Properties and Bounds of Jensen-Type Functionals via Harmonic Convex Functions

Abstract

Dragomir introduced the Jensen-type inequality for harmonic convex functions (HCF) and Baloch et al. studied its different variants, such as Jensen-type inequality for harmonic -convex functions. In this paper, we aim to establish the functional form of inequalities presented by Baloch et al. and prove the superadditivity and monotonicity properties of these functionals. Furthermore, we derive the bound for these functionals under certain conditions. Furthermore, we define more generalized functionals involving monotonic nondecreasing concave function as well as evince superadditivity and monotonicity properties of these generalized functionals.

1. Introduction

Convexity is natural and simple notion which has found applications in business, industry, and medicine. During the study of convexity, many researchers have been fascinated by generalization of this class and have tried to find out those classes of functions which have close relation with this class (but not convex in general). Harmonic convex functions (HCFs) [1], harmonic -convex functions [2], harmonic -convex functions [3, 4], and harmonic -convex functions [5] are among these classes. For a quick glance on importance of these classes and applications, see [6–9] and references therein. The class of harmonic convex functions (HCFs) and its different variants are very important classes that gained prominence in the theory of inequalities and applications as well as in other branches of mathematics. Many researchers have been working on the class of harmonic convex functions (HCFs) due to its significance and have been trying to explore about it more and more. During this study, recently different generalizations of the class of harmonic convex functions (HCFs) have been found, for example, see [10–13] and references therein.

The importance of the class of HCFs continuously encourages us and many other researchers to explore more about it, and the following paper is a link to it. For the better understanding of the results of present paper, we first recall some basic definitions.

Definition 1. Consider . A function is HCF on ifholds, for all and . If we reverse the above inequality, the function becomes harmonic concave.

Remark 1. (see [14]).(i)The function is a HCF on the interval , but it is not a convex function.(ii)The functionis another example of HCF, which is neither convex nor concave.
Baloch et al., in [7], observed some remarkable facts for the class of HCFs.

Proposition 1. For , a function , we have the following facts:(1)If and is nondecreasing and convex function, then is HCF(2)If and is nonincreasing function and harmonically convex, then is convex function(3)If and is nondecreasing function and harmonically convex, then is convex function(4)If and is nonincreasing function and convex, then is HCFVarošanec, in [15], proposed the concept of -convexity (also, see [16–18]) to unify numerous generalized aspects of convex functions. In a similar fashion, harmonic -convexity unifies the various types of harmonic convexities.

Definition 2. (see [19]). Consider a nonnegative function . Then, the function is said to be harmonic -convex on if the inequalityholds, for all and . Furthermore, if we reverse inequality (3), then becomes harmonic -concave.

Remark 2. We provide few examples of harmonic -convex (concave) functions as follows:(i)Obviously, with , the class of nonnegative harmonic convex (concave) functions on become a particular case of the class of harmonic -convex (concave) functions on .(ii)Let and . Consider a function defined by , which is neither nonincreasing nor nondecreasing -convex function. Therefore, is a harmonic -convex function by Proposition 1 given in [7].(iii)Let and be a real-valued function such that on . Then, the following four functions, , , , and , satisfy the conditions of the function mentioned above. Therefore, is a harmonic -convex function for if is a nondecreasing convex function, or harmonic -convex function, or harmonic Godunova–Levin function or harmonic -function.(iv)Let be a nondecreasing continuous function and be a continuous self-concave function such that , for some and all . Then, is a -convex function by Lemma 1 of [20], and hence, is a harmonic -convex function by Proposition 1 of [7].

Definition 3. A function is said to be a submultiplicative function iffor all . If inequality (4) is reversed, then is said to be supermultiplicative function. If just equality holds in relation (4), then is said to be multiplicative function.

Definition 4. A function is said to be a subadditive function iffor all . If inequality (5) is reversed, then is said to be superadditive function. If just the equality holds in relation (5), then is said to be additive function.
Jensen-type inequality for HCFs is proposed by Dragomir [21].

Theorem 1. Let be an interval. If is HCF, thenHolds, for all and with .

In [22], Baloch et al. derived the following results:

Theorem 2. Let . If is HCF, then, for any finite positive sequence and with , we have

Theorem 3. Let be positive real numbers () and . If is a nonnegative supermultiplicative function and if is harmonic -convex function, , then

Theorem 4. Let be a nonnegative supermutiplicative function on . Let be positive real numbers () such that and . If is harmonic -convex on , then, for any finite positive increasing sequence , we have

If is a submultiplicative function, and is harmonic -concave then inequality (9) is reversed.

Remark 3. Importance of the class of HCFs can be guessed by the following applications in the field of mathematics:(i)Harmonic convexity provides a useful analytic tool to calculate several known definite integrals such as , , , and , where , see [7](ii)Inequality (6) provides a very short proof of the discrete form of Hölder’s inequality (see [22])(iii)By inequalities (6) and (7), we can easily prove weighted HGA inequality (see [14])Many researchers considered the functionals related to Jensen’s inequality and tried to find properties and bound for these functionals (for example, see [23–30]). In the sequel, the set of all nonnegative n-tuples , such that , will be denoted with .
The difference between the right-hand and the left-hand side of inequalities (6)–(9) defines the following functionals:For a fixed function and n-tuple , , , , and can be considered as functions on , which is a convex subset in . Furthermore, because of inequalities (6)–(9), we have , , , and , for all .

2. Main Results

In this section, we establish some properties of functionals related to Jensen-type inequalities for HCFs.

Theorem 5. Let and be two n-tuples from . Let be an interval. If is a HCF, is a nonnegative multiplicative and additive function on , and if , , then , for , defined by (10)–(12) are superadditive on , i.e.,for .

Proof. Take in (28) and starting from definition, we havewhile, after arranging and harmonic convexity of , yieldsFinally, combining relation (15) and inequality (16), we obtainNow, taking in (28) and starting from the definition, we havewhile, after arranging and harmonic convexity of , yieldsFinally, combining relation (18) and inequality (19), we obtainTaking in (28) and starting from the definition, we havewhile, after arranging and harmonic -convexity of , yieldsFinally, combining relation (21) and inequality (22), we obtainSimilarly, it can be proved that

Theorem 6. Let and be two n-tuples from such that , (i.e., ). Let be an interval. If is a HCF and if , then , for , defined by (10)–(12) satisfy the following inequality:on .

Proof. The monotonicity property follows directly from superadditivity. Since , can be represented as the sum of two n-tuples: and . Applying (28), we haveFinally, by (10)–(12). So, we have that , which proves the theorem.

Theorem 7. Let and be two n-tuples from . Let and be real constants such that

Let be an interval. If is a HCF and if be any n-tuple, thenfor .

Proof. Since , this implies that and are in . Then, by Theorem 5, we obtainSimilarly, we obtainthat is,

Corollary 1. Let , , , and functional be as in Theorem 5. Then,where

Proof. Let be a constant n-tuple, i.e.,Then, for any , we have . So, by applying Theorem 6, we haveOn the contrary,i.e., . So, it proves the right-hand side of inequality (35). The left-hand inequality is obtained similarly by exchanging the role of min and max.
To present our next results, we need to introduce the following notations: { are such that , for all } {: , for all } {: , for all } {: , for all } {: is finite} the cone of all HCFs on C  the linear space of all real functions on CNow, we consider more general functionals:where , , , , and is a convex function whose properties will determine the behavior of functional , as follows. Obviously, for , we recapture, from functional , the functional considered in Theorem 5.
First of all, we observe that, by Jensen-type inequality, the functional is well defined and positive homogenous in the third variable, that is,for any and .
The following result concerning the superadditivity and the monotonicity of the functional , as function of weights holds.

Theorem 8. Let , , , and be monotonic nondecreasing and concave function.(i)If , then That is, is superadditive as a function of weights.(ii)If , with , meaning that , for each and , thenThat is, is monotonic nondecreasing as function of weights.

Proof. . (i)Let ; then, by the harmonic convexity of on , Since is monotonically nondecreasing and concave, then, by (40), which, by multiplication with , produces the desired result (39) for . Similarly, we can easily verify result (39), for .(ii)If , then by (i),since .

Corollary 2. Let , , , and be a monotonically nondecreasing and concave function.

If there exist the numbers and with such that , then

In particular,

For a function , we now consider the functionalswhere , , , and . Now, if we denote by the sequence , then we observe that, for ,

The following result may be stated.

Corollary 3. Let , , , and be monotonically nondecreasing and concave function. If is concave, then is also concave on , for .

Proof. Utilizing the properties of monotonicity, superadditivity, and positive homogeneity of functional , we havefor any and , which proves the statement.
The following result concerning the superadditivity and monotonicity of the functional , for , as an index set function holds.

Theorem 9. Let , , , and . Assume that is a monotonically nondecreasing and concave function.(i)If with , That is, is superadditive as an index set function on .(ii)If with and , then

That is, is monotonically nondecreasing as an index set function on .

Proof. .(i)Let with . By the harmonic convexity of on , we have Since is monotonically nondecreasing and concave, then, by (50), which, by multiplication with , produces the desired result (48) for . Similarly, we can easily verify result (48), for .(ii)Let with ; then,since .

3. Conclusion

First of all, we have presented the refinement of Jensen-type inequality, and further, we have discussed several important aspect of functionals associated with Jensen-type inequalities for the HCFs. On the basis of ideas discussed in this paper along with the literature present on HCFs, we encourage the interested researcher to explore more interesting results for this class of functions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally in this research article. All authors read and approved the final manuscript.