Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation

Muhammad Bilal Khan; Muhammad Aslam Noor; Jorge E. Macías-Díaz; Mohamed S. Soliman; Hatim Ghazi Zaini

doi:10.1515/dema-2022-0023

Open Access Published by De Gruyter Open Access August 13, 2022

Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation

Muhammad Bilal Khan , Muhammad Aslam Noor , Jorge E. Macías-Díaz , Mohamed S. Soliman and Hatim Ghazi Zaini

From the journal Demonstratio Mathematica

https://doi.org/10.1515/dema-2022-0023

Abstract

It is a well-known fact that inclusion and pseudo-order relations are two different concepts which are defined on the interval spaces, and we can define different types of convexities with the help of both relations. By means of pseudo-order relation, the present article deals with the new notions of convex functions which are known as left and right log- s -convex interval-valued functions (IVFs) in the second sense. The main motivation of this study is to present new inequalities for left and right log- s -convex-IVFs. Therefore, we establish some new Jensen-type, Hermite-Hadamard (HH)-type, and Hermite-Hadamard-Fejér (HH-Fejér)-type inequalities for this kind of IVF, which generalize some known results. To strengthen our main results, we provide nontrivial examples of left and right log- s -convex IVFs.

Keywords: left and right log-s-convex interval-valued functions; Hermite-Hadamard-type inequality; Hermite-Hadamard-Fejér-type inequality; Jensen’s inequality

MSC 2010: 26A33; 26A51; 26D10

1 Introduction

In the last few decades, the concept of classical convexity has been discussed and generalized by several authors. Hanson [1] introduced the most important and significant generalization of convex functions, which is that of invex functions. Hanson discussed basic properties and results of this concept which has greatly expanded the role and applications of invexity in nonlinear optimization and other fields of pure and applied mathematics. Therefore, much attention has been given in studying and characterizing the different approaches of the classical idea of convexity. Therefore, with the help of convexity, the following familiar HH-integral inequality [2,3] is defined in the literature:

(1) S σ + ς 2 ≤ 1 ς − σ ∫ σ ς S ( ϖ ) d ϖ ≤ S ( σ ) + S ( ς ) 2 ,

where S : K → R is a convex function on the interval K = [ σ , ς ] with σ < ς . So the concept of convexity with integral problem is an interesting area for research. Recently, many extensions and generalizations of HH-inequality for generalized convex functions have been established. For more useful details, see [3,4,5,6,7,8,9,10,11,12,13] and references therein.

In recent years, several refinements of the HH-inequalities have been obtained for log-convex functions and their variants. Because, it is well-known that log-convex functions have great importance in convex theory because using these functions we can derive more accurate inequalities as compared to convex functions. Recently, some of the authors discussed different classes and related inequalities of log-convex and log-nonconvex functions.

For log-convex functions and their variants, various refinements of the HH-inequalities have been found in recent years. Several writers recently examined the various classes of log-convex and log-nonconvex functions, for example, log- h -convex [14], s-logarithmically convexity [15], h -convexity [16], and log-preinvexity [17], as well as associated inequalities. The log-convex functions introduced by Pecaric et al. [18] are a significant subclass of convex functions. Furthermore, in both pure and applied mathematics, these inequalities play a crucial role for log-convex functions. We encourage readers to study more about the applications and properties of HH- and Jensen inequalities in the literature [19,20,21,22,23,24,25].

Interval analysis, on the other hand, was mostly forgotten for a long time due to a lack of applicability in other fields. Moore [26] developed and researched the notion of interval analysis. It is the first time in numerical analysis that it is applied to calculate the error boundaries of the numerical solutions of a finite state machine. Since then, a number of analysts have focused on and researched interval analysis and interval-valued functions (IVFs) in mathematics and its applications. Therefore, several authors studied the literature and applications in neural network output optimization, automatic error analysis, computational physics, robotics, computer graphics, and several other well-known areas in science and technology. We encourage readers to conduct more research into fundamental aspects and applications in the literature (see [27,28,29,30] and references therein).

The theory of fuzzy sets and systems has progressed in a number of ways since its introduction five decades ago, as seen in [31]. As a result, it is useful in the study of a variety of issues in pure mathematics and applied sciences, such as operation research, computer science, management sciences, artificial intelligence, control engineering, and decision sciences. Convex analysis has contributed significantly to the advancement of several sectors of practical and pure research. Similarly, the concepts of convexity and non-convexity are important in fuzzy optimization because we obtain fuzzy variational inequalities when we characterize the optimality condition of convexity, so variational inequality theory and fuzzy complementary problem theory established powerful mechanisms of mathematical problems and have a friendly relationship. Costa [32], Costa and Roman-Flores [33], Flores-Franulic et al. [34], Roman-Flores et al. [35,36], and Chalco-Cano et al. [37,38] have recently generalized several classical discrete and integral inequalities not only to the environment of IVF and fuzzy-IVFs but also to more general set valued maps by Nikodem et al. [39], and Zhang et al. [40] used a pseudo-order relation to establish a novel version of Jensen’s inequalities for set-valued and fuzzy set-valued functions, proving that these Jensen’s inequalities are an expanded form of Costa Jensen’s inequalities [32]. Zhao et al. [41], inspired by the previous research, developed h -convex IVFs and have shown that the HH-inequality for h -convex IVFs exists. Guo et al. [42] took a step ahead by introducing the log- h -convex IVF class and establishing the interval Jensen and HH-inequalities for log-h-convex IVFs. We show left and right interval Jensen inequality, HH-inequality, and HH-Fejér inequality for left and right log-s-interval IVFs using a pseudo-order relation, as inspired by Costa and Roman-Flores [33] and Zhang et al. [40]. For more information related to interval inequalities see [43,44,45,46,47,48,49,50] and references therein.

This study is organized as follows: Section 2 presents some preliminary results and some new notions on interval space. Section 3 derives the new interval version of Jensen-, Hadamard-, and Hadamard-Fejér-type inequalities. It also provides some useful examples to strengthen this study. Finally, Section 4 presents our final considerations.

2 Preliminaries

Let K C be the collection of all closed and bounded intervals of R that is K C = { [ U * , U * ] : U * , U * ∈ R and U * ≤ U * } . If U * ≥ 0 , then [ U * , U * ] is named as positive interval. The set of all positive intervals is denoted by K C + and defined as K C + = { [ U * , U * ] ∈ K C : U * ≥ 0 } .

We now discuss some properties of intervals under the arithmetic operations of addition, multiplication and scalar multiplication. If [ Z * , Z * ] , [ U * , U * ] ∈ K C and ρ ∈ R , then arithmetic operations are defined by

[ Z * , Z * ] + [ U * , U * ] = [ Z * + U * , Z * + U * ] ,

[ Z * , Z * ] × [ U * , U * ] = [ min { Z * U * , Z * U * , Z * U * , Z * U * } , max { Z * U * , Z * U * , Z * U * , Z * U * } ] ,

ρ . [ Z * , Z * ] = [ ρ Z * , ρ Z * ] if ρ ≥ 0 , [ ρ Z * , ρ Z * ] if ρ < 0 .

For [ Z * , Z * ] , [ U * , U * ] ∈ K C , the inclusion “ ⊆ ” is defined by

[ Z * , Z * ] ⊆ [ U * , U * ] , if and only if U * ≤ Z * , Z * ≤ U * .

Let f ( Z ) be a real-valued function, we obtain an extension f ( [ Z * , Z * ] ) of f ( Z ) by replacing the real variable Z with an interval variable [ Z * , Z * ] and the real arithmetic operations with the corresponding interval operators. The resulting f ( [ Z * , Z * ] ) is called a natural interval extension of f ( Z ) . In particular, when f ( Z ) is monotonic and continuous, we have

f ( [ Z * , Z * ] ) = [ min { f ( Z * ) , f ( Z * ) } , max { f ( Z * ) , f ( Z * ) } ] ,

e .g.,

If f ( Z ) = e Z , Z ∈ R , then f ( [ Z * , Z * ] ) = e [ Z * , Z * ] = [ e Z * , e Z * ] .
If f ( Z ) = ln ( Z ) , Z > 0 , then f ( [ Z * , Z * ] ) = ln [ Z * , Z * ] = [ ln ( Z * ) , ln ( Z * ) ] , for Z * > 0 .
If ( Z ) = Z Y , Z > 0 , Y > 0 , then f ( [ Z * , Z * ] ) = [ Z * , Z * ] Y = [ Z * Y , Z * Y ] , for Z * > 0 .

Remark 2.1

[40] (i) The relation “ ≤ p ” defined on K C by

[ Z * , Z * ] ≤ p [ U * , U * ] if and only if Z * ≤ U * , Z * ≤ U * ,

for all [ Z * , Z * ] , [ U * , U * ] ∈ K C , is a pseudo-order relation.

(ii) It can be easily seen that “ ≤ p ” looks like “left and right” on the real line R , so we call “ ≤ p ” is “left and right.”

The concept of Riemann integral for IVF first introduced by [25] is defined as follows:

Theorem 2.2

[26] If S : [ σ , ς ] ⊂ R → K C is an IVF such that S ( ϖ ) = [ S * ( ϖ ) , S * ( ϖ ) ] , then S is Riemann integrable over [ σ , ς ] if and only if S * and S * both are Riemann integrable over [ σ , ς ] such that

( IR ) ∫ σ ς S ( ϖ ) d ϖ = ( R ) ∫ σ ς S * ( ϖ ) d ϖ , ( R ) ∫ σ ς S * ( ϖ ) d ϖ ,

where S * , S * : [ σ , ς ] → R . The collection of all Riemann integrable real-valued functions and Riemann integrable IVFs are denoted by R [ σ , ς ] and I R [ σ , ς ] , respectively.

Definition 2.3

[18] A function S : K → R + is named as log-convex function if

(2) S ( ζ ϖ + ( 1 − ζ ) y ) ≤ [ S ( ϖ ) ] ζ [ S ( y ) ] 1 − ζ , ∀ ϖ , y ∈ K , ζ ∈ [ 0 , 1 ] .

If (2) is reversed, then S is named as log-concave function.

Definition 2.4

[14] A function S : K → R + is named as log- P function if

(3) S ( ζ ϖ + ( 1 − ζ ) y ) ≤ S ( ϖ ) S ( y ) , ∀ ϖ , y ∈ K , ζ ∈ [ 0 , 1 ] .

If (3) is reversed, then S is named as log- P -concave.

Definition 2.5

[15] The positive real-valued function S : [ σ , ς ] → R + is named as left and right log- s -convex function in the second sense if for all ϖ , y ∈ [ σ , ς ] and s , ζ ∈ [ 0 , 1 ] we have

(4) S ( ζ ϖ + ( 1 − ζ ) y ) ≤ [ S ( ϖ ) ] ζ s [ S ( y ) ] ( 1 − ζ ) s .

If inequality (4) is reversed, then S is named as log- s -concave on [ σ , ς ] . The set of all log- s -convex (log- s -concave) functions is denoted by

SX ( [ σ , ς ] , R + ) ( SV ( [ σ , ς ] , R + ) ) .

Definition 2.6

The IVF S : [ σ , ς ] → K C + is named as left and right log- s -convex-IVF in the second sense if for all ϖ , y ∈ [ σ , ς ] and s , ζ ∈ [ 0 , 1 ] we have

(5) S ( ζ ϖ + ( 1 − ζ ) y ) ≤ p [ S ( ϖ ) ] ζ s [ S ( y ) ] ( 1 − ζ ) s .

If inequality (5) is reversed, then S is named as left and right log- s -concave on [ σ , ς ] . The set of all left and right log- s -convex IVFs is denoted by

LRSX ( [ σ , ς ] , K C + , s ) ( LRSV ( [ σ , ς ] , K C + , s ) ) .

Remark 2.7

If one takes s = 1 , then from (5) we achieve the inequality:

(6) S ( ζ ϖ + ( 1 − ζ ) y ) ≤ p [ S ( ϖ ) ] ζ [ S ( y ) ] 1 − ζ , ∀ ϖ , y ∈ K , ζ ∈ [ 0 , 1 ] .

If one takes s = 0 , then from (5) we achieve the inequality:

(7) S ( ζ ϖ + ( 1 − ζ ) y ) ≤ p [ S ( ϖ ) ] [ S ( y ) ] , ∀ ϖ , y ∈ K , ζ ∈ [ 0 , 1 ] .

Theorem 2.8

Let S : [ σ , ς ] → K C + be an IVF defined by S ( ϖ ) = [ S * ( ϖ ) , S * ( ϖ ) ] , for all ϖ ∈ [ σ , ς ] . Then S ∈ LRSX ( [ σ , ς ] , K C + , s ) if and only if S * , S * ∈ SX ( [ σ , ς ] , R + ) .

Proof

Assume that S * , S * ∈ SX ( [ σ , ς ] , K C + ) . Then, for all ϖ , y ∈ [ σ , ς ] , ζ ∈ [ 0 , 1 ] , we have

S * ( ζ ϖ + ( 1 − ζ ) y ) ≤ [ S * ( ϖ ) ] ζ s [ S * ( y ) ] ( 1 − ζ ) s

and

S * ( ζ ϖ + ( 1 − ζ ) y ) ≤ [ S * ( ϖ ) ] ζ s [ S * ( y ) ] ( 1 − ζ ) s .

From Definition 2.5 and order relation ≤ p , we have

[ S * ( ζ ϖ + ( 1 − ζ ) y ) , S * ( ζ ϖ + ( 1 − ζ ) y ) ] ≤ p [ [ S * ( ϖ ) ] ζ s [ S * ( y ) ] ( 1 − ζ ) s , [ S * ( ϖ ) ] ζ s [ S * ( y ) ] ( 1 − ζ ) s ]

= [ [ S * ( ϖ ) ] ζ s , [ S * ( ϖ ) ] ζ s ] [ [ S * ( y ) ] ( 1 − ζ ) s , [ S * ( y ) ] ( 1 − ζ ) s ] ,

that is

S ( ζ ϖ + ( 1 − ζ ) y ) ≤ p [ S ( ϖ ) ] ζ s [ S ( y ) ] ( 1 − ζ ) s , ∀ ϖ , y ∈ [ σ , ς ] , ζ ∈ [ 0 , 1 ] .

Hence, S ∈ LRSX ( [ σ , ς ] , K C + , s ) .

Conversely, let S ∈ LRSX ( [ σ , ς ] , K C + , s ) . Then for all ϖ , y ∈ [ σ , ς ] and ζ ∈ [ 0 , 1 ] , we have

S ( ζ ϖ + ( 1 − ζ ) y ) ≤ p ζ [ S ( ϖ ) ] ζ s [ S ( y ) ] ( 1 − ζ ) s .

That is

[ S * ( ζ ϖ + ( 1 − ζ ) y ) , S * ( ζ ϖ + ( 1 − ζ ) y ) ] ≤ p [ [ S * ( ϖ ) ] ζ s [ S * ( y ) ] ( 1 − ζ ) s , [ S * ( ϖ ) ] ζ s [ S * ( y ) ] ( 1 − ζ ) s ] .

It follows that

S * ( ζ ϖ + ( 1 − ζ ) y ) ≤ [ S * ( ϖ ) ] ζ s [ S * ( y ) ] ( 1 − ζ ) s

and

S * ( ζ ϖ + ( 1 − ζ ) y ) ≤ [ S * ( ϖ ) ] ζ s [ S * ( y ) ] ( 1 − ζ ) s .

Hence, the result follows.□

Remark 2.9

If one takes S * ( ϖ ) = S * ( ϖ ) , then from (5), we achieve the inequality (3), see [43].

If one takes S * ( ϖ ) = S * ( ϖ ) with s = 1 , then from (5) we acquire the inequality, see [31]:

S ( ζ ϖ + ( 1 − ζ ) y ) ≤ [ S ( ϖ ) ] ζ [ S ( y ) ] ( 1 − ζ ) .

If one takes S * ( ϖ ) = S * ( ϖ ) with s = 0 , then from (5) we acquire the inequality, see [29]:

S ( ζ ϖ + ( 1 − ζ ) y ) ≤ S ( ϖ ) S ( y ) .

Example 2.10

We consider the IVF S : [ σ , ς ] = [ 0 , 8 ] → K C + defined by

S ( ϖ ) = [ ϖ , 2 e ϖ ] .

Since end point functions S * ( ϖ ) = ϖ , S * ( ϖ ) = 2 e ϖ are log- s -convex functions in the second sense, then, by Theorem 2.8, S ( ϖ ) is left and right log- s -convex-IVF in the second sense.

Theorem 2.11

Let S : [ σ , ς ] → K C + be an IVF defined by S ( ϖ ) = [ S * ( ϖ ) , S * ( ϖ ) ] , for all ϖ ∈ [ σ , ς ] . Then S ∈ LRSV ( [ σ , ς ] , K C + , s ) if and only if, S * , S * ∈ SV ( [ σ , ς ] , R + ) .

Proof

It is similar to the proof of Theorem 2.8.□

Example 2.12

We consider the IVF S : [ σ , ς ] = [ 0 , 4 ] → K C + defined by

S ( ϖ ) = [ − 5 , − 1 ] ϖ .

Since end point functions S * ( ϖ ) = − 5 ϖ , S * ( ϖ ) = − ϖ are log- s -concave functions in the second sense, then, by Theorem 2.11, S ( ϖ ) is left and right log- s -concave-IVF in the second sense.

3 Main results

In this section, we derive some new Jensen, HH, and HH-Fejér inequalities for left and right log- s -concvex-IVFs in the second sense. These inequalities are used to find the lower and upper bounds of the integral mean value of S . First, we give the following result connected with the Jensen inequality.

Theorem 3.1

Let ω j ∈ R + , σ j ∈ [ σ , ς ] ( j = 1 , 2 , 3 , … k , k ≥ 2 ) and let S : [ σ , ς ] → K C + be an IVF such that S ( ϖ ) = [ S * ( ϖ ) , S * ( ϖ ) ] . If S ∈ LRSX ( [ σ , ς ] , K C + , s ) , then

(8) S 1 W k ∑ j = 1 k ω j σ j ≤ p ∏ j = 1 k [ S ( σ j ) ] ω j W k s ,

where W k = ∑ j = 1 k ω j . If S ∈ LRSV ( [ σ , ς ] , K C + , s ) , then equation (8) is reversed.

Proof

When k = 2 , equation (8) is true. Consider equation (8) is true for k = n − 1 , then

S 1 W n − 1 ∑ j = 1 n − 1 ω j σ j ≤ p ∏ j = 1 n − 1 [ S ( σ j ) ] ω j W n − 1 s .

Now, let us prove that equation (46) holds for k = n .

S 1 W n ∑ j = 1 n ω j σ j = S 1 W n − 2 ∑ j = 1 n − 2 ω j σ j + ω n − 1 + ω n W n ω n − 1 ω n − 1 + ω n σ n − 1 + ω n ω n − 1 + ω n σ n .

Then, we have

S ⁎ 1 W n ∑ j = 1 n ω j σ j ≤ S ⁎ 1 W n ∑ j = 1 n − 2 ω j σ j + ω n − 1 + ω n W n ω n − 1 ω n − 1 + ω n σ n − 1 + ω n ω n − 1 + ω n σ n , ≤ ∏ j = 1 n − 2 [ S ⁎ ( σ j ) ] ω j W n s S ⁎ ω n − 1 ω n − 1 + ω n σ n − 1 + ω n ω n − 1 + ω n σ n ω n − 1 + ω n W n s , ≤ ∏ j = 1 n − 2 [ S ⁎ ( σ j ) ] ω j W n s [ S ⁎ ( σ n − 1 ) ] ω n − 1 ω n − 1 + ω n s [ S ⁎ ( σ n ) ] ω n ω n − 1 + ω n s ω n − 1 + ω n W n s , ≤ ∏ j = 1 n − 2 [ S ⁎ ( σ j ) ] ω j W n s [ S ⁎ ( σ n − 1 ) ] ω n − 1 W n s [ S ⁎ ( σ n ) ] ω n W n s , = ∏ j = 1 n [ S ⁎ ( σ j ) ] ω j W n s .

Similarly, for S * ( ϖ ) we have

S * 1 W n ∑ j = 1 n ω j σ j ≤ ∏ j = 1 n [ S * ( σ j ) ] ω j W n s .

From which, we have

S * 1 W n ∑ j = 1 n ω j σ j , S * 1 W n ∑ j = 1 n ω j σ j ≤ p ∏ j = 1 n [ S * ( σ j ) ] ω j W n s , ∏ j = 1 n [ S * ( σ j ) ] ω j W n s ,

that is,

S 1 W n ∑ j = 1 n ω j σ j ≤ p ∏ j = 1 n [ S ( σ j ) ] ω j W n s ,

and the result follows.

If ω 1 = ω 2 = ω 3 = … = ω k = 1 , then Theorem 3.10 reduces to the following result:□

Corollary 3.2

Let σ j ∈ [ σ , ς ] ( j = 1 , 2 , 3 , … k , k ≥ 2 ) and let S : [ σ , ς ] → K C + be an IVF such that S ( ϖ ) = [ S * ( ϖ ) , S * ( ϖ ) ] for all ϖ ∈ [ σ , ς ] . If S ∈ LRSX ( [ σ , ς ] , K C + , s ) , then

(9) S 1 k ∑ j = 1 k σ j ≤ p ∏ j = 1 k [ S ( σ j ) ] 1 k s .

If S ∈ LRSV ( [ σ , ς ] , K C + , s ) , then equation ( 9 ) is reversed.

Theorem 3.3

Let ω j ∈ R + , σ j ∈ [ σ , ς ] ( j = 1 , 2 , 3 , … , k , k ≥ 2 ) and let S : [ σ , ς ] → K C + be an IVF such that S ( ϖ ) = [ S * ( ϖ ) , S * ( ϖ ) ] . If S ∈ LRSX ( [ σ , ς ] , K C + , s ) and ( α , β ) ⊆ [ σ , ς ] , then

(10) ∏ j = 1 k [ S ( σ j ) ] ω j W k s ≤ p ∏ j = 1 k [ S ( α ) ] β − σ j β − α s ω j W k s [ S ( β ) ] σ j − α M − α s ω j W k s ,

where W k = ∑ j = 1 k ω j . If S ∈ LRSV ( [ σ , ς ] , K C + , s ) , then equation (10) is reversed.

Proof

Consider = σ 1 , σ j = σ 2 , ( j = 1 , 2 , 3 , … , k ) , β = σ 3 in equation (48). Then, we have

S * ( σ j ) ≤ [ S * ( α ) ] β − σ j β − α s [ S * ( β ) ] σ j − α M − α s .

The above inequality can be written as follows:

(11) S * ( σ j ) ω j W k s ≤ [ S * ( α ) ] β − σ j β − α s ω j W k s [ S * ( β ) ] σ j − α M − α s ω j W k s .

Taking multiplication of all inequalities (11) for j = 1 , 2 , 3 , … , k , we have

∏ j = 1 k S * ( σ j ) ω j W k s ≤ ∏ j = 1 k [ S * ( α ) ] β − σ j β − α s ω j W k s [ S * ( β ) ] σ j − α M − α s ω j W k s .

Similarly, for S * ( ϖ ) we have

∏ j = 1 k S * ( σ j ) ω j W k s ≤ ∏ j = 1 k [ S * ( α ) ] β − σ j β − α s ω j W k s [ S * ( β ) ] σ j − α M − α s ω j W k s ,

that is,

∏ j = 1 k S ( σ j ) ω j W k s = ∏ j = 1 k S * ( σ j ) ω j W k s , ∏ j = 1 k S * ( σ j ) ω j W k s

≤ p ∏ j = 1 k [ S * ( α ) ] β − σ j β − α s ω j W k s [ S * ( β ) ] σ j − α M − α s ω j W k s , ∏ j = 1 k [ S * ( α ) ] β − σ j β − α s ω j W k s [ S * ( β ) ] σ j − α M − α s ω j W k s , ≤ p ∏ j = 1 k [ S * ( α ) ] β − σ j β − α s ω j W k s , [ S * ( α ) ] β − σ j β − α s ω j W k s . ∏ j = 1 k [ S * ( β ) ] σ j − α M − α s ω j W k s , [ S * ( β ) ] σ j − α M − α s ω j W k s .

= ∏ j = 1 k [ S ( α ) ] β − σ j β − α s ω j W k s . ∏ j = 1 k [ S ( β ) ] σ j − α M − α s ω j W k s .

Thus,

∏ j = 1 k [ S ( σ j ) ] ω j W k s ≤ p ∏ j = 1 k [ S ( α ) ] β − σ j β − α s ω j W k s [ S ( β ) ] σ j − α M − α s ω j W k s ,

this completes the proof.□

Remark 3.4

If one takes s = 1 , then from (8) and (10), we obtain the inequalities:

S 1 W k ∑ j = 1 k ω j σ j ≤ p ∏ j = 1 k [ S ( σ j ) ] ω j W k

and

∏ j = 1 k [ S ( σ j ) ] ω j W k ≤ p ∏ j = 1 k [ S ( α ) ] β − σ j β − α ω j W k [ S ( β ) ] σ j − α M − α ω j W k ,

respectively.

If S * ( ϖ ) = S * ( ϖ ) , then from (8) and (10), we obtain the inequalities, see [16]:

S 1 W k ∑ j = 1 k ω j σ j ≤ p ∏ j = 1 k [ S ( σ j ) ] ω j W k s

and

∏ j = 1 k [ S ( σ j ) ] ω j W k s ≤ p ∏ j = 1 k [ S ( α ) ] β − σ j β − α s ω j W k s [ S ( β ) ] σ j − α M − α s ω j W k s ,

respectively.

If S * ( ϖ ) = S * ( ϖ ) with s = 1 , then from (8) and (10), we obtain the inequalities, see [21]:

S 1 W k ∑ j = 1 k ω j σ j ≤ ∏ j = 1 k [ S ( σ j ) ] ω j W k

and

∏ j = 1 k [ S ( σ j ) ] ω j W k ≤ ∏ j = 1 k [ S ( α ) ] β − σ j β − α ω j W k [ S ( β ) ] σ j − α M − α ω j W k ,

respectively.

We now derive the HH-inequality for left and right log- s -convex-IVFs in the second sense and verify it with the help of nontrivial example.

Theorem 3.5

Let S : [ σ , ς ] → K C + be an IVF such that S ( ϖ ) = [ S * ( ϖ ) , S * ( ϖ ) ] and S ∈ I R ( [ σ , ς ] ) . If S ∈ LRSX ( [ σ , ς ] , K C + , s ) , then

(12) S σ + ς 2 2 s − 1 ≤ p exp 1 ς − σ ( IR ) ∫ σ ς ln S ( ϖ ) d ϖ ≤ p [ S ( σ ) S ( ς ) ] 1 s + 1 .

If S ∈ LRSV ( [ σ , ς ] , K C + , s ) , then

S σ + ς 2 2 s − 1 ≥ p exp 1 ς − σ ( IR ) ∫ σ ς ln S ( ϖ ) d ϖ ≥ p [ S ( σ ) S ( ς ) ] 1 s + 1 .

Proof

Let S : [ σ , ς ] → K C + be a left and right log- s -convex-IVF in the second sense. Then, by hypothesis, we have

S σ + ς 2 ≤ p [ S ( ζ σ + ( 1 − ζ ) ς ) ] 1 2 s [ S ( ( 1 − ζ ) σ + ζ ς ) ] 1 2 s .

Then, we have

(13) S * σ + ς 2 ≤ [ S * ( ζ σ + ( 1 − ζ ) ς ) ] 1 2 s [ S * ( ( 1 − ζ ) σ + ζ ς ) ] 1 2 s .

Taking logarithms on both sides of (13), then we obtain

1 1 2 s ln S * σ + ς 2 ≤ ln S * ( ζ σ + ( 1 − ζ ) ς ) + ln S * ( ( 1 − ζ ) σ + ζ ς ) .

Then,

1 1 2 s ∫ 0 1 ln S * σ + ς 2 d ζ ≤ ∫ 0 1 ln S * ( ζ σ + ( 1 − ζ ) ς ) d ζ + ∫ 0 1 ln S * ( ( 1 − ζ ) σ + ζ ς ) d ζ .

It follows that

1 2 1 2 s ln S * σ + ς 2 ≤ 1 ς − σ ∫ σ ς ln S * ( ϖ ) d ϖ ,

which implies that

S * σ + ς 2 2 s − 1 ≤ exp 1 ς − σ ∫ σ ς ln S * ( ϖ ) d ϖ .

Similarly, for S * ( ϖ ) we have

S * σ + ς 2 2 s − 1 ≤ exp 1 ς − σ ∫ σ ς ln S * ( ϖ ) d ϖ .

That is,

S * σ + ς 2 2 s − 1 , S * σ + ς 2 2 s − 1 ≤ p exp 1 ς − σ ∫ σ ς ln S * ( ϖ ) d ϖ , exp 1 ς − σ ∫ σ ς ln S * ( ϖ ) d ϖ .

Thus,

(14) S σ + ς 2 2 s − 1 ≤ p exp 1 ς − σ ( IR ) ∫ σ ς ln S ( ϖ ) d ϖ .

In a similar way as above, we have

(15) exp 1 ς − σ ( IR ) ∫ σ ς ln S ( ϖ ) d ϖ ≤ p [ S ( σ ) S ( ς ) ] 1 s + 1 .

Combining (14) and (15), we have

S σ + ς 2 2 s − 1 ≤ p exp 1 ς − σ ( IR ) ∫ σ ς ln S ( ϖ ) d ϖ ≤ p [ S ( σ ) S ( ς ) ] 1 s + 1 ,

the required result.□

Remark 3.6

If s = 1 , then from (12), we achieve the inequality:

S σ + ς 2 ≤ p exp 1 ς − σ ( IR ) ∫ σ ς ln S ( ϖ ) d ϖ ≤ p S ( σ ) S ( ς ) .

If s = 0 , then from (12), we achieve the inequality:

S σ + ς 2 1 2 ≤ p exp 1 ς − σ ( IR ) ∫ σ ς ln S ( ϖ ) d ϖ ≤ p S ( σ ) S ( ς ) .

If S * ( ϖ ) = S * ( ϖ ) , then from (12), we acquire the inequality, see [14]:

S σ + ς 2 2 s − 1 ≤ exp 1 ς − σ ( R ) ∫ σ ς ln S ( ϖ ) d ϖ ≤ [ S ( σ ) S ( ς ) ] 1 s + 1 .

If S * ( ϖ ) = S * ( ϖ ) with s = 1 , then from (12), we acquire the inequality, see [20]:

S σ + ς 2 ≤ exp 1 ς − σ ( R ) ∫ σ ς ln S ( ϖ ) d ϖ ≤ S ( σ ) S ( ς ) .

If S * ( ϖ ) = S * ( ϖ ) with s = 0 , then from (12), we achieve the inequality, see [14]:

S σ + ς 2 1 2 ≤ exp 1 ς − σ ( R ) ∫ σ ς ln S ( ϖ ) d ϖ ≤ S ( σ ) S ( ς ) .

Example 3.7

We consider s = 1 , and the IVF S : [ σ , ς ] = [ 1,4 ] → K C + defined by, S ( ϖ ) = [ e ϖ , e ϖ 2 ] , then S ( ϖ ) is left and right log- 1 -convex-IVF in the second sense because both S * ( ϖ ) and S * ( ϖ ) are log- 1 -convex functions. Since S * ( ϖ ) = e ϖ and S * ( ϖ ) = e ϖ 2 , then we have

S * σ + ς 2 2 s − 1 = e 5 2 ,

exp 1 ς − σ ∫ σ ς ln S * ( ϖ ) d ϖ = exp 1 3 ∫ 1 4 ln ( e ϖ ) d ϖ = e 5 2 ,

[ S * ( σ ) S * ( ς ) ] 1 s + 1 = [ ( 1 ) ( 4 ) ] 1 2 = e 5 2 .

That means

e 5 2 ≤ e 5 2 ≤ e 5 2 .

Similarly, it can be easily shown that

S * σ + ς 2 2 s − 1 ≤ exp 1 ς − σ ∫ σ ς ln S * ( ϖ ) d ϖ ≤ [ S * ( σ ) S * ( ς ) ] 1 s + 1 ,

such that

S * σ + ς 2 2 s − 1 = e 5 2 2 = e 25 4 ,

exp 1 ς − σ ∫ σ ς ln S * ( ϖ ) d ϖ = exp 1 3 ∫ 1 4 ln ( e ϖ 2 ) d ϖ = e 7 ,

[ S * ( σ ) S * ( ς ) ] 1 s + 1 = [ e . e 16 ] 1 2 = e 17 2 .

From which, it follows that

e 25 4 ≤ e 7 ≤ e 17 2 ,

that is,

e 5 2 , e 25 4 ≤ p e 5 2 , e 7 ≤ p e 5 2 , e 17 2 .

Hence,

S σ + ς 2 2 s − 1 ≤ p exp 1 ς − σ ( IR ) ∫ σ ς ln S ( ϖ ) d ϖ ≤ p [ S ( σ ) S ( ς ) ] 1 s + 1 .

We now present the following refinements of the second HH-Fejér inequality for left and right log- s -convex-IVF in the second sense.

Theorem 3.8

Let S : [ σ , ς ] → K C + be an IVF with σ < ς , such that S ( ϖ ) = [ S * ( ϖ ) , S * ( ϖ ) ] and S ∈ I R ( [ σ , ς ] ) . If S ∈ LRSX ( [ σ , ς ] , K C + , s ) and W : [ σ , ς ] → R , W ( ϖ ) ≥ 0 , symmetric with respect to σ + ς 2 , then

(16) 1 ς − σ ( IR ) ∫ σ ς [ ln S ( ϖ ) ] W ( ϖ ) d ϖ ≤ p ln [ S ( σ ) S ( ς ) ] ∫ 0 1 ζ s W ( ( 1 − ζ ) σ + ζ ς ) d ζ .

If S ∈ LRSV ( [ σ , ς ] , K C + , s ) , then inequality (16) is reversed.

Proof

Let S be a log- s -convex-IVF in the second sense. Then, for σ , ς ∈ [ σ , ς ] ζ ∈ [ 0 , 1 ] , we have

(17) [ ln S * ( ζ σ + ( 1 − ζ ) ς ) ] W ( ζ σ + ( 1 − ζ ) ς ) ≤ ( ζ s ln S * ( σ ) + ( 1 − ζ ) s ln S * ( ς ) ) W ( ζ σ + ( 1 − ζ ) ς ) .

And

(18) [ ln S * ( ( 1 − ζ ) σ + ζ ς ) ] W ( ( 1 − ζ ) σ + ζ ς ) ≤ ( ( 1 − ζ ) s ln S * ( σ ) + ζ s ln S * ( ς ) ) W ( ( 1 − ζ ) σ + ζ ς ) .

After adding (17) and (18), and integrating over [ 0 , 1 ] , we obtain

∫ 0 1 [ ln S * ( ζ σ + ( 1 − ζ ) ς ) ] W ( ζ σ + ( 1 − ζ ) ς ) d ζ + ∫ 0 1 ln S * ( ( 1 − ζ ) σ + ζ ς ) W ( ( 1 − ζ ) σ + ζ ς ) d ζ ≤ ∫ 0 1 ln S * ( σ ) { ζ s W ( ζ σ + ( 1 − ζ ) ς ) + ( 1 − ζ ) s W ( ( 1 − ζ ) σ + ζ ς ) } + ln S * ( ς ) { ( 1 − ζ ) s W ( ζ σ + ( 1 − ζ ) ς ) + ζ s W ( ( 1 − ζ ) σ + ζ ς ) } d ζ ,

= 2 ln S * ( σ ) ∫ 0 1 ζ s W ( ζ σ + ( 1 − ζ ) ς ) d ζ + 2 ln S * ( ς ) ∫ 0 1 ζ s W ( ( 1 − ζ ) σ + ζ ς ) d ζ .

Since W is symmetric, then

(19) = 2 ln [ S * ( σ ) S * ( ς ) ] ∫ 0 1 ζ s W ( ( 1 − ζ ) σ + ζ ς ) d ζ .

Since

(20) ∫ 0 1 [ ln S * ( ζ σ + ( 1 − ζ ) ς ) ] W ( ζ σ + ( 1 − ζ ) ς ) d ζ = ∫ 0 1 [ ln S * ( ( 1 − ζ ) σ + ζ ς ) ] W ( ( 1 − ζ ) σ + ζ ς ) d ζ = 1 ς − σ ∫ σ ς [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ .

From (19) and (20), we have

1 ς − σ ∫ σ ς [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ ≤ ln [ S * ( σ ) S * ( ς ) ] ∫ 0 1 ζ s W ( ( 1 − ζ ) σ + ζ ς ) d ζ .

Similarly, for S * ( ϖ ) we have

1 ς − σ ∫ σ ς [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ ≤ ln [ S * ( σ ) S * ( ς ) ] ∫ 0 1 ζ s W ( ( 1 − ζ ) σ + ζ ς ) d ζ ,

that is,

1 ς − σ ∫ σ ς [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ , 1 ς − σ ∫ σ ς [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ ≤ p [ ln [ S * ( σ ) S * ( ς ) ] , ln [ S * ( σ ) S * ( ς ) ] ] ∫ 0 1 ζ s W ( ( 1 − ζ ) σ + ζ ς ) d ζ ,

hence

1 ς − σ ( IR ) ∫ σ ς [ ln S ( ϖ ) ] W ( ϖ ) d ϖ ≤ p ln [ S ( σ ) S ( ς ) ] ∫ 0 1 ζ s W ( ( 1 − ζ ) σ + ζ ς ) d ζ .

This concludes the proof.□

The following results find the refinements of first HH-Fejér inequality for left and right log- s -convex-IVF in the second sense.

Theorem 3.10

Let S : [ σ , ς ] → K C + be an IVF with σ < ς , such that S ( ϖ ) = [ S * ( ϖ ) , S * ( ϖ ) ] and S ∈ I R ( [ σ , ς ] ) . If S ∈ LRSX ( [ σ , ς ] , K C + , s ) and W : [ σ , ς ] → R , W ( ϖ ) ≥ 0 , symmetric with respect to σ + ς 2 , and ∫ σ ς W ( ϖ ) d ϖ > 0 , then

(21) ln S σ + ς 2 ≤ p 2 1 − s ∫ σ ς W ( ϖ ) d ϖ ( IR ) ∫ σ ς [ ln S ( ϖ ) ] W ( ϖ ) d ϖ .

If S ∈ LRSV ( [ σ , ς ] , K C + , s ) , then inequality (21) is reversed.

Proof

Since S is a log- s -convex in the second sense, then for σ , ς ∈ [ σ , ς ] , ζ ∈ [ 0 , 1 ] we have

(22) 2 s ln S * σ + ς 2 ≤ ln S * ( ζ σ + ( 1 − ζ ) ς ) + ln S * ( ( 1 − ζ ) σ + ζ ς ) .

By multiplying (22) by W ( ( 1 − ζ ) σ + ζ ς ) = W ( ζ σ + ( 1 − ζ ) ς ) and integrating it by ζ over [ 0 , 1 ] , we obtain

(23) 2 s ln S * σ + ς 2 ∫ 0 1 W ( ( 1 − ζ ) σ + ζ ς ) d ζ ≤ ∫ 0 1 [ ln S * ( ζ σ + ( 1 − ζ ) ς ) ] W ( ζ σ + ( 1 − ζ ) ς ) d ζ + ∫ 0 1 [ ln S * ( ( 1 − ζ ) σ + ζ ς ) ] W ( ( 1 − ζ ) σ + ζ ς ) d ζ .

Since

(24) ∫ 0 1 [ ln S * ( ζ σ + ( 1 − ζ ) ς ) ] W ( ζ σ + ( 1 − ζ ) ς ) d ζ = ∫ 0 1 [ ln S * ( ( 1 − ζ ) σ + ζ ς ) ] W ( ( 1 − ζ ) σ + ζ ς ) d ζ , = 1 ς − σ ∫ σ ς [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ .

From (23) and (24), we have

ln S * σ + ς 2 ≤ 2 1 − s ∫ σ ς W ( ϖ ) d ϖ ∫ σ ς [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ .

Similarly, for S * ( ϖ ) we have

ln S * σ + ς 2 ≤ 2 1 − s ∫ σ ς W ( ϖ ) d ϖ ∫ σ ς [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ .

From which, we have

ln S * σ + ς 2 , ln S * σ + ς 2 ≤ p 2 1 − s ∫ σ ς W ( ϖ ) d ϖ ∫ σ ς [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ , ∫ σ ς [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ ,

that is,

ln S σ + ς 2 ≤ p 2 1 − s ∫ σ ς W ( ϖ ) d ϖ ( IR ) ∫ σ ς [ ln S ( ϖ ) ] W ( ϖ ) d ϖ .

Then we complete the proof.□

Remark 3.10

If one takes W ( ϖ ) = 1 , then by combining (16) and (21), we obtain (12).

If one takes s = 1 , then from (16) and (21), we obtain the inequality for left and right log-convex-IVFs.

If one takes s = 0 , then from (16) and (21), we obtain the inequality for left and right log- P -convex-IVFs.

If one takes S * ( ϖ ) = S * ( ϖ ) , then from (16) and (21), we obtain the inequality for log-s-convex function in the second sense, see [42].

If one takes S * ( ϖ ) = S * ( ϖ ) s = 1 , then from (16) and (21), we obtain the inequality for log-convex function, see [42].

If one takes S * ( ϖ ) = S * ( ϖ ) with s = 0 , then from (16) and (21), we obtain the inequality for log- P -convex function, see [42].

Example 3.11

We consider s = 1 for ζ ∈ [ 0 , 1 ] and the IVF S : [ σ , ς ] = [ π 4 , π 2 ] → K C + defined by S ( ϖ ) = [ e sin ϖ , e 2 sin ϖ ] . Since end point functions S * ( ϖ ) = e sin ϖ , S * ( ϖ ) = e 2 sin ϖ log- 1 -convex functions in the second sense, then, by Theorem 2.8, S ( ϖ ) is log- s -convex-IVF in the second sense. If

W ( ϖ ) = ϖ − π 4 , ϖ ∈ π 4 , 3 π 8 , π 2 − ϖ , ϖ ∈ 3 π 8 , π 2 ,

then, we have

1 ς − σ ∫ σ ς [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ = 4 π ∫ π 4 π 2 [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ = 4 π ∫ π 4 3 π 8 [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ + 4 π ∫ 3 π 8 π 2 [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ ,

(25) = 4 π ∫ π 4 3 π 8 ( sin ϖ ) ϖ − π 4 d ϖ + 4 π ∫ 3 π 8 π 2 ( sin ϖ ) π 2 − ϖ d ϖ ≈ 14 25 π

ln [ S * ( σ ) S * ( ς ) ] ∫ 0 1 ζ s W ( ( 1 − ζ ) σ + ζ ς ) d ζ

(26) = π 4 1 + 2 2 ∫ 0 1 2 ζ 2 d ζ + ∫ 1 2 1 ζ ( 1 + ζ ) d ζ = 17 ( 2 + 2 ) π 96 2 ,

and

(27) = 4 π ∫ π 4 3 π 8 ( 2 sin ϖ ) ϖ − π 2 d ϖ + 4 π ∫ 3 π 8 π 2 ( 2 sin ϖ ) π 2 − ϖ d ϖ ≈ 28 25 π ,

(28) ln [ S * ( σ ) S * ( ς ) ] ∫ 0 1 ζ s W ( ( 1 − ζ ) σ + ζ ς ) d ζ = π 4 ln ( 2 − α ) 2 + 1 + 2 2 ∫ 0 1 2 ζ 2 d ϖ + ∫ 1 2 1 ζ ( 1 + ζ ) d ζ = 17 ( 4 + 2 ) π 192 .

From (25)–(28), we have

14 25 π , 28 25 π ≤ p 17 ( 2 + 2 ) π 96 2 , 17 ( 4 + 2 ) π 192 .

Hence, Theorem 3.8 is verified.

For Theorem 3.9, we have

(29) ln S * σ + ς 2 = ln S * 3 π 8 ≈ 9 10 ,

∫ σ ς W ( ϖ ) d ϖ = ∫ π 4 3 π 8 ϖ − π 4 d ϖ + ∫ 3 π 8 π 2 π 2 − ϖ d ϖ ≈ 3 20 ,

(30) 2 1 − s ∫ σ ς W ( ϖ ) d ϖ ∫ σ ς [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ ≈ 20 3 7 50 = 14 15 ,

and

(31) ln S * σ + ς 2 = ln S * 3 π 8 ≈ 9 5 ,

(32) 2 1 − s ∫ σ ς W ( ϖ ) d ϖ ∫ σ ς [ ln S * ( ϖ ) ] W ( ϖ ) d ϖ ≈ 20 3 7 25 = 28 15 .

From (29)–(32), we have

9 10 , 9 5 ≤ p 14 15 , 28 15 .

Hence, Theorem 3.9 is verified.

4 Conclusion

We have introduced and considered a new class of log-convex-IVFs, which is named as left and right log- s -convex-IVFs in the second sense. We have derived several new types of Jensen and HH-inequalities. With the help of some examples, we have shown that our results include a wide class of new and known inequalities for left and right log- s -convex-IVFs in the second sense and their variants as special cases. In future, we will try to extend this concept by using fractional integral operators.

Acknowledgments

The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research and academic environments.

Funding information: The corresponding author (J.E.M.-D.) wishes to acknowledge the financial support from the National Council for Science and Technology of Mexico (CONACYT) through grant A1-S-45928.
Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.
Conflict of interest: The authors declare that they have no competing interests.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2022-01-18

Revised: 2022-05-02

Accepted: 2022-05-27

Published Online: 2022-08-13

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

Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation

Abstract

1 Introduction

2 Preliminaries

Remark 2.1

Theorem 2.2

Definition 2.3

Definition 2.4

Definition 2.5

Definition 2.6

Remark 2.7

Theorem 2.8

Proof

Remark 2.9

Example 2.10

Theorem 2.11

Proof

Example 2.12

3 Main results

Theorem 3.1

Proof

Corollary 3.2

Theorem 3.3

Proof

Remark 3.4

Theorem 3.5

Proof

Remark 3.6

Example 3.7

Theorem 3.8

Proof

Theorem 3.10

Proof

Remark 3.10

Example 3.11

4 Conclusion

Acknowledgments

References

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