Jensen-type inequalities for m-convex functions

Paul Bosch; Yamilet Quintana; José M. Rodríguez; José M. Sigarreta

doi:10.1515/math-2022-0061

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Jensen-type inequalities for m-convex functions

Paul Bosch , Yamilet Quintana , José M. Rodríguez and José M. Sigarreta

Published/Copyright: September 6, 2022

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$Open Mathematics$

From the journal Open Mathematics Volume 20 Issue 1

Abstract

Inequalities play an important role in pure and applied mathematics. In particular, Jensen’s inequality, one of the most famous inequalities, plays the main role in the study of the existence and uniqueness of initial and boundary value problems for differential equations. In this work, we prove some new Jensen-type inequalities for m-convex functions and apply them to generalized Riemann-Liouville-type integral operators. Furthermore, as a remarkable consequence, some new inequalities for convex functions are obtained.

Keywords: Jensen-type inequalities; convex functions; m-convex functions; fractional derivatives and integrals; fractional integral inequalities

MSC 2010: 26A33; 26A51; 26D15

1 Introduction

As the authors of [1,2,4,6,18,19,29,32] have pointed out, integral inequalities belong to the class of mathematical inequalities playing a significant leading role in different fields of science and technology, such as physical, biological, and engineering sciences, approximation theory and spectral analysis, statistical analysis and theory of distributions, financial economics, and computer science. It is noteworthy that for this last field, and despite the optimization problems in deep learning being generally nonconvex, they often exhibit some properties of convex ones near local minima. This phenomenon allows us to obtain applications of Jensen inequality and some of its generalizations in machine learning (c.f., e.g., [13] and references therein).

In recent years, there has been a growing interest in the study of many classical inequalities applied to integral operators associated with different types of fractional derivatives, since integral inequalities and their applications play a vital role in the theory of differential equations and applied mathematics. Some of the inequalities studied are Gronwall, Chebyshev, Hermite-Hadamard-type, Ostrowski-type, Grüss-type, Hardy-type, Gagliardo-Nirenberg-type, reverse Minkowski, and reverse Hölder inequalities (see, e.g., [3,8,20,21,24,25, 26,27,28]).

We refer the interested reader to the classical books [17,18,23], the monograph [4], and references cited therein for a detailed exposition about the fundamentals of some recent trends of the research in this broad field.

Motivated by some of the aforementioned results and the paper [16], our contributions in the present paper are addressed both to the theory of inequalities and the theory of fractional operators. On one hand, we provide some new Jensen-type inequalities. And, on the other hand, we use such inequalities for obtaining novel results in the current setting of fractional integral inequalities.

2 Preliminaries

One of the first operators that can be called fractional is the Riemann-Liouville fractional derivative of order α ∈ C , with Re ( α ) > 0 , defined as follows (see [7]).

Definition 1

Let a < b and f ∈ L 1 ( ( a , b ) ; R ) . The right and left side Riemann-Liouville fractional integrals of order α , with Re ( α ) > 0 , are defined, respectively, by

(1) J a + α R L f ( t ) = 1 Γ ( α ) ∫ a t ( t − s ) α − 1 f ( s ) d s

and

(2) J b − α R L f ( t ) = 1 Γ ( α ) ∫ t b ( s − t ) α − 1 f ( s ) d s ,

with t ∈ ( a , b ) .

When α ∈ ( 0 , 1 ) , their corresponding Riemann-Liouville fractional derivatives are given by

( D a + α R L f ) ( t ) = d d t ( J a + 1 − α R L f ( t ) ) = 1 Γ ( 1 − α ) d d t ∫ a t f ( s ) ( t − s ) α d s ( D b − α R L f ) ( t ) = − d d t ( J b − 1 − α R L f ( t ) ) = − 1 Γ ( 1 − α ) d d t ∫ t b f ( s ) ( s − t ) α d s .

Other definitions of fractional operators are the following ones.

Definition 2

Let a < b and f ∈ L 1 ( ( a , b ) ; R ) . The right and left side Hadamard fractional integrals of order α , with Re ( α ) > 0 , are defined, respectively, by

(3) H a + α f ( t ) = 1 Γ ( α ) ∫ a t log t s α − 1 f ( s ) s d s

and

(4) H b − α f ( t ) = 1 Γ ( α ) ∫ t b log s t α − 1 f ( s ) s d s ,

with t ∈ ( a , b ) .

When α ∈ ( 0 , 1 ) , Hadamard fractional derivatives are given by the following expressions:

( D a + α H f ) ( t ) = t d d t ( H a + 1 − α f ( t ) ) = 1 Γ ( 1 − α ) t d d t ∫ a t log t s − α f ( s ) s d s , ( D b − α H f ) ( t ) = − t d d t ( H b − 1 − α f ( t ) ) = − 1 Γ ( 1 − α ) t d d t ∫ t b log s t − α f ( s ) s d s ,

with t ∈ ( a , b ) .

Katugampola introduced new fractional integral operators, called Katugampola fractional integrals, in the following way.

Definition 3

Let 0 < a < b , f : [ a , b ] → R an integrable function, and α ∈ ( 0 , 1 ) , ρ > 0 two fixed real numbers. The right and left side Katugampola fractional integrals of order α are defined, respectively, by

(5) K a + α , ρ f ( t ) = ρ 1 − α Γ ( α ) ∫ a t s ρ − 1 ( t ρ − s ρ ) 1 − α f ( s ) d s

and

(6) K b − α , ρ f ( t ) = ρ 1 − α Γ ( α ) ∫ t b t ρ − 1 ( s ρ − t ρ ) 1 − α f ( s ) d s ,

with t ∈ ( a , b ) .

Some generalizations of the Riemann-Liouville and Hadamard fractional derivatives appeared in [10]. These generalizations, called Katugampola fractional derivatives, are defined as

( D a + α , ρ K f ) ( t ) = ρ α Γ ( 1 − α ) t 1 − ρ d d t ∫ a t s ρ − 1 ( t ρ − s ρ ) α f ( s ) d s , ( D b − α , ρ K f ) ( t ) = − ρ α Γ ( 1 − α ) t 1 − ρ d d t ∫ t b s ρ − 1 ( s ρ − t ρ ) α f ( s ) d s ,

with t ∈ ( a , b ) .

The relations between these two fractional operators are the following:

( D a + α , ρ K f ) ( t ) = t 1 − ρ d d t K a + 1 − α , ρ f ( t ) , ( D b − α , ρ K f ) ( t ) = − t 1 − ρ d d t K b − 1 − α , ρ f ( t ) .

Definition 4

Let 0 < a < b , g : [ a , b ] → R an increasing positive function on ( a , b ] with continuous derivative on ( a , b ) , f : [ a , b ] → R an integrable function, and α ∈ ( 0 , 1 ) a fixed real number. The right and left side fractional integrals in [11] of order α of f with respect to g are defined, respectively, by

(7) I g , a + α f ( t ) = 1 Γ ( α ) ∫ a t g ′ ( s ) f ( s ) ( g ( t ) − g ( s ) ) 1 − α d s

and

(8) I g , b − α f ( t ) = 1 Γ ( α ) ∫ t b g ′ ( s ) f ( s ) ( g ( s ) − g ( t ) ) 1 − α d s ,

with t ∈ ( a , b ) .

There are other definitions of integral operators in the global case, but they are slight modifications of the previous ones.

3 General fractional integral of Riemann-Liouville type

Now, we give the definition of a general fractional integral in [2].

Definition 5

Let a < b and α ∈ R + . Let g : [ a , b ] → R be a positive function on ( a , b ] with continuous positive derivative on ( a , b ) and G : [ 0 , g ( b ) − g ( a ) ] × ( 0 , ∞ ) → R a continuous function which is positive on ( 0 , g ( b ) − g ( a ) ] × ( 0 , ∞ ) . Let us define the function T : [ a , b ] × [ a , b ] × ( 0 , ∞ ) → R by

T ( t , s , α ) = G ( ∣ g ( t ) − g ( s ) ∣ , α ) g ′ ( s ) .

The right and left integral operators, denoted, respectively, by J T , a + α and J T , b − α , are defined for each measurable function f on [ a , b ] as

(9) J T , a + α f ( t ) = ∫ a t f ( s ) T ( t , s , α ) d s ,

(10) J T , b − α f ( t ) = ∫ t b f ( s ) T ( t , s , α ) d s ,

with t ∈ [ a , b ] .

We say that f ∈ L T 1 [ a , b ] if J T , a + α ∣ f ∣ ( t ) , J T , b − α ∣ f ∣ ( t ) < ∞ for every t ∈ [ a , b ] .

Note that these operators generalize the integral operators in Definitions 1, 2, and 4:

If we choose
g ( t ) = t , G ( x , α ) = Γ ( α ) x 1 − α , T ( t , s , α ) = Γ ( α ) ∣ t − s ∣ 1 − α ,
then J T , a + α and J T , b − α are the right and left Riemann-Liouville fractional integrals J a + α R L and J b − α R L in (1) and (2), respectively. Its corresponding right and left Riemann-Liouville fractional derivatives are
( D a + α R L f ) ( t ) = d d t ( J a + 1 − α R L f ( t ) ) and ( D b − α R L f ) ( t ) = − d d t ( J b − 1 − α R L f ( t ) ) .
If we choose
g ( t ) = log t , G ( x , α ) = Γ ( α ) x 1 − α , T ( t , s , α ) = Γ ( α ) t log t s 1 − α ,
then J T , a + α and J T , b − α are the right and left Hadamard fractional integrals H a + α and H b − α in (3) and (4), respectively. Its corresponding right and left Hadamard fractional derivatives are
( D a + α H f ) ( t ) = t d d t ( H a + 1 − α f ( t ) ) , and ( D b − α H f ) ( t ) = − t d d t ( H b − 1 − α f ( t ) ) .
If we choose
g ( t ) = t ρ , G ( x , α ) = Γ ( α ) ρ α x 1 − α , T ( t , s , α ) = Γ ( α ) ρ 1 − α ∣ t ρ − s ρ ∣ 1 − α s ρ − 1 ,
then J T , a + α and J T , b − α are the right and left Katugampola fractional integrals K a + α , ρ and K b − α , ρ in (5) and (6), respectively. Its corresponding right and left Katugampola fractional derivatives are
( D a + α , ρ K f ) ( t ) = t 1 − ρ d d t ( K a + 1 − α , ρ f ( t ) ) , and ( D b − α , ρ K f ) ( t ) = − t 1 − ρ d d t ( K b − 1 − α , ρ f ( t ) ) .
If we choose a function g with the properties in Definition 5 and
G ( x , α ) = Γ ( α ) x 1 − α , and T ( t , s , α ) = Γ ( α ) ∣ g ( t ) − g ( s ) ∣ 1 − α g ′ ( s ) ,
then J T , a + α and J T , b − α are the right and left fractional integrals I g , a + α and I g , b − α in (7) and (8), respectively.

Definition 6

Let a < b and α ∈ R + . Let g : [ a , b ] → R be a positive function on ( a , b ] with continuous positive derivative on ( a , b ) and G : [ 0 , g ( b ) − g ( a ) ] × ( 0 , ∞ ) → R a continuous function, which is positive on ( 0 , g ( b ) − g ( a ) ] × ( 0 , ∞ ) . For each function f ∈ L T 1 [ a , b ] , its right and left generalized derivatives of order α are defined, respectively, by

(11) D T , a + α f ( t ) = 1 g ′ ( t ) d d t ( J T , a + 1 − α f ( t ) ) , D T , b − α f ( t ) = − 1 g ′ ( t ) d d t ( J T , b − 1 − α f ( t ) ) .

for each t ∈ ( a , b ) .

Note that if we choose

g ( t ) = t , G ( x , α ) = Γ ( α ) x 1 − α , T ( t , s , α ) = Γ ( α ) ∣ t − s ∣ 1 − α ,

then D T , a + α f ( t ) = D a + α R L f ( t ) and D T , b − α f ( t ) = D b − α R L f ( t ) . Also, we can obtain Hadamard and other fractional derivatives as particular cases of this generalized derivative.

4 Jensen-type inequalities for m -convex functions

The property of m -convexity for functions on [ 0 , b ] , b > 0 was introduced in [30] as an intermediate property between the usual convexity and star-shaped property. Since then many properties, especially inequalities, have been obtained for them (cf. [5,12,15,22,31]). One of the classical integral inequalities frequently studied in this setting is Jensen’s inequality, which relates the value of a convex function of an integral to the integral of the convex function. It was proved in 1906 [9], and it can be stated as follows:

Let μ be a probability measure on the space X . If f : X → ( a , b ) is μ -integrable and φ is a convex function on ( a , b ) , then

φ ∫ X f d μ ≤ ∫ X φ ∘ f d μ .

Definition 7

Let I ⊆ R be an interval containing zero, and let m ∈ ( 0 , 1 ] . A function φ : I → R is said to be m -convex if the inequality

(12) φ ( t x + m ( 1 − t ) y ) ≤ t φ ( x ) + m ( 1 − t ) φ ( y )

holds for every pair of points x , y ∈ I and every coefficient t ∈ [ 0 , 1 ] .

If m ∈ ( 0 , 1 ) , then the hypothesis 0 ∈ I guarantees that t x + m ( 1 − t ) y ∈ I .

It is clear that taking m = 1 in Definition 7 we recover the concept of classical convex functions on I . Note that in this case it is not necessary the hypothesis 0 ∈ I , since t x + ( 1 − t ) y ∈ I for every x , y ∈ I .

Note that if we choose the coefficient t = 0 in (12), we obtain the inequality φ ( m y ) ≤ m φ ( y ) .

Also, Definition 7 is equivalent to

(13) φ ( m t x + ( 1 − t ) y ) ≤ m t φ ( x ) + ( 1 − t ) φ ( y ) ,

for all x , y ∈ I and t ∈ [ 0 , 1 ] .

The following discrete Jensen-type inequality for m -convex functions was established in [22, Theorem 3.2]:

Theorem 1

Let I ⊆ R be an interval containing the zero, and let ∑ k = 1 n w k x k be a convex combination of points x k ∈ I with coefficients w k ∈ [ 0 , 1 ] . If φ is an m-convex function on I, with m ∈ ( 0 , 1 ] , then

(14) φ m ∑ k = 1 n w k x k ≤ m ∑ k = 1 n w k φ ( x k ) .

This inequality is a discrete version of the following one for continuous m -convex functions [22, Corollary 4.2]:

Theorem 2

Let μ be a probability measure on the space X . If I ⊆ R is an interval containing zero, f : X → I is μ -integrable, and φ is a continuous m-convex function on I , with m ∈ ( 0 , 1 ] , then

(15) φ m ∫ X f d μ ≤ m ∫ X φ ∘ f d μ .

The following discrete Jensen-type inequality for convex functions appears in [16, Theorem 1.2]:

Theorem 3

Let x 1 ≤ x 2 ≤ ⋯ ≤ x n and let { w k } k = 1 n be positive weights whose sum is 1. If φ is a convex function on [ x 1 , x n ] , then

φ x 1 + x n − ∑ k = 1 n w k x k ≤ φ ( x 1 ) + φ ( x n ) − ∑ k = 1 n w k φ ( x k ) .

Our purpose is to prove continuous versions of the above discrete inequality in the setting of m -convexity (see Theorems 7 and 8). Before stating such a result, we require some properties of the m -convex functions.

Lemma 4

Let I ⊆ R be an interval containing the zero, and let φ be an m-convex function on I with m ∈ ( 0 , 1 ] . For { x k } k = 1 n ⊂ I such that x 1 ≤ x 2 ≤ ⋯ ≤ x n , the following inequalities hold:

(16) φ ( x 1 + m x n − m x k ) ≤ φ ( x 1 ) + m φ ( x n ) − φ ( m x k ) , 1 ≤ k ≤ n .

Proof

Let us consider y k = x 1 + m x n − m x k . Then x 1 + m x n = y k + m x k and so the pairs x 1 , m x n and y k , m x k have the same midpoint. Since x 1 ≤ y k and m x k ≤ m x n , we have x 1 ≤ y k , m x k ≤ m x n and there exists λ ∈ [ 0 , 1 ] such that

m x k = λ x 1 + m ( 1 − λ ) x n , y k = ( 1 − λ ) x 1 + m λ x n ,

for 1 ≤ k ≤ n . From Definition 7 and its equivalent form (13) we obtain

φ ( y k ) ≤ m λ φ ( x n ) + ( 1 − λ ) φ ( x 1 ) = φ ( x 1 ) + m φ ( x n ) − [ λ φ ( x 1 ) + m ( 1 − λ ) φ ( x n ) ] ≤ φ ( x 1 ) + m φ ( x n ) − φ ( λ x 1 + m ( 1 − λ ) x n ) = φ ( x 1 ) + m φ ( x n ) − φ ( m x k ) ,

and (16) follows.

Note that since m ∈ ( 0 , 1 ] , the hypothesis 0 ∈ I guarantees that m x k ∈ I .□

The following two results generalize Theorem 3 in the setting of m -convexity.

Theorem 5

Let I ⊆ R be an interval containing the zero, let { x k } k = 1 n ⊂ I with x 1 ≤ x 2 ≤ ⋯ ≤ x n , and let { w k } k = 1 n be positive weights whose sum is 1. If φ is an m-convex function on I, with m ∈ ( 0 , 1 ] , then

(17) φ m x 1 + m 2 x n − m 2 ∑ k = 1 n w k x k ≤ m φ ( x 1 ) + m 2 φ ( x n ) − m ∑ k = 1 n w k φ ( m x k ) .

Remark 1

Theorem 3 gives that if m = 1 , the inequality in Theorem 5 also holds if we remove the hypothesis 0 ∈ I .

Proof

First, note that

x 1 + m x n − m ∑ k = 1 n w k x k = ∑ k = 1 n w k ( x 1 + m x n − m x k ) ,

and thus

m x 1 + m 2 x n − m 2 ∑ k = 1 n w k x k = m ∑ k = 1 n w k ( x 1 + m x n − m x k ) .

Then it follows from (14) and (16) that

φ m x 1 + m 2 x n − m 2 ∑ k = 1 n w k x k = φ m ∑ k = 1 n w k ( x 1 + m x n − m x k ) ≤ m ∑ k = 1 n w k φ ( x 1 + m x n − m x k ) ≤ m ∑ k = 1 n w k ( φ ( x 1 ) + m φ ( x n ) − φ ( m x k ) ) = m φ ( x 1 ) + m 2 φ ( x n ) − m ∑ k = 1 n w k φ ( m x k ) ,

and this concludes the proof of the inequality.

Let us check that the hypothesis 0 ∈ I guarantees that m x 1 + m 2 x n − m 2 ∑ k = 1 n w k x k ∈ I :

Assume that I = [ a , b ] . Then

m x 1 + m 2 x n − m 2 ∑ k = 1 n w k x k ≥ m x 1 + m 2 x n − m 2 ∑ k = 1 n w k x n = m x 1 ≥ min { 0 , x 1 } ≥ a .

Also,

m x 1 + m 2 x n − m 2 ∑ k = 1 n w k x k ≤ m x 1 + m 2 x n − m 2 ∑ k = 1 n w k x 1 = m x 1 − m 2 x 1 + m 2 x n .

If x 1 ≤ 0 , then m x 1 − m 2 x 1 ≤ 0 and so

m x 1 + m 2 x n − m 2 ∑ k = 1 n w k x k ≤ m x 1 − m 2 x 1 + m 2 x n ≤ m 2 x n ≤ max { 0 , x n } ≤ b .

Assume now that x 1 > 0 . Let us consider the function v ( t ) = t x 1 − t 2 x 1 + t 2 x n . Since v ′ ( t ) = x 1 + 2 t ( x n − x 1 ) > 0 and m ∈ ( 0 , 1 ] , we have

v ( m ) ≤ v ( 1 ) = x n ≤ b , m x 1 + m 2 x n − m 2 ∑ k = 1 n w k x k ≤ m x 1 − m 2 x 1 + m 2 x n = v ( m ) ≤ b .

If I is not a closed interval [ a , b ] , a similar argument gives the result.□

If φ is a continuous m -convex function, we can obtain the following improvement of Theorem 5.

Theorem 6

Let a ≤ 0 ≤ b , let { y k } k = 1 n ⊂ [ a , b ] , and let { w k } k = 1 n be positive weights whose sum is 1. If φ is a continuous m-convex function on [ a , b ] , with m ∈ ( 0 , 1 ] , then

(18) φ m a + m 2 b − m 2 ∑ k = 1 n w k y k ≤ m φ ( a ) + m 2 φ ( b ) − m ∑ k = 1 n w k φ ( m y k ) .

Proof

If we consider 0 < ε < 1 , y 0 = a , y n + 1 = b , w k ′ = ( 1 − ε ) w k ( 1 ≤ k ≤ n ) , w 0 ′ = ε / 2 , and w n + 1 ′ = ε / 2 , then ∑ k = 0 n + 1 w k ′ = 1 and Theorem 5 gives

φ m a + m 2 b − m 2 ε 2 a − m 2 ε 2 b − m 2 ∑ k = 1 n ( 1 − ε ) w k y k ≤ m φ ( a ) + m 2 φ ( b ) − m ε 2 φ ( m a ) − m ε 2 φ ( m b ) − m ∑ k = 1 n ( 1 − ε ) w k φ ( m y k ) .

Since φ is a continuous function on [ a , b ] , if we take ε → 0 + , we obtain (18).□

Next, we present a continuous version of the above discrete inequality.

Theorem 7

Let μ be a probability measure on the space X and a ≤ 0 ≤ b real constants. If f : X → [ a , b ] is a measurable function and φ is a continuous m-convex function on [ a , b ] , with m ∈ ( 0 , 1 ] , then f and φ ( m f ) are μ -integrable functions and

(19) φ m a + m 2 b − m 2 ∫ X f d μ ≤ m φ ( a ) + m 2 φ ( b ) − m ∫ X φ ( m f ) d μ .

If m = 1 , this inequality also holds if we remove the hypothesis 0 ∈ [ a , b ] .

Proof

Since a ≤ f ≤ b and φ is a continuous function on [ a , b ] , we have that f and φ ( m f ) are bounded measurable functions on X . And using that μ is a probability measure on X , we conclude that f and φ ( m f ) are μ -integrable functions.

For each n ≥ 1 and 0 ≤ k ≤ 2 n , let us consider the sets

E n , k = { x ∈ X : a + k 2 − n ( b − a ) ≤ f ( x ) < a + ( k + 1 ) 2 − n ( b − a ) } .

Since f is a measurable function satisfying a ≤ f ≤ b , we have that { E n , k } k = 0 2 n are pairwise disjoint measurable sets and X = ∪ k = 0 2 n E n , k for each n . Thus,

∑ k = 0 2 n μ ( E n , k ) = 1

for each n .

Since f is a measurable function satisfying a ≤ f ≤ b and { E n , k } k = 0 2 n is a partition of X , the sequence of simple functions

f n = ∑ k = 0 2 n ( a + k 2 − n ( b − a ) ) χ E n , k

satisfies a ≤ f n ≤ b and f − 2 − n ( b − a ) < f n ≤ f for every n and so

lim n → ∞ f n = f .

Note that

∫ X f n d μ = ∑ k = 0 2 n ( a + k 2 − n ( b − a ) ) μ ( E n , k ) .

Since { E n , k } k = 0 2 n is a partition of X , we have

φ ( m f ) = ∑ k = 0 2 n φ ( m a + m k 2 − n ( b − a ) ) χ E n , k , ∫ X φ ( m f ) d μ = ∑ k = 0 2 n φ ( m a + m k 2 − n ( b − a ) ) μ ( E n , k ) .

Hence, Theorem 6 gives

(20) φ m a + m 2 b − m 2 ∫ X f n d μ ≤ m φ ( a ) + m 2 φ ( b ) − m ∫ X φ ( m f ) d μ .

If m = 1 , Theorem 3 gives the above inequality without the hypothesis 0 ∈ [ a , b ] .

Since a ≤ f n ≤ b for every n , μ is a finite measure and lim n → ∞ f n = f , dominated convergence theorem gives

lim n → ∞ ∫ X f n d μ = ∫ X f d μ .

If m ∈ ( 0 , 1 ) , then the hypothesis 0 ∈ [ a , b ] guarantees that m a + m 2 b − m 2 ∫ X f n d μ ∈ [ a , b ] :

Since a ≤ 0 ≤ b , we have

m a + m 2 b − m 2 ∫ X f n d μ ≤ m a + m 2 b − m 2 a = m ( 1 − m ) a + m 2 b ≤ m 2 b ≤ b , m a + m 2 b − m 2 ∫ X f n d μ ≥ m a + m 2 b − m 2 b = m a ≥ a .

If m = 1 , then

a + b − ∫ X f n d μ ≤ a + b − a = b , a + b − ∫ X f n d μ ≥ a + b − b = a ,

and so, we do not need the hypothesis 0 ∈ [ a , b ] .

Since

a ≤ m a + m 2 b − m 2 ∫ X f n d μ ≤ b

for every n and m ∈ ( 0 , 1 ] , and φ is a continuous function on [ a , b ] ,

lim n → ∞ φ m a + m 2 b − m 2 ∫ X f n d μ = φ m a + m 2 b − m 2 ∫ X f d μ .

Since a ≤ m f n ≤ b for every n , lim n → ∞ f n = f and φ is a continuous function on [ a , b ] , lim n → ∞ φ ( m f n ) = φ ( m f ) .

Again, from the continuity of φ on [ a , b ] , there exists a positive constant K with ∣ φ ∣ ≤ K on [ a , b ] and so, ∣ φ ( m f n ) ∣ ≤ K for every n .

In view of the finiteness of μ , dominated convergence theorem guarantees that

lim n → ∞ ∫ X φ ( m f ) d μ = ∫ X φ ( m f ) d μ .

Combining the foregoing facts with (20), we obtain (19).□

If m = 1 , it is possible to improve Theorem 7, by removing the hypothesis of continuity.

Theorem 8

Let μ be a probability measure on the space X and a ≤ b real constants. If f : X → [ a , b ] is a measurable function and φ is a convex function on [ a , b ] , then f and φ ∘ f are μ -integrable functions and

φ a + b − ∫ X f d μ ≤ φ ( a ) + φ ( b ) − ∫ X φ ∘ f d μ .

Proof

Since φ is a convex function on [ a , b ] , φ is continuous on ( a , b ) and there exist the limits

lim s → a + φ ( s ) , and lim s → b − φ ( s ) .

Define φ ∗ as follows:

φ ∗ ( t ) = φ ( t ) if t ∈ ( a , b ) , lim s → a + φ ( s ) if t = a , lim s → b − φ ( s ) if t = b .

Hence, φ ∗ is a continuous convex function on [ a , b ] , and by Theorem 7 with m = 1 , we have

φ ∗ a + b − ∫ X f d μ ≤ φ ∗ ( a ) + φ ∗ ( b ) − ∫ X φ ∗ ∘ f d μ .

Assume that f = a μ -a.e. or f = b μ -a.e.; in the first case,

φ a + b − ∫ X f d μ = φ ( a + b − a ) = φ ( b ) = φ ( a ) + φ ( b ) − φ ( a ) = φ ( a ) + φ ( b ) − ∫ X φ ∘ f d μ ;

in the second case,

φ a + b − ∫ X f d μ = φ ( a + b − b ) = φ ( a ) = φ ( a ) + φ ( b ) − φ ( b ) = φ ( a ) + φ ( b ) − ∫ X φ ∘ f d μ .

Otherwise, a < ∫ X f d μ < b and a < a + b − ∫ X f d μ < b . Consequently,

φ a + b − ∫ X f d μ = φ ∗ a + b − ∫ X f d μ .

If we define

Δ a = φ ( a ) − φ ∗ ( a ) ≥ 0 , Δ b = φ ( b ) − φ ∗ ( b ) ≥ 0 ,

then

φ = φ ∗ + Δ a χ { a } + Δ b χ { b } , φ ∘ f = φ ∗ ∘ f + Δ a χ { f = a } + Δ b χ { f = b } ,

where χ A is the function with value 1 on the set A and 0 otherwise (i.e., the characteristic function of A ).

Hence,

∫ X φ ∘ f d μ = ∫ X φ ∗ ∘ f d μ + Δ a μ ( { f = a } ) + Δ b μ ( { f = b } ) ,

and we have

φ a + b − ∫ X f d μ = φ ∗ a + b − ∫ X f d μ ≤ φ ∗ ( a ) + φ ∗ ( b ) − ∫ X φ ∗ ∘ f d μ = φ ( a ) − Δ a + φ ( b ) − Δ b − ∫ X φ ∘ f d μ + Δ a μ ( { f = a } ) + Δ b μ ( { f = b } ) = φ ( a ) + φ ( b ) − ∫ X φ ∘ f d μ − Δ a [ 1 − μ ( { f = a } ) ] − Δ b [ 1 − μ ( { f = b } ) ] ≤ φ ( a ) + φ ( b ) − ∫ X φ ∘ f d μ . □

Theorem 8 has the following direct consequence.

Corollary 9

Let a ≤ b , let { y k } k = 1 n ⊂ [ a , b ] , and let { w k } k = 1 n be positive weights whose sum is 1. If φ is a convex function on [ a , b ] , then

φ a + b − ∑ k = 1 n w k y k ≤ φ ( a ) + φ ( b ) − ∑ k = 1 n w k φ ( y k ) .

Note that Theorem 8 provides a kind of converse of the classical Jensen’s inequality for convex functions.

Proposition 10

φ ∫ X f d μ ≤ ∫ X φ ∘ f d μ ≤ φ ( a ) + φ ( b ) − φ a + b − ∫ X f d μ .

Theorems 8 and 7 have, respectively, the following direct consequences for general fractional integrals of Riemann-Liouville type.

Proposition 11

Let c < d and a ≤ b be real constants. If f : [ c , d ] → [ a , b ] is a measurable function, φ is a convex function on [ a , b ] , and

T ( α ) = ∫ c d 1 T ( d , s , α ) d s = ∫ 0 g ( d ) − g ( c ) d x G ( x , α ) < ∞ ,

then f ( s ) / T ( d , s , α ) , φ ( f ( s ) ) / T ( d , s , α ) ∈ L 1 [ c , d ] , and

φ a + b − 1 T ( α ) ∫ c d f ( s ) T ( d , s , α ) d s ≤ φ ( a ) + φ ( b ) − 1 T ( α ) ∫ c d φ ( f ( s ) ) T ( d , s , α ) d s .

Proposition 12

Let c < d and a ≤ 0 ≤ b be real constants. If f : [ c , d ] → [ a , b ] is a measurable function, φ is a continuous m-convex function on [ a , b ] , with m ∈ ( 0 , 1 ] , and

T ( α ) = ∫ c d 1 T ( d , s , α ) d s = ∫ 0 g ( d ) − g ( c ) d x G ( x , α ) < ∞ ,

then f ( s ) / T ( d , s , α ) , φ ( m f ( s ) ) / T ( d , s , α ) ∈ L 1 [ c , d ] and

φ m a + m 2 b − m 2 T ( α ) ∫ c d f ( s ) T ( d , s , α ) d s ≤ m φ ( a ) + m 2 φ ( b ) − m T ( α ) ∫ c d φ ( m f ( s ) ) T ( d , s , α ) d s .

5 Conclusion

The main goal of our research is to determine new Jensen-type inequalities for m-convex functions and apply them to generalized Riemann-Liouville-type integral operators. In particular, we prove continuous versions of the discrete Jensen-type inequality in [16, Theorem 1.2], in the setting of m -convexity. Furthermore, our results provide new inequalities for convex functions.

Funding information: The research of Yamilet Quintana, José M. Rodríguez, and José M. Sigarreta is supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00/AEI/10.13039/501100011033), Spain. The research of Yamilet Quintana and José M. Rodríguez is supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23) and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation).
Conflict of interest: The authors state no conflict of interest.

References

[1] M. Bohner, A. Kashuri, P. Mohammed, and J. E. Nápoles Valdés, Hermite-Hadamard-type inequalities for conformable integrals, Hacet. J. Math. Stat. 2022 (2022), 1–12, https://doi.org/10.15672/hujms.946069. Search in Google Scholar

[2] P. Bosch, H. J. Carmenate, J. M. Rodrıguez, and J. M. Sigarreta, Generalized inequalities involving fractional operators of the Riemann-Liouville type, AIMS Math. 7 (2022), no. 1, 1470–1485, https://doi.org/10.3934/math.2022087. Search in Google Scholar

[3] Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integral, Ann. Funct. Anal. 1 (2010), no. 1, 51–58, https://doi.org/10.15352/afa/1399900993. Search in Google Scholar

[4] S. S. Dragomir and C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University, Melbourne, 2001, https://rgmia.org/papers/monographs/Master2.pdf. Search in Google Scholar

[5] S. S. Dragomir and G. H. Toader, Some inequalities for m-convex functions, Studia Univ. Babes-Bolyai Math. 38 (1993), no. 1, 21–28.Search in Google Scholar

[6] A. Fernandez and P. Mohammed, Hermite-Hadamard inequalities in fractional calculus defined using Mittag-Leffler kernels, Math. Methods Appl. Sci. 44 (2021), 8414–8431, https://doi.org/10.1002/mma.6188. Search in Google Scholar

[7] R. Gorenflo and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, 1st ed., Springer, Vienna, 1997. 10.1007/978-3-7091-2664-6_5Search in Google Scholar

[8] J. Han, P. O. Mohammed, and H. Zeng, Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function, Open Math. 18 (2020), 794–806, https://doi.org/10.1515/math-2020-0038. Search in Google Scholar

[9] J. L. W. V. Jensen, Sur les fonctions convexes et les inégalités entre les valeurs moyennes, Acta Math. 30 (1906), no. 1, 175–193, https://doi.org/10.1007/BF02418571. Search in Google Scholar

[10] V. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. App. 6 (2014), no. 4, 1–15. Search in Google Scholar

[11] A. A. Kilbas, O. I. Marichev, and S. G. Samko, Fractional Integrals and Derivatives. Theory and Applications, 1st ed., Gordon & Breach, Pennsylvania, 1993. Search in Google Scholar

[12] M. K. Bakula, J. Pečarić, and M. Ribičić, Companion inequalities to Jensen’s inequality for m-convex and (α,m)-convex functions, J. Ineq. Pure Appl. Math. 7 (2008), no. 5, 194, https://www.emis.de/journals/JIPAM/volumes.html. Search in Google Scholar

[13] P. Izmailov, D. Podoprikhin, T. Garipov, D. Vetrov, and A. G. Wilson, Averaging weights leads to wider optima and better generalization, Uncertainty in Artificial Intelligence. Proceedings of the Thirty-Fourth Conference, Paper presented at Thirty-Fourth Conference, California, USA, 2018, August 6–10, AUAI Press Corvallis, Oregon, 2018, pp. 876–885. Search in Google Scholar

[14] P. O. Mohammed and I. Brevik, A new version of the Hermite-Hadamard inequality for Riemann-Liouville fractional integrals, Symmetry 12 (2021), no. 4, 610, https://doi.org/10.3390/sym12040610. Search in Google Scholar

[15] T. Lara, N. Merentes, R. Quintero, and E. Rosales, On strongly m-convex functions, Pure Math. Sci. 6 (2017), no. 1, 87–94, https://doi.org/10.12988/pms.2017.61018. Search in Google Scholar

[16] A. McD. Mercer, A variant of Jensen’s inequality, J. Ineq. Pure Appl. Math. 4 (2003), no. 4, 73, http://emis.icm.edu.pl/journals/JIPAM/v4n4/116_03.html.Search in Google Scholar

[17] D. S. Mitrinović, Analytic Inequalities, Springer-Verlag, Berlin Heidelberg, 1970. 10.1007/978-3-642-99970-3Search in Google Scholar

[18] D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht, 1993.10.1007/978-94-017-1043-5Search in Google Scholar

[19] P. O. Mohammed and I. Brevik, A new version of the Hermite-Hadamard inequality for Riemann-Liouville fractional integrals dl, Symmetry 12 (2021), no. 4, 610, https://doi.org/10.3390/sym12040610. Search in Google Scholar

[20] S. Mubeen, S. Habib, and M. N. Naeem, The Minkowski inequality involving generalized k-fractional conformable integral, J. Inequal. Appl. 2019 (2019), 81, https://doi.org/10.1186/s13660-019-2040-8. Search in Google Scholar

[21] K. S. Nisar, F. Qi, G. Rahman, S. Mubeen, and M. Arshad, Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric K-function, J. Inequal. Appl. 2018 (2018), 135, https://doi.org/10.1186/s13660-018-1717-8. Search in Google Scholar

[22] Z. Paviccc and M. Avci Ardiç, The most important inequalities of m-convex functions, Turkish J. Math. 41 (2017), 625–635, https://doi.org/10.3906/mat-1604-45. Search in Google Scholar

[23] J. E. Pečarić, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, 1st ed., Academic Press Inc., San Diego, 1992. 10.1016/S0076-5392(08)62813-1Search in Google Scholar

[24] G. Rahman, G. Rahman, T. Abdeljawad, F. Jarad, A. Khan, and K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral operators, Adv. Differ. Equ. 2019 (2019), 454, https://doi.org/10.1186/s13662-019-2381-0. Search in Google Scholar

[25] G. Rahman, K. S. Nisar, B. Ghanbari, and T. Abdeljawad, On generalized fractional integral inequalities for the monotone weighted Chebyshev functionals, Adv. Differ. Equ. 2020 (2020), 368, https://doi.org/10.1186/s13662-020-02830-7. Search in Google Scholar

[26] S. Rashid, M. A. Noor, K. I. Noor, and Y.-M. Chu, Ostrowski type inequalities in the sense of generalized K-fractional integral operator for exponentially convex functions, AIMS Math. 5 (2020), no. 3, 2629–2645, https://doi.org/10.3934/math.2020171. Search in Google Scholar

[27] Y. Sawano and H. Wadade, On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space, J. Fourier Anal. Appl. 19 (2013), no. 1, 20–47, https://doi.org/10.1007/s00041-012-9223-8. Search in Google Scholar

[28] E. Set, M. Tomar, and M. Z. Sarikaya, On generalized Grüss type inequalities for k-fractional integrals, Appl. Math. Comput. 269 (2015), 29–34, https://doi.org/10.1016/j.amc.2015.07.026. Search in Google Scholar

[29] H. M. Srivastava, D. Raghavan, and S. Nagarajan, Generalized inequalities involving fractional operators of the Riemann-Liouville type, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 116 (2022), no. 3, 98, https://doi.org/10.1007/s13398-022-01239-z. Search in Google Scholar

[30] G. Toader, Some generalizations of the convexity, In: Proceedings of the Colloquium on Approximation and Optimization, (Cluj-Naploca, Romania), University of Cluj-Napoca, Romania, 1984, October 25–27, pp. 329–338. Search in Google Scholar

[31] G. Toader, The hierarchy of convexity and some classic inequalities, J. Math. Inequal. 3 (2009), no. 3, 3–30, https://dx.doi.org/10.7153/jmi-03-30. 10.7153/jmi-03-30Search in Google Scholar

[32] S. Yu, P. O. Mohammed, L. Xu, and T. Du, An improvement of the power-mean integral inequality in frame of fractal space and certain related midpoint-type integral inequalities, Fractals 30 (2022), no. 4, 1–23, https://doi.org/10.1142/S0218348X22500852. Search in Google Scholar

Received: 2022-03-19

Revised: 2022-04-28

Accepted: 2022-06-23

Published Online: 2022-09-06

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

Articles in the same Issue

https://doi.org/10.1515/math-2022-0061

Keywords for this article

Jensen-type inequalities; convex functions; m-convex functions; fractional derivatives and integrals; fractional integral inequalities

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