On inequalities involving n-polynomial exponential-type convex functions

Muhammad Samraiz; Ghada Alobaidi; Muhammad Hammad; Gauhar Rahman; Yasser Elmasry

doi:10.1515/dema-2025-0176

Article Open Access

On inequalities involving n-polynomial exponential-type convex functions

Muhammad Samraiz , Ghada Alobaidi , Muhammad Hammad , Gauhar Rahman and Yasser Elmasry

Published/Copyright: October 9, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

This article deals with new generalized and more refined classes of Hermite-Hadamard and Fejér Hermite-Hadamard-type inequalities via n -polynomial exponential-type convex functions. The illustrative graphs confirm the validity of new inequalities. Some applications are provided for generalized means to accommodate different user preferences, device capabilities, and technological advancements. To establish the main results, we primarily use Hölder’s inequality, the power mean inequality, and some other generalized inequalities. These inequalities have strong applicability in the theory of optimization.

Keywords: n-polynomial exponential-type convex; Hölder’s inequality; Power-mean inequality; Fejér-type inequality; means

MSC 2020: 26A51

1 Introduction

The concept of convexity holds a significant value in mathematical areas. It describes the property of a function wherein the graph of a line segment joining any two points is located above the graph between the two points. While the primary source of convexity lies in geometry, it also possesses numerous useful qualities that render it applicable in a variety of scientific and mathematical domains. Convexity plays a helpful role in optimization problems [1], as well as in exploring other fields of study such as finance [2], machine learning [3,4], data science [5,6], control theory [7], and linear programming [8]. Furthermore, convexity is applied for the development of statistical models and efficient estimates [9,10].

The study of inequalities is essential in mathematics for resolving different types of problems. The necessity for inequalities arises since not all mathematical expressions are in the form of equality. Several kinds of inequalities exist in the literature, which include linear inequalities, quadratic inequalities, and rational inequalities. Some widely used inequalities include the triangle inequality [11,12], the inequality of arithmetic and geometric means [13,14], Jensen’s inequality [15,16], Bell’s inequality [17,18], and the Cauchy-Schwarz inequality [19,20]. Inequalities are extensively utilized in different mathematical branches such as analysis [21,22], number theory [23], and other scientific fields like engineering [24], economics [25], environmental science [26], and operation research [27]. Inequalities are also relevant to the field of convexity, especially in optimization theory. There are important convex inequalities in the literature, like the Jensen inequality, the Hermite-Hadamard inequality [28–30], and the Fejér-type inequality [31]. The widely known Hermite-Hadamard inequality, named after Charles Hermite and Jacques Hadamard,is defined as follows:

If f : [ s 1 , s 2 ] → R is a convex function, then the Hermite-Hadamard inequality is defined as

f s 1 + s 2 2 ≤ 1 s 2 − s 1 ∫ s 1 s 2 f ( s ) d s ≤ f ( s 1 ) + f ( s 2 ) 2 .

Due to important properties and usefulness of the Hermite-Hadamard inequality, many novel and generalized forms of Hermite-Hadamard-type inequality are being explored by researchers. Some of its refinements include its generalized forms to n -intervals, weighted Hermite-Hadamard inequality [32]. Furthermore, Fejér-type inequality, which was discovered by Lipót Fejér in 1906, is

f s 1 + s 2 2 ∫ s 1 s 2 w ( s ) d s ≤ ∫ s 1 s 2 f ( s ) w ( s ) d s ≤ f ( s 1 ) + f ( s 2 ) 2 ∫ s 1 s 2 w ( s ) d s ,

where f is a convex function in the interval ( s 1 , s 2 ) and w is a positive and symmetric function in the same interval. This inequality is a weighted version of the Hermite-Hadamard inequality.

In this article, we investigate a new generalized class of Hermite-Hadamard inequalities via n -polynomial exponential-type convexity. Moreover, we will also derive Fejér-type inequalities for the n -polynomial exponential-type convex function. First, we present some foundational definitions and concepts below.

Definition 1

A function f : I → R is called convex if

f ( t s 1 + ( 1 − t ) s 2 ) ≤ t f ( s 1 ) + ( 1 − t ) f ( s 2 ) ,

for all s 1 , s 2 ∈ I and for all t ∈ [ 0 , 1 ] .

Definition 2

A function f : I → R is called exponential-type convex if

f ( t s 1 + ( 1 − t ) s 2 ) ≤ ( e t − 1 ) f ( s 1 ) + ( e 1 − t − 1 ) f ( s 2 ) ,

where t ∈ [ 0 , 1 ] and s 1 , s 2 ∈ I .

The n -polynomial convex function is defined as follows.

Definition 3

A function f : I → R is called n -polynomial convex if

f ( t s 1 + ( 1 − t ) s 2 ) ≤ 1 n ∑ i = 1 n [ 1 − ( 1 − t ) i ] f ( s 1 ) + 1 n ∑ i = 1 n [ 1 − t i ] f ( s 2 ) ,

where s 1 , s 2 ∈ I , n ∈ N , and t ∈ [ 0 , 1 ] .

Now, we present the definition of the n -polynomial exponential-type convex function.

Definition 4

[33] A function f : I → R is called n -polynomial exponential-type convex if

f ( t s + ( 1 − t ) x ) ≤ 1 n ∑ i = 1 n ( e t − 1 ) i f ( s ) + 1 n ∑ i = 1 n ( e 1 − t − 1 ) i f ( x ) ,

holds for all s , x ∈ I , t ∈ [ 0 , 1 ] , and n ∈ N .

Theorem 1

(Hölder’s integral inequality [34]). Let f and g be the real-valued functions defined on the domain [ s 1 , s 2 ] . If ∣ f ∣ p and ∣ g ∣ q are integrable over [ s 1 , s 2 ] , then

∫ s 1 s 2 ∣ f ( s ) g ( s ) ∣ d s ≤ ∫ s 1 s 2 ∣ f ( s ) ∣ p d s 1 p ∫ s 1 s 2 ∣ g ( s ) ∣ q d s 1 q ,

where p > 1 and 1 p + 1 q = 1 .

Theorem 2

(Power-mean integral inequality [34]). Let f and g be the real-valued functions defined on the domain [ s 1 , s 2 ] . If ∣ f ∣ , ∣ f ∣ ∣ g ∣ q are integrable on [ s 1 , s 2 ] , then

∫ s 1 s 2 ∣ f ( s ) g ( s ) ∣ d s ≤ ∫ s 1 s 2 ∣ f ( s ) ∣ d s 1 − 1 q ∫ s 1 s 2 ∣ f ( s ) ∣ ∣ g ( s ) ∣ q d s 1 q .

The following Hölder-İscan inequality is the refined form of Hölder’s inequality.

Theorem 3

[35] Let f and g be the real-valued mappings defined on the domain [ s 1 , s 2 ] . If ∣ f ∣ p and ∣ g ∣ q are integrable over [ s 1 , s 2 ] , then

∫ s 1 s 2 ∣ f ( s ) g ( s ) ∣ d s ≤ 1 s 2 − s 1 ∫ s 1 s 2 ( s 2 − s ) ∣ f ( s ) ∣ p d s 1 p ∫ s 1 s 2 ( s 2 − s ) ∣ g ( s ) ∣ q d s 1 q + ∫ s 1 s 2 ( s − s 1 ) ∣ f ( s ) ∣ q d s 1 p ∫ s 1 s 2 ( s − s 1 ) ∣ g ( s ) ∣ q d s 1 q ,

where p > 1 and 1 p + 1 q = 1 .

The following theorem presents an improved form of the power-mean integral inequality.

Theorem 4

[36] Let f and g be the real-valued mappings on the interval [ s 1 , s 2 ] . If ∣ f ∣ , ∣ f ∣ ∣ g ∣ q are integrable on [ s 1 , s 2 ] , then

∫ s 1 s 2 ∣ f ( s ) g ( s ) ∣ d s ≤ 1 s 2 − s 1 ∫ s 1 s 2 ( s 2 − s ) ∣ f ( s ) ∣ d s 1 − 1 q ∫ s 1 s 2 ( s 2 − s ) ∣ f ( s ) ∣ ∣ g ( s ) ∣ q d s 1 q + ∫ s 1 s 2 ( s − s 1 ) ∣ f ( s ) ∣ d s 1 − 1 q ∫ s 1 s 2 ( s − s 1 ) ∣ f ( s ) ∣ ∣ g ∣ q d s 1 q ,

where q ≥ 1 .

Now, we present the following useful lemma which was proved by İscan.

Lemma 1

[37] Let the function f : I ⊂ R → R be differentiable on interior I 0 of I, and s 1 , s 2 ∈ I 0 with s 1 < s 2 , n ∈ N , and m ∈ { 0 , 1 , … , n − 1 } . If f ′ ∈ L [ s 1 , s 2 ] , then the following equality holds true

I n ( f , s 1 , s 2 ) = ∑ m = 0 n − 1 1 2 n f ( n − m ) s 1 + m s 2 n + f ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n − 1 s 2 − s 1 ∫ s 1 s 2 f ( s ) d s = ∑ m = 0 n − 1 s 2 − s 1 2 n 2 ∫ 0 1 ( 1 − 2 t ) f ′ t ( n − m ) s 1 + m s 2 n + ( 1 − t ) ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n d t .

Studying generalized versions of inequalities allows for broader applicability and flexibility in various mathematical and scientific areas. It often leads to the discovery of new results and applications. Keeping in view the importance and extensive use of Hermite-Hadamard and Fejér Hermite-Hadamard in analysis and convexity theory, the present study is to explore generalized and refined Hermite-Hadamard inequalities via n -polynomial exponential-type convexity. Some applications of the explored results are provided in terms of means to enhance the impact and relevance of research, fostering innovation and facilitating its uptake in practical settings.

2 Main results

In this section, we study some new generalized inequalities of Hermite-Hadamard type for n -polynomial exponential-type convex functions by implementing Hölder’s inequality, power mean inequality, and their improved forms. The first result of this section is as follows.

Theorem 5

Let f : I ⊂ R → R be a differentiable function on interior I 0 of I and s 1 , s 2 ∈ I 0 with s 1 < s 2 . If ∣ f ′ ∣ q is an n-polynomial exponential-type convex mapping on [ s 1 , s 2 ] for some p > 1 where n ∈ N , then the inequality

(1) ∣ I n ( f , s 1 , s 2 ) ∣ ≤ ∑ m = 0 n − 1 s 2 − s 1 2 n 2 1 1 + p 1 p ( e − 1 ) ( 1 − ( e − 1 ) n ) ( 2 − e ) n 1 q × f ′ ( n − m ) s 1 + m s 2 n q + f ′ ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q 1 q

holds for 1 p + 1 q = 1 .

Proof

By applying Lemma 1, we obtain

∣ I n ( f , s 1 , s 2 ) ∣ ≤ ∑ m = 0 n − 1 s 2 − s 1 2 n 2 ∫ 0 1 ( 1 − 2 t ) f ′ t ( n − m ) s 1 + m s 2 n + ( 1 − t ) ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n d t .

By using Hölder’s inequality, we obtain

I n ( f , s 1 , s 2 ) ≤ ∑ m = 0 n − 1 s 2 − s 1 2 n 2 ∫ 0 1 ∣ 1 − 2 t ∣ p d t 1 p ∫ 0 1 f ′ t ( n − m ) s 1 + m s 2 n + ( 1 − t ) ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q d t 1 q .

Since ∣ f ′ ∣ q is the n -polynomial exponential-type convex function,

I n ( f , s 1 , s 2 ) ≤ ∑ m = 0 n − 1 s 2 − s 1 2 n 2 ∫ 0 1 ∣ 1 − 2 t ∣ p d t 1 p ∫ 0 1 1 n ∑ i = 1 n ( e t − 1 ) i f ′ ( n − m ) s 1 + m s 2 n q + 1 n ∑ i = 1 n ( e 1 − t − 1 ) i f ′ ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q d t 1 q = ∑ m = 0 n − 1 s 2 − s 1 2 n 2 ∫ 0 1 ∣ 1 − 2 t ∣ p d t 1 p 1 n ∑ i = 1 n ∫ 0 1 ( e t − 1 ) i f ′ ( n − m ) s 1 + m s 2 n q d t + ∫ 0 1 ( e 1 − t − 1 ) i f ′ ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q d t 1 q .

Now, using the facts that e t ≤ e and e 1 − t ≤ e for any t ∈ [ 0 , 1 ] , so

I n ( f , s 1 , s 2 ) ≤ ∑ m = 0 n − 1 s 2 − s 1 2 n 2 ∫ 0 1 ∣ 1 − 2 t ∣ p d t 1 p 1 n ∑ i = 1 n ∫ 0 1 ( e − 1 ) i f ′ ( n − m ) s 1 + m s 2 n q d t + ∫ 0 1 ( e − 1 ) i f ′ ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q d t 1 q = ∑ m = 0 n − 1 s 2 − s 1 2 n 2 1 1 + p 1 p 1 n ∑ i = 1 n ( e − 1 ) i 1 q f ′ ( n − m ) s 1 + m s 2 n q + f ′ ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q 1 q = ∑ m = 0 n − 1 s 2 − s 1 2 n 2 1 1 + p 1 p ( e − 1 ) ( 1 − ( e − 1 ) n ) ( 2 − e ) n 1 q × f ′ ( n − m ) s 1 + m s 2 n q + f ′ ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q 1 q .

Hence, the required result is proved.□

Example 1

Here, we present the graphical representation to show the validity of Theorem 5. For this purpose, we make substitution f ( s ) = e s and f ′ ( s ) = e s to arrive at

(2) ∑ m = 0 n − 1 1 2 n e ( n − m ) s 1 + m s 2 n + e ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n − ∑ m = 0 n − 1 s 2 − s 1 2 n 2 1 1 + p 1 p 1 n ∑ i = 1 n ( e − 1 ) i 1 q e ( ( n − m ) s 1 + m s 2 ) q n + e ( ( n − m − 1 ) s 1 + ( m + 1 ) s 2 ) q n 1 q ≤ 1 s 2 − s 1 ∫ s 1 s 2 f ( s ) d s ≤ ∑ m = 0 n − 1 1 2 n e ( n − m ) s 1 + m s 2 n + e ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n + ∑ m = 0 n − 1 s 2 − s 1 2 n 2 1 1 + p 1 p 1 n ∑ i = 1 n ( e − 1 ) i 1 q e ( ( n − m ) s 1 + m s 2 ) q n + e ( ( n − m − 1 ) s 1 + ( m + 1 ) s 2 ) q n 1 q .

By choosing the parameters n = 1 , s 1 = 0 , s 2 = 1 , p = 2 , q = 2 in (2), we achieve

p o ( n ) = e + 1 2 − 0.8362 1 n ∑ i = 1 n ( e − 1 ) i 1 2 , p 1 ( n ) = e − 1 , p 2 ( n ) = e + 1 2 + 0.8362 1 n ∑ i = 1 n ( e − 1 ) i 1 2 .

Figure 1 shows the validity of inequality (2) for the above functions with 1.1 ≤ n ≤ 10 .

$Figure 1 2D graph exhibiting inequality (2) for 1.1 ≤ n ≤ 10 1.1\le n\le 10 .$

Figure 1

2D graph exhibiting inequality (2) for 1.1 ≤ n ≤ 10 .

In Table 1, a comparative analysis is provided for the functions p 0 ( n ) , p 1 ( n ) , and p 2 ( n ) .

Table 1

Comparison of the results between the double inequality in Example 1

Functions	1.1	2	4	6	8	10
p 0 ( n )	0.76302	0.58126	0.06271	− 0.76696	− 2.10059	− 4.25326
p 1 ( n )	1.71828	1.71828	1.71828	1.71828	1.71828	1.71828
p 2 ( n )	2.95526	3.13702	3.65557	4.48525	5.81887	7.97154

Now, for 3D representation, we choose n = 1 , p = 2 , q = 2 , and n = 1 over the interval [ s 1 , s 2 ] with 0.1 ≤ s 1 ≤ 0.5 , 1.1 ≤ s 2 ≤ 1.5 to write

e s 1 + e s 2 2 − 0.2887 ( e − 1 ) 1 2 ( s 2 − s 1 ) e s 1 2 + e s 2 2 1 2

(3) ≤ e s 2 − e s 1 s 2 − s 1 ≤ e s 1 + e s 2 2 + 0.2887 ( e − 1 ) 1 2 ( s 2 − s 1 ) e s 1 2 + e s 2 2 1 2 .

Figure 2 indicates the validity of inequality (2) with the choice of the parameters 0.1 ≤ s 1 ≤ 0.5 , 1.1 ≤ s 2 ≤ 1.5 in a 3D format.

$Figure 2 3D graph exhibiting inequality (3) for 0.1 ≤ s 1 ≤ 0.5 0.1\le {s}_{1}\le 0.5 , 1.1 ≤ s 2 ≤ 1.5 1.1\le {s}_{2}\le 1.5 .$

Figure 2

3D graph exhibiting inequality (3) for 0.1 ≤ s 1 ≤ 0.5 , 1.1 ≤ s 2 ≤ 1.5 .

Corollary 1

If we apply n-polynomial exponential-type convexity of ∣ f ′ ∣ q in inequality (1) again, then we obtain

∣ I n ( f , s 1 , s 2 ) ∣ ≤ ∑ m = 0 n − 1 s 2 − s 1 2 1 − 1 q n 2 1 1 + p 1 p ( e − 1 ) ( 1 − ( e − 1 ) n ) ( 2 − e ) n 2 q [ ∣ f ′ ( s 1 ) ∣ q + ∣ f ′ ( s 2 ) ∣ q ] 1 q .

Corollary 2

If we specify n = 1 in Corollary 1, we obtain

f ( s 1 ) + f ( s 2 ) 2 − 1 s 2 − s 1 ∫ s 1 s 2 f ( s ) d s ≤ s 2 − s 1 2 1 − 1 q 1 1 + p 1 p ( e − 1 ) ( 1 − ( e − 1 ) n ) ( 2 − e ) n 2 q [ ∣ f ′ ( s 1 ) ∣ q + ∣ f ′ ( s 2 ) ∣ q ] 1 q .

Theorem 6

Let the function f : I ⊂ R → R be differentiable on interior I 0 of I and s 1 , s 2 ∈ I 0 with s 1 < s 2 . If ∣ f ′ ∣ q is n-polynomial exponential-type convex on the closed interval [ s 1 , s 2 ] for some q ≥ 1 , then the following inequality

(4) ∣ I n ( f , s 1 , s 2 ) ∣ ≤ ∑ m = 0 n − 1 s 2 − s 1 4 n 2 ( e − 1 ) ( 1 − ( e − 1 ) n ) ( 2 − e ) n 1 q × f ′ ( n − m ) s 1 + m s 2 n q + f ′ ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q 1 q .

is true.

Proof

By utilizing Lemma 1 and then power mean inequality, we obtain

∣ I n ( f , s 1 , s 2 ) ∣ ≤ ∑ m = 0 n − 1 s 2 − s 1 2 n 2 ∫ 0 1 ( 1 − 2 t ) f ′ t ( n − m ) s 1 + m s 2 n + ( 1 − t ) ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n d t ≤ ∑ m = 0 n − 1 s 2 − s 1 2 n 2 ∫ 0 1 ∣ 1 − 2 t ∣ d t 1 − 1 q × ∫ 0 1 ∣ 1 − 2 t ∣ f ′ t ( n − m ) s 1 + m s 2 n + ( 1 − t ) ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q d t 1 q .

Since ∣ f ′ ∣ q is n -polynomial exponential-type convex on [ s 1 , s 2 ] , so

∣ I n ( f , s 1 , s 2 ) ∣ ≤ ∑ m = 0 n − 1 s 2 − s 1 2 n 2 ∫ 0 1 ∣ 1 − 2 t ∣ d t 1 − 1 q ∫ 0 1 ∣ 1 − 2 t ∣ 1 n ∑ i = 1 n ( e t − 1 ) i f ′ ( n − m ) s 1 + m s 2 n q + ( e 1 − t − 1 ) i f ′ ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q d t 1 q .

Now, using the facts that e t ≤ e and e 1 − t ≤ e for any t ∈ [ 0 , 1 ] , so

∣ I n ( f , s 1 , s 2 ) ∣ ≤ ∑ m = 0 n − 1 s 2 − s 1 2 n 2 ∫ 0 1 ∣ 1 − 2 t ∣ d t 1 − 1 q ∫ 0 1 ∣ 1 − 2 t ∣ 1 n ∑ i = 1 n ( e − 1 ) i f ′ ( n − m ) s 1 + m s 2 n q + ( e − 1 ) i f ′ ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q d t 1 q = ∑ m = 0 n − 1 s 2 − s 1 2 2 − 1 q n 2 1 n ∑ i = 1 n ( e − 1 ) i 1 q f ′ ( n − m ) s 1 + m s 2 n q + f ′ ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q 1 q ∫ 0 1 ∣ 1 − 2 t ∣ d t 1 q = ∑ m = 0 n − 1 s 2 − s 1 4 n 2 ( e − 1 ) ( 1 − ( e − 1 ) n ) ( 2 − e ) n 1 q × f ′ ( n − m ) s 1 + m s 2 n q + f ′ ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q 1 q .

This proves the desired result.□

Example 2

Here, we present the graphical representation to show the validity of Theorem 6. For this purpose, we make substitution f ( s ) = e s and f ′ ( s ) = e s to arrive at

(5) ∑ m = 0 n − 1 1 2 n e ( n − m ) s 1 + m s 2 n + e ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n − ∑ m = 0 n − 1 s 2 − s 1 4 n 2 1 n ∑ i = 1 n ( e − 1 ) i 1 q e ( ( n − m ) s 1 + m s 2 ) q n + e ( ( n − m − 1 ) s 1 + ( m + 1 ) s 2 ) q n 1 q ≤ 1 s 2 − s 1 ∫ s 1 s 2 f ( s ) d s ≤ ∑ m = 0 n − 1 1 2 n e ( n − m ) s 1 + m s 2 n + e ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n + ∑ m = 0 n − 1 s 2 − s 1 4 n 2 1 n ∑ i = 1 n ( e − 1 ) i 1 q e ( ( n − m ) s 1 + m s 2 ) q n + e ( ( n − m − 1 ) s 1 + ( m + 1 ) s 2 ) q n 1 q .

Now, choosing the parameters n = 1 , s 1 = 0 , s 2 = 1 , p = 2 , q = 2 in (5), we achieve

p o ( n ) = e + 1 2 − 0.7241 1 n ∑ i = 1 n ( e − 1 ) i 1 2 , p 1 ( n ) = e − 1 , p 2 ( n ) = e + 1 2 + 0.7241 1 n ∑ i = 1 n ( e − 1 ) i 1 2 .

Figure 3 shows the validity of inequality (5) for the above functions with 1.1 ≤ n ≤ 10 .

$Figure 3 2D graph exhibiting inequality (5) for 1.1 ≤ n ≤ 10 1.1\le n\le 10 .$

Figure 3

2D graph exhibiting inequality (5) for 1.1 ≤ n ≤ 10 .

In Table 2, a comparative analysis is provided for the functions p 0 ( n ) , p 1 ( n ) , and p 2 ( n ) .

Table 2

Comparison of the results between the double inequality in Example 2

Functions	1.1	2	4	6	8	10
p 0 ( n )	0.90997	0.75257	0.30354	− 0.41491	− 1.56976	− 3.43384
p 1 ( n )	1.71828	1.71828	1.71828	1.71828	1.71828	1.71828
p 2 ( n )	2.80831	2.96571	3.41474	4.13319	5.28804	7.15212

Now, for 3D representation, we chose n = 1 , p = 2 , q = 2 , and n = 1 over the interval [ s 1 , s 2 ] with 0.1 ≤ s 1 ≤ 0.5 , 1.1 ≤ s 2 ≤ 1.5 to achieve

(6) e s 1 + e s 2 2 − ( e − 1 ) 1 2 ( s 2 − s 1 ) 4 e s 1 2 + e s 2 2 1 2 ≤ e s 2 − e s 1 s 2 − s 1 ≤ e s 1 + e s 2 2 + ( e − 1 ) 1 2 ( s 2 − s 1 ) 4 e s 1 2 + e s 2 2 1 2 .

Figure 4 indicates the validity of inequality (6) with the choice of the parameters 0.1 ≤ s 1 ≤ 0.5 , 1.1 ≤ s 2 ≤ 1.5 in a 3D format.

$Figure 4 3D graph exhibiting inequality (6) for 0.1 ≤ s 1 ≤ 0.5 0.1\le {s}_{1}\le 0.5 , 1.1 ≤ s 2 ≤ 1.5 1.1\le {s}_{2}\le 1.5 .$

Figure 4

3D graph exhibiting inequality (6) for 0.1 ≤ s 1 ≤ 0.5 , 1.1 ≤ s 2 ≤ 1.5 .

Corollary 3

If n-polynomial exponential-type convexity of ∣ f ′ ∣ q is applied in Theorem 2.5 again, we obtain

∣ I n ( f , s 1 , s 2 ) ∣ ≤ ∑ m = 0 n − 1 s 2 − s 1 2 2 − 1 q n 2 ( e − 1 ) ( 1 − ( e − 1 ) n ) ( 2 − e ) n 2 q [ ∣ f ′ ( s 1 ) ∣ q + ∣ f ′ ( s 2 ) ∣ q ] 1 q .

Corollary 4

If we specify n = 1 in Corollary 3, we arrive at

f ( s 1 ) + f ( s 2 ) 2 − 1 s 2 − s 1 ∫ s 1 s 2 f ( s ) d s ≤ s 2 − s 1 2 2 − 1 q ( e − 1 ) ( 1 − ( e − 1 ) n ) ( 2 − e ) n 2 q [ ∣ f ′ ( s 1 ) ∣ q + ∣ f ′ ( s 2 ) ∣ q ] 1 q .

Theorem 7

Let the mapping f : I ⊂ R → R be differentiable on interior I 0 of I and s 1 , s 2 ∈ I 0 with s 1 < s 2 . If ∣ f ′ ∣ q is the n-polynomial exponential-type convex function on [ s 1 , s 2 ] , then the following inequality Let the mapping f : I ⊂ R → R be differentiable on interior I 0 of I and s 1 , s 2 ∈ I 0 with s 1 < s 2 . If ∣ f ′ ∣ q is the n-polynomial exponential-type convex function on [ s 1 , s 2 ] , then the following inequality

(7) ∣ I n ( f , s 1 , s 2 ) ∣ ≤ ∑ m = 0 n − 1 s 2 − s 1 2 n 2 ( e − 1 ) ( 1 − ( e − 1 ) n ) ( 2 − e ) n 1 q 1 ( p + 1 ) 1 p × f ′ ( n − m ) s 1 + m s 2 n q + f ′ ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q 1 q

is satisfied, where 1 p + 1 q = 1 .

Proof

If we apply the Hölder-İscan inequality upon Lemma 1 and then use the same steps as in Theorem 5, we arrive at inequality (7) which is the same as inequality (1).□

Theorem 8

Let the mapping f : I ⊂ R → R be differentiable on I 0 , s 1 , s 2 ∈ I 0 with s 1 < s 2 . If ∣ f ′ ∣ q is n-polynomial exponential-type convex mapping on [ s 1 , s 2 ] , then the following inequality

(8) ∣ I n ( f , s 1 , s 2 ) ∣ ≤ ∑ m = 0 n − 1 s 2 − s 1 4 n 2 ( e − 1 ) ( 1 − ( e − 1 ) n ) ( 2 − e ) n 1 q × f ′ ( n − m ) s 1 + m s 2 n q + f ′ ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n q 1 q

is true for q ≥ 1 .

Proof

If we apply the improved power-mean inequality upon Lemma 1 and then use the same steps as in Theorem 6, we arrive at inequality (8) which is the same as inequality (4).□

In the next theorem, we explore a new result for the Fejér-Hermite-Hadamard-type inequality.

Theorem 9

If n ∈ N and w is a non-negative, symmetric, and integrable function, then n-polynomial exponential-type convex function f : I = [ s 1 , s 2 ] ⊂ R → R satisfies the following inequality:

(9) ( 2 − e ) n ( e − 1 ) ( 1 − ( e − 1 ) n ) f s 1 + s 2 2 ∫ s 1 s 2 w ( s ) d s − f ( x ) ∫ s 1 s 2 w ( s ) d s ≤ ∫ s 1 s 2 f ( s ) w ( s ) d s ≤ 1 n ∑ i = 1 n f ( s 1 ) ∫ s 1 s 2 ( e s 2 − s s 2 − s 1 − 1 ) i w ( s ) d s + f ( s 2 ) ∫ s 1 s 2 ( e s − s 1 s 2 − s 1 − 1 ) i w ( s ) d s ,

where s , x ∈ I .

Proof

Substituting t = 1 2 into Definition 4, we obtain

f s + x 2 ≤ 1 n ∑ i = 1 n ( e 1 2 − 1 ) i f ( s ) + 1 n ∑ i = 1 n ( e 1 2 − 1 ) i f ( x ) = 1 n ∑ i = 1 n ( e 1 2 − 1 ) i ( f ( s ) + f ( x ) ) .

Now, substituting s = t s 1 + ( 1 − t ) s 2 , x = t s 2 + ( 1 − t ) s 1 , we have

f s 1 + s 2 2 ≤ 1 n ∑ i = 1 n ( e − 1 ) i [ f ( t s 1 + ( 1 − t ) s 2 ) + f ( t s 2 + ( 1 − t ) s 1 ) ] .

Since w is non-negative, symmetric, and integrable function, we have

(10) f s 1 + s 2 2 w ( t s 1 + ( 1 − t ) s 2 ) ≤ ( e − 1 ) ( 1 − ( e − 1 ) n ) ( 2 − e ) n [ f ( t s 1 + ( 1 − t ) s 2 ) w ( t s 1 + ( 1 − t ) s 2 ) + f ( t s 2 + ( 1 − t ) s 1 ) w ( t s 1 + ( 1 − t ) s 2 ) ] .

Integrating inequality (10) with respect to t over [ 0 , 1 ] , we obtain

∫ 0 1 f s 1 + s 2 2 w ( t s 1 + ( 1 − t ) s 2 ) d t ≤ ( e − 1 ) ( 1 − ( e − 1 ) n ) ( 2 − e ) n ∫ 0 1 f ( t s 1 + ( 1 − t ) s 2 ) w ( t s 1 + ( 1 − t ) s 2 ) d t + ∫ 0 1 f ( t s 2 + ( 1 − t ) s 1 ) w ( t s 1 + ( 1 − t ) s 2 ) d t .

Now, we again substitute s = t s 1 + ( 1 − t ) s 2 , x = t s 2 + ( 1 − t ) s 1 to achieve

f s 1 + s 2 2 ∫ s 2 s 1 w ( s ) s 1 − s 2 d s ≤ ( e − 1 ) ( 1 − ( e − 1 ) n ) ( 2 − e ) n ∫ s 2 s 1 f ( s ) w ( s ) s 1 − s 2 d s + ∫ s 2 s 1 f ( x ) w ( s ) s 1 − s 2 d s

This can also be written as

(11) ( 2 − e ) n ( e − 1 ) ( 1 − ( e − 1 ) n ) f s 1 + s 2 2 ∫ s 1 s 2 w ( s ) d s − f ( x ) ∫ s 1 s 2 w ( s ) d s ≤ ∫ s 1 s 2 f ( s ) w ( s ) d s .

which is the left side of inequality (9). Now, for the right side of inequality (9), we set s = s 1 , x = s 2 in Definition 4 to obtain

f ( t s 1 + ( 1 − t ) s 2 ) ≤ 1 n ∑ i = 1 n ( e t − 1 ) i f ( s 1 ) + 1 n ∑ i = 1 n ( e 1 − t − 1 ) i f ( s 2 ) .

Since w is symmetric and integrable function, we have

(12) f ( t s 1 + ( 1 − t ) s 2 ) w ( t s 1 + ( 1 − t ) s 2 ) ≤ 1 n ∑ i = 1 n ( e t − 1 ) i f ( s 1 ) w ( t s 1 + ( 1 − t ) s 2 ) + 1 n ∑ i = 1 n ( e 1 − t − 1 ) i f ( s 2 ) w ( t s 1 + ( 1 − t ) s 2 ) .

Integrating inequality (12) with respect to t over [0, 1], we arrive at

∫ 0 1 f ( t s 1 + ( 1 − t ) s 2 ) w ( t s 1 + ( 1 − t ) s 2 ) d t ≤ ∫ 0 1 1 n ∑ i = 1 n ( e t − 1 ) i f ( s 1 ) w ( t s 1 + ( 1 − t ) s 2 ) d t + ∫ 0 1 1 n ∑ i = 1 n ( e 1 − t − 1 ) i f ( s 2 ) w ( t s 1 + ( 1 − t ) s 2 ) d t .

By substituting s = t s 1 + ( 1 − t ) s 2 , we acquire

∫ s 2 s 1 f ( s ) w ( s ) s 1 − s 2 d s ≤ 1 n ( s 1 − s 2 ) ∑ i = 1 n f ( s 1 ) ∫ s 2 s 1 ( e s 2 − s s 2 − s 1 − 1 ) i w ( s ) d s + f ( s 2 ) ∫ s 2 s 1 ( e s − s 1 s 2 − s 1 − 1 ) i w ( s ) d s

This can also be written as

(13) ∫ s 1 s 2 f ( s ) w ( s ) d s ≤ 1 n ∑ i = 1 n f ( s 1 ) ∫ s 1 s 2 ( e s 2 − s s 2 − s 1 − 1 ) i w ( s ) d s + f ( s 2 ) ∫ s 1 s 2 ( e s − s 1 s 2 − s 1 − 1 ) i w ( s ) d s ,

which is the right side of inequality (9).

Ultimately, we combine inequalities (11) and (13) to achieve

( 2 − e ) n ( e − 1 ) ( 1 − ( e − 1 ) n ) f s 1 + s 2 2 ∫ s 1 s 2 w ( s ) d s − f ( s 2 ) ∫ s 1 s 2 w ( s ) d s ≤ ∫ s 1 s 2 f ( s ) w ( s ) d s ≤ 1 n ∑ i = 1 n f ( s 1 ) ∫ s 1 s 2 ( e s 2 − s s 2 − s 1 − 1 ) i w ( s ) d s + f ( s 2 ) ∫ s 1 s 2 ( e s − s 1 s 2 − s 1 − 1 ) i w ( s ) d s .

Thus, the proof is completed.□

3 Applications to generalized average values

Mean is the average value that is used to sum up the statistical data. These are extensively utilized in mathematics, statistics, economics, and other numerical fields. In this section, to gain the relations for the following means, we apply the novel inequalities of Section 2.

Arithmetic mean:

Let s 1 , s 2 ∈ R ,

A = A ( s 1 , s 2 ) = s 1 + s 2 2 , s 1 , s 2 ≥ 0 .

r-Logarithmic mean:

Let s 1 , s 2 ∈ R ,

L r ( s 1 , s 2 ) = s 1 if s 1 = s 2 ; s 2 r + 1 − s 1 r + 1 ( r + 1 ) ( s 2 − s 1 ) 1 r if s 1 ≠ s 2 , r ∈ R \ { − 1,0 } , s 1 , s 2 > 0 .

Proposition 1

Let s 1 , s 2 ∈ R , 0 < s 1 < s 2 , h ∈ N , where h ≥ 2 . Then, for all q > 1 , the inequality

∑ m = 0 n − 1 1 n A ( n − m ) s 1 + m s 2 n h , ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n h − ( L h ( s 1 , s 2 ) ) h ≤ ∑ m = 0 n − 1 h ( s 2 − s 1 ) 2 1 − 1 q n 2 1 1 + p 1 p ( e − 1 ) ( 1 − ( e − 1 ) h ) ( 2 − e ) h 2 q [ s 1 ( h − 1 ) q + s 2 ( h − 1 ) q ] 1 q

holds.

Proof

In Corollary 1, if we substitute f ( s ) = s h where s ∈ [ s 1 , s 2 ] , h ∈ N , and h ≥ 2 , the result is obtained.□

Proposition 2

Let s 1 , s 2 ∈ R , 0 < s 1 < s 2 , h ∈ N , where h ≥ 2 . Then, for all q > 1 , the inequality

∑ m = 0 n − 1 1 n A ( n − m ) s 1 + m s 2 n h , ( n − m − 1 ) s 1 + ( m + 1 ) s 2 n h − ( L h ( s 1 , s 2 ) ) h ≤ ∑ m = 0 n − 1 s 2 − s 1 2 2 − 1 q n 2 ( e − 1 ) ( 1 − ( e − 1 ) h ) ( 2 − e ) h 2 q [ s 1 ( h − 1 ) q + s 2 ( h − 1 ) q ] 1 q

holds.

Proof

In Corollary 3, if we substitute f ( s ) = s h where s ∈ [ s 1 , s 2 ] , h ∈ N , and h ≥ 2 , the result is obtained.□

4 Conclusions

In this article, we have built generalized inequalities of the Hermite-Hadamard type with regard to n -polynomials exponential-type convex functions. In this regard, we employed Hölder’s inequality and power-mean inequality. These inequalities are extremely useful in investigating generalized inequalities as well as in other fields of study such as statistics and engineering. To validate the results, examples are also rendered. Graphical representation is also brought forth for further explication. Special cases of these inequalities are also derived. Moreover, we attained a generalized form of Fejér-type inequality utilizing n -polynomials exponential-type convex functions. Finally, we demonstrated the relation of these new results to some special means. Furthermore, these inequalities can prove worthwhile in dealing with other mathematical problems. Moreover, we hope that these new results will prove fruitful to further refinements using different convex functions.

msamraizuos@gmail.com, drgauhar.rahman@hu.edu.pk

Acknowledgments

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/43/46.

Funding information: The authors state no funding involved.
Author contributions: The authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this research as no datasets were generated or analysed during the current study.

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Received: 2024-06-24

Revised: 2025-03-11

Accepted: 2025-08-20

Published Online: 2025-10-09

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0176

Keywords for this article

n-polynomial exponential-type convex; Hölder’s inequality; Power-mean inequality; Fejér-type inequality; means

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