell }\mathsf{\Lambda }\left(\sigma \right)d\sigma \le \frac{\mathsf{\Lambda }\left(\hslash \right)+\mathsf{\Lambda }(\ell )}{2}.$$

(2)

In recent years, there has been considerable interest in the generalization of convex functions and H–H inequality. Researchers in this field are focusing on expanding the boundaries of traditional convex functions and related inequalities to discover new mathematical insights and relationships. Various articles published in recent years reflect the increasing interest and research efforts in this area. Some refinements, generalizations and enhancements of convex functions and inequality (2) can be found in recent papers [14,15,16,17,18,19,20,21,22,23]. These examples highlight the growing interest and importance of generalizing convex functions and related inequalities in contemporary mathematical research. By exploring new avenues of generalization, researchers aim to enhance our understanding of fundamental mathematical concepts and develop more powerful analytical tools with applications across various scientific disciplines.

2. Preliminaries

Throughout this study, ℵ will denote an interval and j will denote an integer between 1 and n.

Definition 1 

([24]). Let ℵ be an interval and $p\in \mathbb{R}-\left\{0\right\}$. Then, ℵ is called a p-convex set if ${\left(\mu {\sigma }^p+(1-\mu ){\xi }^p\right)}^{\frac{1}{p}}\in \aleph$ for all $\sigma ,\xi \in \aleph ,$ $p>0$ and $\mu \in \left[0,1\right].$

Definition 2 

([24]). A function $\mathsf{\Lambda }:\aleph \to \mathbb{R}$ is said to be p-convex if

$$\mathsf{\Lambda }\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)\le \mu \mathsf{\Lambda }\left(\sigma \right)+\left(1-\mu \right)\mathsf{\Lambda }\left(\xi \right)$$

holds for all $\sigma ,\xi \in \aleph =\left[\hslash ,\ell \right]$ and $\mu \in \left[0,1\right]$, where $p>0.$

It is obvious that, for $p=1,$ p-convexity coincides with the classical convexity on $\aleph \subset \left(0,\infty \right).$

Remark 1. 

An interval ℵ is called a p-convex set if ${\left(\mu {\sigma }^p+(1-\mu ){\xi }^p\right)}^{\frac{1}{p}}\in \aleph$ for all $\sigma ,\xi \in \aleph$ and $\mu \in \left[0,1\right],$ where $p=2r+1$ or $p=\frac{s}{t},$ $t=2u+1,$ $s=2v+1$ and $r,u,v\in \mathbb{N}.$

Remark 2. 

Let $\aleph \subset \left(0,\infty \right)$ be a real interval and $p\in \mathbb{R}-\left\{0\right\}.$ Then, ${\left(\mu {\sigma }^p+(1-\mu ){\xi }^p\right)}^{\frac{1}{p}}\in \aleph$ for all $\sigma ,\xi \in \aleph$ and $\mu \in \left[0,1\right].$

According to Remark 2, in [25], İşcan introduced a different version of p-convexity as follows:

Definition 3. 

Let $\aleph \subset \left(0,\infty \right)$ be a real interval and $p\in \mathbb{R}-\left\{0\right\}.$ A function $\mathsf{\Lambda }:\aleph \to \mathbb{R}$ is called p-convex if

$$\mathsf{\Lambda }\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)\le \mu \mathsf{\Lambda }\left(\sigma \right)+\left(1-\mu \right)\mathsf{\Lambda }\left(\xi \right)$$

(3)

for all $\sigma ,\xi \in \aleph$ and $\mu \in \left[0,1\right].$

Definition 4 

([26]). Let $\aleph \subseteq \mathbb{R}-\left\{0\right\}$ be an interval. A function $\mathsf{\Lambda }:\aleph \to \mathbb{R}$ is called harmonically convex if

$$\mathsf{\Lambda }\left(\frac{\sigma \xi }{\mu \sigma +(1-\mu )\xi }\right)\le \mu \mathsf{\Lambda }\left(\xi \right)+\left(1-\mu \right)\mathsf{\Lambda }\left(\sigma \right)$$

holds for all $\sigma ,\xi \in \aleph =\left[\hslash ,\ell \right]$ and $\mu \in \left[0,1\right].$

Definition 5 

([27]). Let $n\in \mathbb{N}.$ A non-negative function $\mathsf{\Lambda }:\aleph \to \mathbb{R}$ is called n-polynomial convex if

$$\mathsf{\Lambda }\left(\mu \sigma +(1-\mu )\xi \right)\le \frac{1}{n}\sum _{j=1}^n\left[1-{(1-\mu )}^j\right]\mathsf{\Lambda }\left(\sigma \right)+\frac{1}{n}\sum _{j=1}^n\left[1-{\mu }^j\right]\mathsf{\Lambda }\left(\xi \right)$$

(4)

for all $\sigma ,\xi \in \aleph$ and $\mu \in \left[0,1\right].$

Definition 6 

([28]). Let $n\in \mathbb{N}$ and $\aleph \subseteq \left(0,\infty \right)$ be an interval. A non-negative function $\mathsf{\Lambda }:\aleph \to \left[0,\infty \right)$ is called n-polynomial harmonically convex if

$$\mathsf{\Lambda }\left(\frac{\sigma \xi }{\mu \sigma +(1-\mu )\xi }\right)\le \frac{1}{n}\sum _{j=1}^n\left[1-{(1-\mu )}^j\right]\mathsf{\Lambda }\left(\xi \right)+\frac{1}{n}\sum _{j=1}^n\left[1-{\mu }^j\right]\mathsf{\Lambda }\left(\sigma \right)$$

(5)

for all $\sigma ,\xi \in \aleph$ and $\mu \in \left[0,1\right].$

Definition 7 

([29]). Let $n\in \mathbb{N}.$ A non-negative function $\mathsf{\Lambda }:\aleph \to \mathbb{R}$ is called n-polynomial p-convex if

$$\mathsf{\Lambda }\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)\le \frac{1}{n}\sum _{j=1}^n\left[1-{(1-\mu )}^j\right]\mathsf{\Lambda }\left(\sigma \right)+\frac{1}{n}\sum _{j=1}^n\left[1-{\mu }^j\right]\mathsf{\Lambda }\left(\xi \right)$$

(6)

for all $\sigma ,\xi \in \aleph ,$ $p>0$ and $\mu \in \left[0,1\right].$

Remark 3. 

For $n=1$ and $p=1,$ Definition 7 reduces to the definition of classical convexity.

Remark 4. 

For $n=1$ and $p=-1,$ Definition 7 reduces to Definition 4.

Remark 5. 

For $p=1,$ Definition 7 reduces to Definition 5.

Remark 6. 

For $p=-1,$ Definition 7 reduces to Definition 6.

Definition 8 

([30]). Let $n\in \mathbb{N}$ and ${\alpha }_j\ge 0$ $\left(j=\overline{1,n}\right)$ such that ${\sum }_{j=1}^n{\alpha }_j>0.$ A non-negative function $\mathsf{\Lambda }:\aleph \subset \mathbb{R}\to \mathbb{R}$ is called generalized n-polynomial convex function if, for every $\sigma ,\xi \in \aleph$ and $\mu \in \left[0,1\right]$,

$$\mathsf{\Lambda }\left(\mu \sigma +(1-\mu )\xi \right)\le \frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\xi \right).$$

(7)

We note that every non-negative convex function is also generalized n-polynomial convex function [30].

3. Main Results

In this section, we first introduce the class of generalized n-polynomial p-convex functions and investigate some of their algebraic properties. Following this introduction, we present the H–H inequality for this newly defined class of functions. Subsequently, we establish several refinements of the H–H inequality for this novel generalization.

Definition 9. 

Let $n\in \mathbb{N}$, ${\alpha }_j\ge 0$ $\left(j=\overline{1,n}\right)$ such that ${\sum }_{j=1}^n{\alpha }_j>0$ and $p\in \mathbb{R}-\left\{0\right\}$. A non-negative function $\mathsf{\Lambda }:\aleph \subseteq \left(0,\infty \right)\to \mathbb{R}$ is called generalized n-polynomial p-convex $(GPOLC-p)$ if for every $\sigma ,\xi \in \aleph$ and $\mu \in \left[0,1\right]$,

$$\begin{array}{c}\hfill \mathsf{\Lambda }\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)\le \frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\xi \right).\end{array}$$

(8)

Remark 7. 

If $\mathsf{\Lambda }:\aleph \subseteq \left(0,\infty \right)\to \mathbb{R}$ is a $GPOLC-p$ function, then $\mathsf{\Lambda }$ is a non-negative function. Indeed, since $\mathsf{\Lambda }$ is a $GPOLC-p$ function, one can write

$$\begin{array}{ccc}\hfill \mathsf{\Lambda }\left(\sigma \right)& =& \mathsf{\Lambda }\left({\left[\mu {\sigma }^p+(1-\mu ){\sigma }^p\right]}^{\frac{1}{p}}\right)\hfill \\ & \le & \left[\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\right]\mathsf{\Lambda }\left(\sigma \right)\hfill \end{array}$$

for all $\sigma \in \aleph$ and $\mu \in \left[0,1\right].$ Therefore, one has

$$\left[\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}-1\right]\mathsf{\Lambda }\left(\sigma \right)\ge 0$$

(9)

for all $\sigma \in \aleph$ and $\mu \in \left[0,1\right].$ But, since

$$\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\ge \mu \mathrm{and}\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\ge 1-\mu$$

for all $\mu \in \left[0,1\right]$ in inequality (9), one can write

$$\left[\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}-1\right]\ge 0.$$

So, one obtains $\mathsf{\Lambda }\left(\sigma \right)\ge 0$ for all $\sigma \in \aleph .$

Remark 8. 

For $n=1$ and $p=1,$ Definition 9 reduces to the definition of classical convexity.

Remark 9. 

For ${\alpha }_j=1$ $\left(j=\overline{1,n}\right)$ and $p=1,$ Definition 9 reduces to Definition 5.

Remark 10. 

For $p=1,$ Definition 9 reduces to Definition 8.

Remark 11. 

For ${\alpha }_j=1$ $\left(j=\overline{1,n}\right),$ Definition 9 reduces to Definition 7.

Remark 12. 

If one takes $p=-1$ in Definition 9, then one obtains

$$\mathsf{\Lambda }\left(\frac{\sigma \xi }{\mu \xi +\left(1-\mu \right)\sigma }\right)\le \frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\xi \right).$$

Such functions are called generalized n-polynomial harmonically convex functions.

Remark 13. 

For $n=1$, Definition 9 reduces to Definition 3. So, one can say every p-convex function is generalized 1-polynomial p-convex.

Proposition 1. 

Let $n\in \mathbb{N}$, ${\alpha }_j\ge 0$ $\left(j=\overline{1,n}\right)$ such that ${\sum }_{j=1}^n{\alpha }_j>0,$ $p\in \mathbb{R}-\left\{0\right\}$ and $\mathsf{\Lambda },\mathsf{\Theta }:\aleph \subseteq \left(0,\infty \right)\to \mathbb{R}$ be two non-negative $GPOLC-p$ functions. Then, the function $\mathsf{\Lambda }+\mathsf{\Theta }$ is $GPOLC-p$.

Proof. 

Assume $\mathsf{\Lambda }$ and $\mathsf{\Theta }$ are two $GPOLC-p$ functions, and then one has

$$\begin{array}{ccc}& & \left(\mathsf{\Lambda }+\mathsf{\Theta }\right)\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)\hfill \\ & =& \mathsf{\Lambda }\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)+\mathsf{\Theta }\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)\hfill \\ & \le & \frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\xi \right)\hfill \\ & & +\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Theta }\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Theta }\left(\xi \right)\hfill \\ & =& \frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\left(\mathsf{\Lambda }+\mathsf{\Theta }\right)\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\left(\mathsf{\Lambda }+\mathsf{\Theta }\right)\left(\xi \right)\hfill \end{array}$$

for all $\sigma ,\xi \in \aleph$ and $\mu \in \left[0,1\right].$ This shows that $\mathsf{\Lambda }+\mathsf{\Theta }$ is a $GPOLC-p$ function. □

Proposition 2. 

Let $n\in \mathbb{N}$, ${\alpha }_j\ge 0$ $\left(j=\overline{1,n}\right)$ such that ${\sum }_{j=1}^n{\alpha }_j>0,$ $p\in \mathbb{R}-\left\{0\right\}$ and $\mathsf{\Lambda }:\aleph \subseteq \left(0,\infty \right)\to \mathbb{R}$ be a non-negative $GPOLC-p$ function and $\theta \ge 0$. Then, the function $\theta \mathsf{\Lambda }:\aleph \to \mathbb{R}$ is a $GPOLC-p$ function.

Proof. 

Assume $\mathsf{\Lambda }$ is a $GPOLC-p$ function, and then one has

$$\begin{array}{ccc}& & \left(\theta \mathsf{\Lambda }\right)\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)\hfill \\ & =& \theta \left(\mathsf{\Lambda }{\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)\hfill \\ & \le & \theta \left[\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\xi \right)\right]\hfill \\ & =& \frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\left(\theta \mathsf{\Lambda }\right)\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\left(\theta \mathsf{\Lambda }\right)\left(\xi \right)\hfill \end{array}$$

for all $\sigma ,\xi \in \aleph$ and $\mu \in \left[0,1\right].$ This ensures that $\theta \mathsf{\Lambda }$ is a $GPOLC-p$ function. □

Proposition 3. 

Let $n\in \mathbb{N}$, ${\alpha }_j\ge 0$ $\left(j=\overline{1,n}\right)$ such that ${\sum }_{j=1}^n{\alpha }_j>0,$ $p\in \mathbb{R}-\left\{0\right\}$ and the function $\mathsf{\Lambda }:\aleph \subseteq \left(0,\infty \right)\to \mathbb{R}$ be non-negative $GPOLC-p$ function. Then,

$$\mathsf{\Lambda }=max\left\{{\mathsf{\Lambda }}_k,k=1,2,\dots ,n\right\}$$

is also a $GPOLC-p$ function.

Proof. 

Take any $\sigma ,\xi \in \aleph$ and $\mu \in \left[0,1\right].$ Denote $\mathsf{\Lambda }=max{\mathsf{\Lambda }}_k,$ where $k=1,2,\dots ,n.$ Then, one obtains

$$\begin{array}{ccc}& & \mathsf{\Lambda }\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)\hfill \\ & =& max\left\{{\mathsf{\Lambda }}_k\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right),k=1,2,\dots ,n\right\}\hfill \\ & =& {\mathsf{\Lambda }}_i\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)\hfill \\ & \le & \frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}{\mathsf{\Lambda }}_i\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}{\mathsf{\Lambda }}_i\left(\xi \right)\hfill \\ & =& \frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}max{\mathsf{\Lambda }}_k\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}max{\mathsf{\Lambda }}_k\left(\xi \right)\hfill \\ & =& \frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\xi \right),\hfill \end{array}$$

which is the desired outcome. □

Proposition 4. 

Let $n\in \mathbb{N}$, ${\alpha }_j\ge 0$ $\left(j=\overline{1,n}\right)$ such that ${\sum }_{j=1}^n{\alpha }_j>0,$ $p\in \mathbb{R}-\left\{0\right\}$ and $\mathsf{\Lambda }:\aleph \subseteq \left(0,\infty \right)\to \mathbb{R}$ be a non-negative $GPOLC-p$ function. Then,

$$\mathsf{\Lambda }=sup{\mathsf{\Lambda }}_k,k=1,2,\dots ,n$$

is also a $GPOLC-p$ function.

Proof. 

Take any $\sigma ,\xi \in \aleph$ and $\mu \in \left[0,1\right].$ Denote $\mathsf{\Lambda }=sup{\mathsf{\Lambda }}_k,$ $k=1,2,\dots ,n.$ Then, one obtains

$$\begin{array}{ccc}& & \mathsf{\Lambda }\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)\hfill \\ & =& sup{\mathsf{\Lambda }}_k\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right),k=1,2,\dots ,n\hfill \\ & \le & \frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}sup{\mathsf{\Lambda }}_k\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}sup{\mathsf{\Lambda }}_k\left(\xi \right)\hfill \\ & =& \frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\xi \right).\hfill \end{array}$$

This justifies the supremum function is a $GPOLC-p$ function. □

Proposition 5. 

Let $\aleph \subseteq \left(0,\infty \right)$ be a real interval, $p\in \mathbb{R}-\left\{0\right\}$ and $\mathsf{\Lambda }:\aleph \to \mathbb{R}$ is a function. Then,

1.

If $p\le 1$ and $\mathsf{\Lambda }$ is $GPOLC-p$ and non-decreasing function, then $\mathsf{\Lambda }$ is $GPOLC-p$ function.

2.

If $p\ge 1$ and $\mathsf{\Lambda }$ is $GPOLC-p$ and non-decreasing function, then $\mathsf{\Lambda }$ is generalized n-polynomial convex function.

3.

If $p\ge 1$ and $\mathsf{\Lambda }$ is generalized n-polynomial convex and non-decreasing function, then $\mathsf{\Lambda }$ is $GPOLC-p$ function.

4.

If $p\le 1$ and $\mathsf{\Lambda }$ is $GPOLC-p$ and non-decreasing function, then $\mathsf{\Lambda }$ is generalized n-polynomial convex function.

Proof. 

1. If $p\le 1$ then,

$${\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\le \mu \sigma +(1-\mu )\xi .$$

Also, if the function $\mathsf{\Lambda }$ is non-decreasing and generalized n-polynomial convex, then one has

$$\begin{array}{ccc}& & \mathsf{\Lambda }\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)\hfill \\ & \le & \mathsf{\Lambda }\left(\mu \sigma +(1-\mu )\xi \right)\hfill \\ & \le & \frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\xi \right).\hfill \end{array}$$

So, $\mathsf{\Lambda }$ is a $GPOLC-p$ function.

2. If $p\ge 1$, then

$${\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\ge \mu \sigma +(1-\mu )\xi .$$

Also, if the function $\mathsf{\Lambda }$ is non-decreasing and $GPOLC-p$ function, then one has

$$\begin{array}{ccc}& & \mathsf{\Lambda }\left(\mu \sigma +(1-\mu )\xi \right)\hfill \\ & \le & \mathsf{\Lambda }\left({\left[\mu {\sigma }^p+(1-\mu ){\xi }^p\right]}^{\frac{1}{p}}\right)\hfill \\ & \le & \frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\sigma \right)+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left(\xi \right).\hfill \end{array}$$

So, the function $\mathsf{\Lambda }$ is generalized n-polynomial convex.

One can obtain the proof of other items in a similar way. □

Now, we establish a novel generalization of H–H inequality for the $GPOLC-p$ function $\mathsf{\Lambda }$.

Theorem 1. 

Let $0<\hslash <\ell$ and the function $\mathsf{\Lambda }:\left[\hslash ,\ell \right]\to \mathbb{R}$ be $GPOLC-p$ function. If $\mathsf{\Lambda }\in L\left[\hslash ,\ell \right]$, where $p\in \mathbb{R}-\left\{0\right\},$ then

$$\begin{array}{ccc}& & \frac{1}{2}\left(\frac{{\sum }_{j=1}^n{\alpha }_j}{{\sum }_{j=1}^n{\alpha }_j\left[1-{\left(\frac{1}{2}\right)}^j\right]}\right)\mathsf{\Lambda }\left[{\left(\frac{{\hslash }^p+{\ell }^p}{2}\right)}^{\frac{1}{p}}\right]\hfill \\ & \le & \frac{p}{{\ell }^p-{\hslash }^p}{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^{1-p}}d\sigma \le \frac{\mathsf{\Lambda }\left(\hslash \right)+\mathsf{\Lambda }(\ell )}{{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j\left(\frac{j}{j+1}\right).\hfill \end{array}$$

(10)

Proof. 

Since $\mathsf{\Lambda }$ is a $GPOLC-p$ function, one has

$$\begin{array}{ccc}\hfill \mathsf{\Lambda }\left[{\left(\frac{{\hslash }^p+{\ell }^p}{2}\right)}^{\frac{1}{p}}\right]& =& \mathsf{\Lambda }\left[{\left(\frac{\left[\mu {\hslash }^p+(1-\mu ){\ell }^p\right]+\left[(1-\mu ){\hslash }^p+\mu {\ell }^p\right]}{2}\right)}^{\frac{1}{p}}\right]\hfill \\ & =& \mathsf{\Lambda }\left[{\left(\frac{\left[\mu {\hslash }^p+(1-\mu ){\ell }^p\right]}{2}+\frac{\left[(1-\mu ){\hslash }^p+\mu {\ell }^p\right]}{2}\right)}^{\frac{1}{p}}\right]\hfill \\ & \le & \frac{{\sum }_{j=1}^n{\alpha }_j\left[1-{\left(1-\frac{1}{2}\right)}^j\right]}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left[{\left(\mu {\hslash }^p+(1-\mu ){\ell }^p\right)}^{\frac{1}{p}}\right]\hfill \\ & & +\frac{{\sum }_{j=1}^n{\alpha }_j\left[1-{\left(\frac{1}{2}\right)}^j\right]}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }\left[{\left((1-\mu ){\hslash }^p+\mu {\ell }^p\right)}^{\frac{1}{p}}\right].\hfill \end{array}$$

Integrating the last inequality with respect to $\mu \in \left[0,1\right]$, one obtains

$$\begin{array}{ccc}& & \mathsf{\Lambda }\left[{\left(\frac{{\hslash }^p+{\ell }^p}{2}\right)}^{\frac{1}{p}}\right]\hfill \\ & \le & \frac{{\sum }_{j=1}^n{\alpha }_j\left[1-{\left(\frac{1}{2}\right)}^j\right]}{{\sum }_{j=1}^n{\alpha }_j}{\int }_0^1\mathsf{\Lambda }\left({\left[\mu {\hslash }^p+(1-\mu ){\ell }^p\right]}^{\frac{1}{p}}\right)d\mu \hfill \\ & & +\frac{{\sum }_{j=1}^n{\alpha }_j\left[1-{\left(\frac{1}{2}\right)}^j\right]}{{\sum }_{j=1}^n{\alpha }_j}{\int }_0^1\mathsf{\Lambda }\left({\left[(1-\mu ){\hslash }^p+\mu {\ell }^p\right]}^{\frac{1}{p}}\right)d\mu \hfill \\ & =& \frac{{\sum }_{j=1}^n{\alpha }_j\left[1-{\left(\frac{1}{2}\right)}^j\right]}{{\sum }_{j=1}^n{\alpha }_j}\hfill \\ & & \times {\int }_0^1\left(\mathsf{\Lambda }\left[{\left(\mu {\hslash }^p+(1-\mu ){\ell }^p\right)}^{\frac{1}{p}}\right]+\mathsf{\Lambda }\left({\left[(1-\mu ){\hslash }^p+\mu {\ell }^p\right]}^{\frac{1}{p}}\right)\right)d\mu .\hfill \end{array}$$

So, one obtains

$$\frac{1}{2}\left(\frac{{\sum }_{j=1}^n{\alpha }_j}{{\sum }_{j=1}^n{\alpha }_j\left[1-{\left(\frac{1}{2}\right)}^j\right]}\right)\mathsf{\Lambda }\left[{\left(\frac{{\hslash }^p+{\ell }^p}{2}\right)}^{\frac{1}{p}}\right]\le \frac{1}{{\ell }^p-{\hslash }^p}{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^{1-p}}d\sigma .$$

On the other hand, one has

$$\begin{array}{ccc}& & \frac{p}{{\ell }^p-{\hslash }^p}{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^{1-p}}d\sigma \hfill \\ & =& {\int }_0^1\mathsf{\Lambda }\left({\left[\mu {\hslash }^p+(1-\mu ){\ell }^p\right]}^{\frac{1}{p}}\right)d\mu \hfill \\ & \le & {\int }_0^1\left[\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }(\hslash )+\frac{{\sum }_{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)}{{\sum }_{j=1}^n{\alpha }_j}\mathsf{\Lambda }(\ell )\right]d\mu \hfill \\ & =& \frac{\mathsf{\Lambda }(\hslash )}{{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j{\int }_0^1\left[1-{(1-\mu )}^j\right]d\mu +\frac{\mathsf{\Lambda }(\ell )}{{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j{\int }_0^1\left[1-{\mu }^j\right]d\mu \hfill \\ & =& \frac{\mathsf{\Lambda }(\hslash )+\mathsf{\Lambda }(\ell )}{{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j\left(\frac{j}{j+1}\right).\hfill \end{array}$$

This completes the proof. □

Remark 14. 

For $n=1$ and $p=1,$ inequality (10) reduces to inequality (2) for classical convex functions.

Remark 15. 

If one takes ${\alpha }_j=1$ $\left(j=\overline{1,n}\right)$ and $p=1$ in inequality (10), then one obtains H–H inequality for n-polynomial convex functions in ([27], Theorem 4).

Remark 16. 

If one takes $p=1$ in inequality (10), then one obtains H–H inequality for generalized n-polynomial convex functions in ([30], Theorem 3.1).

Remark 17. 

If one takes $n=1$ in inequality (10), then one obtains H–H inequality for p-convex functions in ([25], Theorem 6).

Remark 18. 

If one takes ${\alpha }_j=1$ $\left(j=\overline{1,n}\right)$ in inequality (10), then one obtains H–H inequality for n-polynomial p-convex functions in ([29], Theorem 4.1).

Corollary 1. 

For $p=-1$ in Theorem 1, one obtains H–H type inequality for generalized n-polynomial harmonically convex functions as follows:

$$\begin{array}{ccc}& & \frac{1}{2}\left(\frac{{\sum }_{j=1}^n{\alpha }_j}{{\sum }_{j=1}^n{\alpha }_j\left[1-{\left(\frac{1}{2}\right)}^j\right]}\right)\mathsf{\Lambda }\left[\left(\frac{2\hslash \ell }{\hslash +\ell }\right)\right]\hfill \\ & \le & \frac{\hslash \ell }{\ell -\hslash }{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^2}d\sigma \le \frac{\mathsf{\Lambda }\left(\hslash \right)+\mathsf{\Lambda }(\ell )}{{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j\left(\frac{j}{j+1}\right).\hfill \end{array}$$

(11)

Remark 19. 

If one takes $n=1$ in inequality (11), then one obtains H–H inequality for harmonically convex functions in ([26], Theorem 2.4).

Now, we establish new refinements of H–H inequality for functions whose first derivatives in absolute value at certain power are $GPOLC-p$.

Lemma 1 

([20]). Let $\mathsf{\Lambda }:\aleph =\left[\hslash ,\ell \right]\subset \mathbb{R}\to \mathbb{R}$ be a differentiable mapping on ${\aleph }^{\circ }$ with $\hslash <\ell .$ If ${\mathsf{\Lambda }}^{\prime }\in L\left[\hslash ,\ell \right]$, then

$$\begin{array}{ccc}& & \frac{\mathsf{\Lambda }\left(\hslash \right)+\mathsf{\Lambda }(\ell )}{2}-\frac{p}{{\ell }^p-{\hslash }^p}{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^{1-p}}d\sigma \hfill \\ & =& \frac{{\ell }^p-{\hslash }^p}{2p}{\int }_0^1(1-2\mu ){\left(\mu {\hslash }^p+(1-\mu ){\ell }^p\right)}^{\frac{1}{p}-1}{\mathsf{\Lambda }}^{\prime }\left({\left[\mu {\hslash }^p+(1-\mu ){\ell }^p\right]}^{\frac{1}{p}}\right)d\mu .\hfill \end{array}$$

Theorem 2. 

Let $\mathsf{\Lambda }:\aleph \subseteq \left(0,\infty \right)\to \mathbb{R}$ be a differentiable function on ${\aleph }^{\circ }$, $\hslash ,\ell \in {\aleph }^{\circ }$ with $\hslash <\ell ,$ $p\in \mathbb{R}-\left\{0\right\}$ and ${\mathsf{\Lambda }}^{\prime }\in L\left[\hslash ,\ell \right]$. If $\left|{\mathsf{\Lambda }}^{\prime }\right|$ is a generalized n-polynomial p-convex function on $\left[\hslash ,\ell \right]$, then for $\mu \in \left[0,1\right]:$

$$\begin{array}{ccc}& & \left|\frac{\mathsf{\Lambda }\left(\hslash \right)+\mathsf{\Lambda }(\ell )}{2}-\frac{p}{{\ell }^p-{\hslash }^p}{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^{1-p}}d\sigma \right|\hfill \\ & \le & \frac{{\ell }^p-{\hslash }^p}{2p{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j\left[\left|{\mathsf{\Lambda }}^{\prime }\left(\hslash \right)\right|{\kappa }_p\left(j\right)+\left|{\mathsf{\Lambda }}^{\prime }\left(\ell \right)\right|{\eta }_p\left(j\right)\right],\hfill \end{array}$$

(12)

where

$${\kappa }_p\left(j\right)={\int }_0^1\left|1-2\mu \right|\left(1-{\left(1-\mu \right)}^j\right){\left|\mu {\hslash }^p+(1-\mu ){\ell }^p\right|}^{\frac{1}{p}-1}d\mu$$

and

$${\eta }_p\left(j\right)={\int }_0^1\left|1-2\mu \right|\left(1-{\mu }^j\right){\left|\mu {\hslash }^p+(1-\mu ){\ell }^p\right|}^{\frac{1}{p}-1}d\mu .$$

Proof. 

Using Lemma 1 and the definition $GPOLC-p$ function, one obtains

$$\begin{array}{ccc}& & \left|\frac{\mathsf{\Lambda }\left(\hslash \right)+\mathsf{\Lambda }(\ell )}{2}-\frac{p}{{\ell }^p-{\hslash }^p}{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^{1-p}}d\sigma \right|\hfill \\ & \le & \frac{{\ell }^p-{\hslash }^p}{2p}{\int }_0^1\left|1-2\mu \right|{\left|\mu {\hslash }^p+(1-\mu ){\ell }^p\right|}^{\frac{1}{p}-1}\left|{\mathsf{\Lambda }}^{\prime }\left({\left[\mu {\hslash }^p+(1-\mu ){\ell }^p\right]}^{\frac{1}{p}}\right)\right|d\mu \hfill \\ & \le & \frac{{\ell }^p-{\hslash }^p}{2p{\sum }_{j=1}^n{\alpha }_j}\left|{\mathsf{\Lambda }}^{\prime }\left(\hslash \right)\right|{\int }_0^1\left|1-2\mu \right|{\left|\mu {\hslash }^p+(1-\mu ){\ell }^p\right|}^{\frac{1}{p}-1}\sum _{j=1}^n{\alpha }_j\left(1-{\left(1-\mu \right)}^j\right)d\mu \hfill \\ & & +\frac{{\ell }^p-{\hslash }^p}{2p{\sum }_{j=1}^n{\alpha }_j}\left|{\mathsf{\Lambda }}^{\prime }\left(\ell \right)\right|{\int }_0^1\left|1-2\mu \right|{\left|\mu {\hslash }^p+(1-\mu ){\ell }^p\right|}^{\frac{1}{p}-1}\sum _{j=1}^n{\alpha }_j\left(1-{\mu }^j\right)d\mu \hfill \\ & =& \frac{{\ell }^p-{\hslash }^p}{2p{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j\left[\left|{\mathsf{\Lambda }}^{\prime }\left(\hslash \right)\right|{\kappa }_p\left(j\right)+\left|{\mathsf{\Lambda }}^{\prime }\left(\ell \right)\right|{\eta }_p\left(j\right)\right].\hfill \end{array}$$

□

Remark 20. 

For $n=1$ and $p=1,$ inequality (12) reduces to the inequality in ([31], Theorem 2.2) for classical convex functions.

Remark 21. 

For ${\alpha }_j=1$ $\left(j=\overline{1,n}\right)$ and $p=1,$ inequality (12) reduces to the inequality in ([27], Theorem 5) for n-polynomial convex functions.

Remark 22. 

For $p=1,$ inequality (10) reduces to the inequality in ([30], Theorem 4.2) for generalized n-polynomial convex functions.

Remark 23. 

For ${\alpha }_j=1$ $\left(j=\overline{1,n}\right),$ inequality (10) reduces to the inequality in ([29], Theorem 5.2) for n-polynomial p-convex functions.

Corollary 2. 

For $p=-1$ in Theorem 2, one obtains the following inequality for generalized n-polynomial harmonically convex functions:

$$\begin{array}{ccc}& & \left|\frac{\mathsf{\Lambda }\left(\hslash \right)+\mathsf{\Lambda }(\ell )}{2}-\frac{\hslash \ell }{\ell -\hslash }{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^2}d\sigma \right|\hfill \\ & \le & \frac{\left(\ell -\hslash \right)\hslash \ell }{2{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j\left[\left|{\mathsf{\Lambda }}^{\prime }\left(\hslash \right)\right|\kappa \left(j\right)+\left|{\mathsf{\Lambda }}^{\prime }\left(\ell \right)\right|\eta \left(j\right)\right],\hfill \end{array}$$

where

$$\kappa \left(j\right)={\int }_0^1\left|1-2\mu \right|\left(1-{\left(1-\mu \right)}^j\right){\left|\mu \ell +(1-\mu )\hslash \right|}^{-2}d\mu ,$$

$$\eta \left(j\right)={\int }_0^1\left|1-2\mu \right|\left(1-{\mu }^j\right){\left|\mu \ell +(1-\mu )\hslash \right|}^{-2}d\mu .$$

Theorem 3. 

Let $\mathsf{\Lambda }:\aleph \subseteq \left(0,\infty \right)\to \mathbb{R}$ be a differentiable function on ${\aleph }^{\circ }$, $\hslash ,\ell \in {\aleph }^{\circ }$ with $\hslash <\ell ,p\in \mathbb{R}-\left\{0\right\},$ $q>1,\frac{1}{r}+\frac{1}{q}=1$ and assume that ${\mathsf{\Lambda }}^{\prime }\in L\left[\hslash ,\ell \right]$. If ${\left|{\mathsf{\Lambda }}^{\prime }\right|}^q$ is a $GPOLC-p$ function on $\left[\hslash ,\ell \right]$; then, for $\mu \in \left[0,1\right]:$

$$\begin{array}{ccc}& & \left|\frac{\mathsf{\Lambda }\left(\hslash \right)+\mathsf{\Lambda }(\ell )}{2}-\frac{p}{{\ell }^p-{\hslash }^p}{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^{1-p}}d\sigma \right|\hfill \\ & \le & \frac{{\ell }^p-{\hslash }^p}{2p}{\left({\varrho }_{p,r}\right)}^{\frac{1}{r}}{\left(\frac{2}{{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j\frac{j}{j+1}\right)}^{\frac{1}{q}}{A}^{\frac{1}{q}}\left({\left|{\mathsf{\Lambda }}^{\prime }\left(\hslash \right)\right|}^q,{\left|{\mathsf{\Lambda }}^{\prime }\left(\ell \right)\right|}^q\right),\hfill \end{array}$$

(13)

where

$${\varrho }_{p,r}={\int }_0^1\frac{{\left|1-2\mu \right|}^r}{{\left|{\left(\mu {\hslash }^p+(1-\mu ){\ell }^p\right)}^{1-\frac{1}{p}}\right|}^r}d\mu ,$$

and A is the arithmetic mean.

Proof. 

Using Lemma 1, the definition of $GPOLC-p$ function and Hölder’s integral inequality, one obtains

$$\begin{array}{ccc}& & \left|\frac{\mathsf{\Lambda }\left(\hslash \right)+\mathsf{\Lambda }(\ell )}{2}-\frac{p}{{\ell }^p-{\hslash }^p}{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^{1-p}}d\sigma \right|\hfill \\ & \le & \frac{{\ell }^p-{\hslash }^p}{2p}{\int }_0^1\left|1-2\mu \right|{\left|\mu {\hslash }^p+(1-\mu ){\ell }^p\right|}^{\frac{1}{p}-1}\left|{\mathsf{\Lambda }}^{\prime }\left({\left[\mu {\hslash }^p+(1-\mu ){\ell }^p\right]}^{\frac{1}{p}}\right)\right|d\mu \hfill \\ & \le & \frac{{\ell }^p-{\hslash }^p}{2p}{\left({\int }_0^1\frac{{\left|1-2\mu \right|}^r}{{\left|{\left(\mu {\hslash }^p+(1-\mu ){\ell }^p\right)}^{1-\frac{1}{p}}\right|}^r}d\mu \right)}^{\frac{1}{r}}\hfill \\ & & \times {\left({\int }_0^1{\left|{\mathsf{\Lambda }}^{\prime }\left(\mu {\hslash }^p+(1-\mu ){\ell }^p\right)\right|}^{q}d\mu \right)}^{\frac{1}{q}}\hfill \\ & \le & \frac{{\ell }^p-{\hslash }^p}{2p}{\left({\varrho }_{p,r}\right)}^{\frac{1}{r}}\left(\frac{{\left|{\mathsf{\Lambda }}^{\prime }\left(\hslash \right)\right|}^q}{{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j{\int }_0^1\left(1-{\left(1-\mu \right)}^j\right)d\mu \right.\hfill \\ & & {\left.+\frac{{\left|{\mathsf{\Lambda }}^{\prime }\left(\ell \right)\right|}^q}{{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j{\int }_0^1\left(1-{\mu }^j\right)d\mu \right)}^{\frac{1}{q}}\hfill \\ & =& \frac{{\ell }^p-{\hslash }^p}{2p}{\left({\varrho }_{p,r}\right)}^{\frac{1}{r}}{\left(\frac{2}{{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j\frac{j}{j+1}\right)}^{\frac{1}{q}}{A}^{\frac{1}{q}}\left({\left|{\mathsf{\Lambda }}^{\prime }\left(\hslash \right)\right|}^q,{\left|{\mathsf{\Lambda }}^{\prime }\left(\ell \right)\right|}^q\right),\hfill \end{array}$$

where

$${\int }_0^1\left[1-{\left(1-\mu \right)}^j\right]d\mu ={\int }_0^1\left[1-{\mu }^j\right]d\mu =\frac{j}{j+1}.$$

□

Remark 24. 

For $n=1$ and $p=1,$ inequality (13) reduces to the inequality in ([31], Theorem 2.3) for classical convex functions.

Remark 25. 

For ${\alpha }_j=1$ $\left(j=\overline{1,n}\right)$ and $p=1,$ inequality (13) reduces to the inequality in ([27], Theorem 6) for n-polynomial convex functions.

Remark 26. 

For $p=1,$ inequality (13) reduces to the inequality in ([30], Theorem 4.5) for generalized n-polynomial convex functions.

Remark 27. 

For ${\alpha }_j=1$ $\left(j=\overline{1,n}\right),$ inequality (13) reduces to the inequality in ([29], Theorem 5.7) for n-polynomial p-convex functions.

Corollary 3. 

If one takes $p=-1$ in Theorem 3, then one obtains the following inequality for generalized n-polynomial harmonically convex functions:

$$\begin{array}{ccc}& & \left|\frac{\mathsf{\Lambda }\left(\hslash \right)+\mathsf{\Lambda }(\ell )}{2}-\frac{\hslash \ell }{\ell -\hslash }{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^2}d\sigma \right|\hfill \\ & \le & \frac{\left(\ell -\hslash \right)\hslash \ell }{2}{\left({\varrho }_r\right)}^{\frac{1}{r}}{\left(\frac{2}{{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n\frac{j}{j+1}\right)}^{\frac{1}{q}}{A}^{\frac{1}{q}}\left({\left|{\mathsf{\Lambda }}^{\prime }\left(\hslash \right)\right|}^q,{\left|{\mathsf{\Lambda }}^{\prime }\left(\ell \right)\right|}^q\right),\hfill \end{array}$$

where

$${\varrho }_r={\int }_0^1\frac{{\left|1-2\mu \right|}^r}{{\left|{\left(\mu \ell +(1-\mu )\hslash \right)}^2\right|}^r}d\mu .$$

Theorem 4. 

Let $\mathsf{\Lambda }:\aleph \subseteq \left(0,\infty \right)\to \mathbb{R}$ be a differentiable function on ${\aleph }^{\circ }$, $\hslash ,\ell \in {\aleph }^{\circ }$ with $\hslash <\ell ,$ $p\in \mathbb{R}-\left\{0\right\},$ $q\ge 1$ and assume that ${\mathsf{\Lambda }}^{\prime }\in L\left[\hslash ,\ell \right]$. If ${\left|{\mathsf{\Lambda }}^{\prime }\right|}^q$ is a $GPOLC-p$ function on $\left[\hslash ,\ell \right]$; then, for $\mu \in \left[0,1\right]:$

$$\begin{array}{ccc}& & \left|\frac{\mathsf{\Lambda }\left(\hslash \right)+\mathsf{\Lambda }(\ell )}{2}-\frac{p}{{\ell }^p-{\hslash }^p}{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^{1-p}}d\sigma \right|\hfill \\ & \le & \frac{{\ell }^p-{\hslash }^p}{2p}{\left({\varrho }_p^{*}\right)}^{1-\frac{1}{q}}{\left(\frac{{\left|{\mathsf{\Lambda }}^{\prime }\left(\hslash \right)\right|}^q}{{\sum }_{j=1}^n{\alpha }_j}{\omega }_p\left(j\right)+\frac{{\left|{\mathsf{\Lambda }}^{\prime }\left(\ell \right)\right|}^q}{{\sum }_{j=1}^n{\alpha }_j}{\varpi }_p\left(j\right)\right)}^{\frac{1}{q}},\hfill \end{array}$$

(14)

where

$${\varrho }_p^{*}={\int }_0^1\frac{\left|1-2\mu \right|}{\left|{\left(\mu {\hslash }^p+(1-\mu ){\ell }^p\right)}^{1-\frac{1}{p}}\right|}d\mu ,$$

$${\omega }_p\left(j\right)=\sum _{j=1}^n{\alpha }_j{\int }_0^1\frac{\left|1-2\mu \right|\left[1-{(1-\mu )}^j\right]}{\left|{\left(\mu {\hslash }^p+(1-\mu ){\ell }^p\right)}^{1-\frac{1}{p}}\right|}d\mu ,$$

$${\varpi }_p\left(j\right)=\sum _{j=1}^n{\alpha }_j{\int }_0^1\frac{\left|1-2\mu \right|\left[1-{\mu }^j\right]}{\left|{\left(\mu {\hslash }^p+(1-\mu ){\ell }^p\right)}^{1-\frac{1}{p}}\right|}d\mu$$

and A is the arithmetic mean.

Proof. 

Using the definition of $GPOLC-p$ function and Lemma 1, one obtains

$$\begin{array}{ccc}& & \left|\frac{\mathsf{\Lambda }\left(\hslash \right)+\mathsf{\Lambda }(\ell )}{2}-\frac{p}{{\ell }^p-{\hslash }^p}{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^{1-p}}d\sigma \right|\hfill \\ & \le & \left|\frac{{\ell }^p-{\hslash }^p}{2p}{\int }_0^1\left|1-2\mu \right|{\left|\mu {\hslash }^p+(1-\mu ){\ell }^p\right|}^{\frac{1}{p}-1}{\mathsf{\Lambda }}^{\prime }\left({\left[\mu {\hslash }^p+(1-\mu ){\ell }^p\right]}^{\frac{1}{p}}\right)d\mu \right|\hfill \\ & \le & \frac{{\ell }^p-{\hslash }^p}{2p}{\left({\int }_0^1\frac{\left|1-2\mu \right|}{\left|{\left(\mu {\hslash }^p+(1-\mu ){\ell }^p\right)}^{1-\frac{1}{p}}\right|}d\mu \right)}^{1-\frac{1}{q}}\hfill \\ & & \times {\left({\int }_0^1\frac{\left|1-2\mu \right|}{\left|{\left(\mu {\hslash }^p+(1-\mu ){\ell }^p\right)}^{1-\frac{1}{p}}\right|}{\left|{\mathsf{\Lambda }}^{\prime }\left({\left[\mu {\hslash }^p+(1-\mu ){\ell }^p\right]}^{\frac{1}{p}}\right)\right|}^{q}d\mu \right)}^{\frac{1}{q}}\hfill \\ & \le & \frac{{\ell }^p-{\hslash }^p}{2p}{\left({\varrho }_p^{*}\right)}^{1-\frac{1}{q}}\left[\frac{{\left|{\mathsf{\Lambda }}^{\prime }\left(\hslash \right)\right|}^q}{{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j{\int }_0^1\frac{\left|1-2\mu \right|\left[1-{(1-\mu )}^j\right]}{{\left|\left(\mu {\hslash }^p+(1-\mu ){\ell }^p\right)\right|}^{1-\frac{1}{p}}}d\mu \right.\hfill \\ & & {\left.+\frac{{\left|{\mathsf{\Lambda }}^{\prime }\left(\ell \right)\right|}^q}{{\sum }_{j=1}^n{\alpha }_j}\sum _{j=1}^n{\alpha }_j{\int }_0^1\frac{\left|1-2\mu \right|\left[1-{\mu }^j\right]}{{\left|\left(\mu {\hslash }^p+(1-\mu ){\ell }^p\right)\right|}^{1-\frac{1}{p}}}d\mu \right]}^{\frac{1}{q}}\hfill \\ & =& \frac{{\ell }^p-{\hslash }^p}{2p}{\left({\varrho }_p^{*}\right)}^{1-\frac{1}{q}}{\left(\frac{{\left|{\mathsf{\Lambda }}^{\prime }\left(\hslash \right)\right|}^q}{{\sum }_{j=1}^n{\alpha }_j}{\omega }_p\left(j\right)+\frac{{\left|{\mathsf{\Lambda }}^{\prime }\left(\ell \right)\right|}^q}{{\sum }_{j=1}^n{\alpha }_j}{\varpi }_p\left(j\right)\right)}^{\frac{1}{q}}.\hfill \end{array}$$

□

Remark 28. 

For $n=1$ and $p=1,$ inequality (14) reduces to the inequality in ([32], Theorem 1) for classical convex functions.

Remark 29. 

For ${\alpha }_j=1$ $\left(j=\overline{1,n}\right)$ and $p=1,$ inequality (14) reduces to the inequality in ([27], Theorem 7) for n-polynomial convex functions.

Remark 30. 

For $p=1,$ inequality (14) reduces to the inequality in ([30], Theorem 4.8) for generalized n-polynomial convex functions.

Remark 31. 

For ${\alpha }_j=1$ $\left(j=\overline{1,n}\right),$ inequality (14) reduces to the inequality in ([29], Theorem 5.10) for n-polynomial p-convex functions.

Corollary 4. 

For $p=-1$ in Theorem 4, one obtains the following inequality for generalized n-polynomial harmonically convex functions:

$$\begin{array}{ccc}& & \left|\frac{\mathsf{\Lambda }\left(\hslash \right)+\mathsf{\Lambda }(\ell )}{2}-\frac{\hslash \ell }{\ell -\hslash }{\int }_{\hslash }^{\ell }\frac{\mathsf{\Lambda }\left(\sigma \right)}{{\sigma }^2}d\sigma \right|\hfill \\ & \le & \frac{\left(\ell -\hslash \right)\hslash \ell }{2}{\left({\varrho }^{*}\right)}^{1-\frac{1}{q}}{\left(\frac{{\left|{\mathsf{\Lambda }}^{\prime }\left(\hslash \right)\right|}^q}{{\sum }_{j=1}^n{\alpha }_j}\omega \left(j\right)+\frac{{\left|{\mathsf{\Lambda }}^{\prime }\left(\ell \right)\right|}^q}{{\sum }_{j=1}^n{\alpha }_j}\varpi \left(j\right)\right)}^{\frac{1}{q}},\hfill \end{array}$$

where

$${\varrho }^{*}={\int }_0^1\frac{\left|1-2\mu \right|}{{\left(\mu \ell +(1-\mu )\hslash \right)}^2}d\mu ,$$

$$\omega \left(j\right)={\int }_0^1\frac{\left|1-2\mu \right|}{{\left(\mu \ell +(1-\mu )\hslash \right)}^2}\sum _{j=1}^n{\alpha }_j\left[1-{(1-\mu )}^j\right]d\mu ,$$

$$\varpi \left(j\right)={\int }_0^1\frac{\left|1-2\mu \right|}{{\left(\mu \ell +(1-\mu )\hslash \right)}^2}\sum _{j=1}^n{\alpha }_j\left[1-{\mu }^j\right]d\mu .$$

4. Conclusions

In this work, the class of generalized n-polynomial p-convex functions is introduced and related properties are provided. Relationships between these functions and other kinds of convex functions are provided. New refinements of the well-known Hermite–Hadamard inequality are established. Several special cases have been discussed for the newly obtained results. For some special cases, the definition and results of generalized n-polynomial p-convex functions reduce to a new definition and new results for the class of convex functions, called generalized n-polynomial harmonically convex functions. Our methodology could have additional applications in the realm of convexity theory. It would be intriguing to broaden these discoveries to encompass other forms of convexities documented in the literature. Through the utilization of generalized convexities, novel integral inequalities can be derived. The concepts elucidated in this manuscript aim to inspire researchers with an interest in exploring alternative classes of convex functions and constructing related integral inequalities.