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.
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.,  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 , 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 , 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.
One of the first operators that can be called fractional is the Riemann-Liouville fractional derivative of order , with , defined as follows (see ).
Let and . The right and left side Riemann-Liouville fractional integrals of order , with , are defined, respectively, by
When , their corresponding Riemann-Liouville fractional derivatives are given by
Other definitions of fractional operators are the following ones.
Let and . The right and left side Hadamard fractional integrals of order , with , are defined, respectively, by
When , Hadamard fractional derivatives are given by the following expressions:
Katugampola introduced new fractional integral operators, called Katugampola fractional integrals, in the following way.
Let , an integrable function, and two fixed real numbers. The right and left side Katugampola fractional integrals of order are defined, respectively, by
Some generalizations of the Riemann-Liouville and Hadamard fractional derivatives appeared in . These generalizations, called Katugampola fractional derivatives, are defined as
The relations between these two fractional operators are the following:
Let , an increasing positive function on with continuous derivative on , an integrable function, and a fixed real number. The right and left side fractional integrals in  of order of with respect to are defined, respectively, by
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 .
Let and . Let be a positive function on with continuous positive derivative on and a continuous function which is positive on . Let us define the function by
The right and left integral operators, denoted, respectively, by and , are defined for each measurable function on as
We say that if for every .
Let and . Let be a positive function on with continuous positive derivative on and a continuous function, which is positive on . For each function , its right and left generalized derivatives of order are defined, respectively, by
for each .
Note that if we choose
then and . Also, we can obtain Hadamard and other fractional derivatives as particular cases of this generalized derivative.
4 Jensen-type inequalities for -convex functions
The property of -convexity for functions on , was introduced in  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 , and it can be stated as follows:
Let be a probability measure on the space . If is -integrable and is a convex function on , then
Let be an interval containing zero, and let . A function is said to be -convex if the inequality
holds for every pair of points and every coefficient .
If , then the hypothesis guarantees that .
It is clear that taking in Definition 7 we recover the concept of classical convex functions on . Note that in this case it is not necessary the hypothesis , since for every .
Note that if we choose the coefficient in (12), we obtain the inequality .
Also, Definition 7 is equivalent to
for all and .
The following discrete Jensen-type inequality for -convex functions was established in [22, Theorem 3.2]:
Let be an interval containing the zero, and let be a convex combination of points with coefficients . If is an m-convex function on I, with , then
This inequality is a discrete version of the following one for continuous -convex functions [22, Corollary 4.2]:
Let be a probability measure on the space . If is an interval containing zero, is -integrable, and is a continuous m-convex function on , with , then
The following discrete Jensen-type inequality for convex functions appears in [16, Theorem 1.2]:
Let and let be positive weights whose sum is 1. If is a convex function on , then
Our purpose is to prove continuous versions of the above discrete inequality in the setting of -convexity (see Theorems 7 and 8). Before stating such a result, we require some properties of the -convex functions.
Let be an interval containing the zero, and let be an m-convex function on with . For such that , the following inequalities hold:
Let us consider . Then and so the pairs and have the same midpoint. Since and , we have and there exists such that
and (16) follows.
Note that since , the hypothesis guarantees that .□
The following two results generalize Theorem 3 in the setting of -convexity.
Let be an interval containing the zero, let with , and let be positive weights whose sum is 1. If is an m-convex function on I, with , then
First, note that
and this concludes the proof of the inequality.
Let us check that the hypothesis guarantees that :
Assume that . Then
If , then and so
Assume now that . Let us consider the function . Since and , we have
If is not a closed interval , a similar argument gives the result.□
If is a continuous -convex function, we can obtain the following improvement of Theorem 5.
Let , let , and let be positive weights whose sum is 1. If is a continuous m-convex function on , with , then
If we consider , , , , , and , then and Theorem 5 gives
Since is a continuous function on , if we take , we obtain (18).□
Next, we present a continuous version of the above discrete inequality.
Let be a probability measure on the space and real constants. If is a measurable function and is a continuous m-convex function on , with , then and are -integrable functions and
If , this inequality also holds if we remove the hypothesis .
Since and is a continuous function on , we have that and are bounded measurable functions on . And using that is a probability measure on , we conclude that and are -integrable functions.
For each and , let us consider the sets
Since is a measurable function satisfying , we have that are pairwise disjoint measurable sets and for each . Thus,
for each .
Since is a measurable function satisfying and is a partition of , the sequence of simple functions
satisfies and for every and so
Since is a partition of , we have
Hence, Theorem 6 gives
If , Theorem 3 gives the above inequality without the hypothesis .
Since for every , is a finite measure and , dominated convergence theorem gives
If , then the hypothesis guarantees that :
Since , we have
If , then
and so, we do not need the hypothesis .
for every and , and is a continuous function on ,
Since for every , and is a continuous function on , .
Again, from the continuity of on , there exists a positive constant with on and so, for every .
In view of the finiteness of , dominated convergence theorem guarantees that
If , it is possible to improve Theorem 7, by removing the hypothesis of continuity.
Let be a probability measure on the space and real constants. If is a measurable function and is a convex function on , then and are -integrable functions and
Since is a convex function on , is continuous on and there exist the limits
Define as follows:
Hence, is a continuous convex function on , and by Theorem 7 with , we have
Assume that -a.e. or -a.e.; in the first case,
in the second case,
Otherwise, and . Consequently,
If we define
where is the function with value 1 on the set and 0 otherwise (i.e., the characteristic function of ).
and we have
Theorem 8 has the following direct consequence.
Let , let , and let be positive weights whose sum is 1. If is a convex function on , then
Note that Theorem 8 provides a kind of converse of the classical Jensen’s inequality for convex functions.
Let be a probability measure on the space and real constants. If is a measurable function and is a convex function on , then and are -integrable functions and
Let and be real constants. If is a measurable function, is a convex function on , and
then , and
Let and be real constants. If is a measurable function, is a continuous m-convex function on , with , and
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 -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.
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