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

Authors: Aqeel Ahmad Mughal; Hassan Almusawa; Absar Ul Haq; Imran Abbas Baloch
Journal: Journal of Mathematics
 (2021)

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## 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 harmonich-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.


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## Body

## 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 α,m-convex functions [2], harmonic s,m-convex functions [3, 4], and harmonic p,s,m-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.
ConsiderI⊂ℝ\0. A function f:I⟶ℝ is HCF on I if(1)fw1w2tw1+1−tw2≤tfw2+1−tfw1,holds, for all w1,w2∈I and t∈0,1. If we reverse the above inequality, the function f becomes harmonic concave.Remark 1.
(see [14]).(i)
The functionfw=lnw is a HCF on the interval 0,∞, but it is not a convex function.(ii)
The function(2)fw=1−ww,0<w≤1,0,1<w≤2,w−2w,w>2,is 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.
ForI⊆ℝ\0, a function f:I⟶ℝ, we have the following facts:(1)
IfI⊂0,∞ and f is nondecreasing and convex function, then f is HCF(2)
IfI⊂0,∞ and f is nonincreasing function and harmonically convex, then f is convex function(3)
IfI⊂−∞,0 and f is nondecreasing function and harmonically convex, then f is convex function(4)
IfI⊂−∞,0 and f is nonincreasing function and convex, then f is HCF
Varošanec, in [15], proposed the concept of h-convexity (also, see [16–18]) to unify numerous generalized aspects of convex functions. In a similar fashion, harmonic h-convexity unifies the various types of harmonic convexities.Definition 2.
(see [19]). Consider a nonnegative function h:0,1⟶ℝ+. Then, the function f:I⊆ℝ\0⟶0,∞ is said to be harmonic h-convex on I if the inequality(3)fw1w2tw1+1−tw2≤htfw2+h1−tfw1,holds, for all w1,w2∈I and t∈0,1. Furthermore, if we reverse inequality (3), then f becomes harmonic h-concave.Remark 2.
We provide few examples of harmonich-convex (concave) functions as follows:(i)
Obviously, withht=t, the class of nonnegative harmonic convex (concave) functions on I become a particular case of the class of harmonic h-convex (concave) functions on I.(ii)
Lett∈0,1 and ht=t2. Consider a function f:−1,0∪0,1⟶ℝ defined by fx=1, which is neither nonincreasing nor nondecreasing h-convex function. Therefore, f is a harmonic h-convex function by Proposition 1 given in [7].(iii)
Lett∈0,1 and h:0,1⟶0,∞ be a real-valued function such that ht≥t on 0,1. Then, the following four functions, h1t=t, h2t=tss∈0,1, h3t=1/t, and h4t=1, satisfy the conditions of the function h mentioned above. Therefore, f is a harmonic hα-convex function for α=1,2,3,4 if f:I⊆0,∞⟶0,∞ is a nondecreasing convex function, or harmonic s-convex function, or harmonic Godunova–Levin function or harmonic P-function.(iv)
Letf:0,∞⟶0,∞ be a nondecreasing continuous function and h:0,1⟶0,∞ be a continuous self-concave function such that ftw1+1−tw2≤htfw1+1−tfw2, for some t∈0,1 and all w1,w2∈0,∞. Then, f is a h-convex function by Lemma 1 of [20], and hence, f is a harmonic h-convex function by Proposition 1 of [7].Definition 3.
A functionh:I⊆ℝ⟶ℝ is said to be a submultiplicative function if(4)hw1w2≤hw1hw2,for all w1,w2∈I. If inequality (4) is reversed, then h is said to be supermultiplicative function. If just equality holds in relation (4), then h is said to be multiplicative function.Definition 4.
A functionh:I⊆ℝ⟶ℝ is said to be a subadditive function if(5)hw1+w2≤hw1+hw2,for all w1,w2∈I. If inequality (5) is reversed, then h is said to be superadditive function. If just the equality holds in relation (5), then h is said to be additive function.
Jensen-type inequality for HCFs is proposed by Dragomir [21].Theorem 1.
LetI⊆0,∞ be an interval. If f:I⟶ℝ is HCF, then(6)f11/Bn∑α=1nbα/wα≤1Bn∑α=1nbαfwα,Holds, for all w1,…,wn∈I and b1,…,bn≥0 with ∑α=1nbα=Bn.In [22], Baloch et al. derived the following results:Theorem 2.
LetI⊆ℝ\0. If f:I⟶ℝ is HCF, then, for any finite positive sequence wαα=1n∈I and b1,…,bn≥0 with ∑α=1nbα=Bn, we have(7)f11/w1+1/wn−1/Bn∑α=1nbα/wα≤fw1+fwn−1/Bn∑α=1nbαfwα.Theorem 3.
Letb1,…,bn be positive real numbers (n≥2) and Bn=∑α=1nbα. If h:I⊇0,1⟶ℝ is a nonnegative supermultiplicative function and if f is harmonic h-convex function, wαα=1n∈I, then(8)f11/Bn∑α=1nbα/wα≤∑α=1nhbαBnfwα.Theorem 4.
Leth:I⊇0,1⟶ℝ be a nonnegative supermutiplicative function on I. Let b1,…,bn be positive real numbers (n≥2) such that Bn=∑α=1nbα and ∑α=1nhbα/Bn≤1. If f is harmonic h-convex on I⊆ℝ\0, then, for any finite positive increasing sequence wαk=1n∈I, we have(9)f11/w1+1/wn−1/Bn∑α=1nbα/wα≤fw1+fwn−∑α=1nhbαBnfwα.Ifh is a submultiplicative function, ∑α=1nhbα/Bn≥1 and f is harmonic h-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∫abew/xndw, ∫abew2dw, ∫absinw/wndw, and ∫abcosw/wndx∀n∈ℕ, where a,b∈0,∞, 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 b=b1,…,bn, such that Bn≔∑α=1nbα>0, will be denoted with ℬn0.
The difference between the right-hand and the left-hand side of inequalities (6)–(9) defines the following functionals:(10)ℳ1f,w,b≔∑α=1nbαfwα−Bnf11/Bn∑α=1nbα/wα,(11)ℳ2f,w,b≔Bnfw1+fwn−∑α=1nbαfwα−Bnf11/w1+1/wn−1/Bn∑α=1nbα/wα,(12)ℳ3f,w,b≔∑α=1nhbαBnfwα−f11/Bn∑α=1nbα/wα,(13)ℳ4f,w,b≔fw1+fwn−∑α=1nhbαBnfwα−f11/w1+1/wn−1/Bn∑α=1nbα/wα.
For a fixed functionf and n-tuple w, ℳ1f,w,., ℳ2f,w,., ℳ3f,w,., and ℳ4f,w,. can be considered as functions on ℬn0, which is a convex subset in ℝn. Furthermore, because of inequalities (6)–(9), we have ℳ1f,w,b≥0, ℳ2f,w,b≥0, ℳ3f,w,b≥0, and ℳ4f,w,b≥0, for all b∈ℬn0.


## 2. Main Results

In this section, we establish some properties of functionals related to Jensen-type inequalities for HCFs.Theorem 5.
Letb=b1,…,bn and c=c1,…,cn be two n-tuples from ℬn0. Let I⊆0,∞ be an interval. If f:I⟶ℝ is a HCF, h:I⊇0,1⟶ℝ is a nonnegative multiplicative and additive function on J, and if w=w1,…,wn∈In, hα+h1−α≥2, then ℳif,w,., for i=1,2,3,4, defined by (10)–(12) are superadditive on ℬn0, i.e.,(14)ℳif,w,b+c≥ℳif,w,b+ℳif,w,c≥0,for i=1,2,3,4.Proof.
Takei=1 in (28) and starting from definition, we have(15)ℳ1f,w,b+c=∑α=1nbα+cαfwα−Bn+Cnf11/Bn+Cn∑α=1nbα+cα/wα=∑α=1nbαfwα+∑α=1ncαfwα−Bn+Cnf11/Bn+Cn∑α=1nbα+cα/wα,while, after arranging and harmonic convexity of f, yields(16)f11/Bn+Cn∑α=1nbα+cα/wα=f1Bn/Bn+Cn.1/Bn∑α=1nbα/wα+Cn/Bn+Cn.1/Cn∑α=1ncα/wα≤BnBn+Cnf11/Bn∑α=1nbα/wα+CnBn+Cnf11/Cn∑α=1ncα/wα.
Finally, combining relation (15) and inequality (16), we obtain(17)ℳ1f,w,b+c≥∑α=1nbαfwα+∑α=1ncαfwα−Bnf11/Bn∑α=1nbα/wα−Cnf11/Cn∑α=1ncα/wα=ℳ1f,w,b+ℳ1f,w,c.
Now, takingi=2 in (28) and starting from the definition, we have(18)ℳ2f,w,b+c=Bn+Cnfw1+fwn−∑α=1nbα+cαfwα−Bn+Cnf11/w1+1/wn−1/Bn+Cn∑α=1nbα+cα/wα=Bnfw1+fwn+Cnfw1+fwn−∑α=1nbαfwα−∑α=1ncαfwα−Bn+Cnf11/w1+1/wn−1/Bn+Cn∑α=1nbα+cα/wα,while, after arranging and harmonic convexity of f, yields(19)f11/w1+1/wn−1/Bn+Cn∑α=1nbα+cα/wα=f11/Bn+Cn∑α=1nbα+cα1/w1+1/wn−1/wα=f1Bn/Bn+Cn∑α=1nbα/Bn1/w1+1/wn−1/wα+Cn/Bn+Cn∑α=1ncα/Cn1/w1+1/wn−1/wα≤BnBn+Cnf1∑α=1nbα/Bn1/w1+1/wn−1/wα+CnBn+Cnf1∑α=1ncα/Cn1/w1+1/wn−1/wα=BnBn+Cnf11/w1+1/wn−1/Bn∑α=1nbα/wα+CnBn+Cnf11/w1+1/wn−1/Cn∑α=1ncα/wα.
Finally, combining relation (18) and inequality (19), we obtain(20)ℳ2f,w,b+c≥Bnfw1+fwn+Cnfw1+fwn−∑α=1nbαfwα−∑α=1ncαfwα−Bn+Cn.BnBn+Cnf11/w1+1/wn−1/Bn∑α=1nbα/wα−Bn+Cn.CnBn+Cnf11/w1+1/wn−1/Cn∑α=1ncα/wα=ℳ2f,w,b+ℳ2f,w,c.
Takingi=3 in (28) and starting from the definition, we have(21)ℳ3f,w,b+c=∑α=1nhbα+cαBn+Cnfwα−f11/Bn+Cn∑α=1nbα+cα/wα=∑α=1nhbαBn+Cnfwα+∑α=1nhcαBn+Cnfwα−f11/Bn+Cn∑α=1nbα+cα/wα=∑α=1nhBnBn+CnhbαBnfwα+∑α=1nhCnBn+CnhcαCnfwα−f11/Bn+Cn∑α=1nbα+cα/wα,while, after arranging and harmonic h-convexity of f, yields(22)f11/Bn+Cn∑α=1npi+cα/wα=f1Bn/Bn+Cn.1/Bn∑α=1nbα/wα+Cn/Bn+Cn.1/Cn∑α=1ncα/wα≤hBnBn+Cnf11/Bn∑α=1nbα/wα+hCnBn+Cnf11/Cn∑α=1ncα/wα=hBnBn+Cnf11/Bn∑α=1nbα/wα+hCnBn+Cnf11/Cn∑α=1ncα/wα.
Finally, combining relation (21) and inequality (22), we obtain(23)ℳ3f,w,b+c≥hBnBn+Cn∑α=1nhbαBnfwα+hCnBn+Cn∑α=1nhcαCnfwα−hBnBn+Cnf11/Bn∑α=1nbα/wα−hCnBn+Cnf11/Cn∑α=1ncα/wα=hBnBn+Cn∑α=1nhbαBnfwα−f11/Bn∑α=1nbα/wα+hCnBn+Cn∑α=1nhcαCnfwα−f11/Cn∑i=1ncα/wα≥∑α=1nhbαBnfwα−f11/Bn∑α=1nbα/wα+∑α=1nhcαCnfwα−f11/Cn∑α=1ncα/wα=ℳ3f,w,b+ℳ3f,w,c.
Similarly, it can be proved that(24)ℳ4f,w,b+c≥ℳ4f,w,b+ℳ4f,w,c.Theorem 6.
Letb=b1,…,bn and c=c1,…,cn be two n-tuples from ℬn0 such that b≥c, (i.e., bα≥cα,α=1,…,n). Let I⊆0,∞ be an interval. If f:I⟶ℝ is a HCF and if w=w1,…,wn∈In, then ℳif,w,., for i=1,2,3,4, defined by (10)–(12) satisfy the following inequality:(25)ℳif,w,b≥ℳif,w,c,on ℬn0.Proof.
The monotonicity property follows directly from superadditivity. Sinceb≥c, b can be represented as the sum of two n-tuples: b−c and c. Applying (28), we have(26)ℳif,w,b=ℳif,w,b−c+c≥ℳif,w,b−c+ℳif,w,c.
Finally,ℳif,w,b−c≥0 by (10)–(12). So, we have that ℳif,w,b≥ℳif,w,c, which proves the theorem.Theorem 7.
Letb=b1,…,bn and c=c1,…,cn be two n-tuples from ℬn0. Let m and M be real constants such that(27)m≥0,bα−mcα≥0,Mcα−bα≥0,α=1,…n.LetI⊆0,∞ be an interval. If f:I⟶ℝ is a HCF and if w=w1,…,wn∈In be any n-tuple, then(28)Mℳif,w,c≥ℳif,w,b≥mℳif,w,c,for i=1,2,3,4.Proof.
Sincem≥0,bα−mcα≥0andMcα−bα≥0,α=1,…n, this implies that b−mc and Mc−b are in ℬn0. Then, by Theorem 5, we obtain(29)ℳif,w,b≥ℳif,w,b−mc+ℳif,w,mc≥mℳif,w,c.
Similarly, we obtain(30)ℳif,w,b≤Mℳif,w,c,that is,(31)Mℳif,w,c≥ℳif,w,b≥mℳif,w,c.Corollary 1.
Letb, w, f, and functional ℳi be as in Theorem 5. Then,(32)max1≤k≤nbαℳiNf,w≥ℳif,w,b≥min1≤k≤nbαℳiNf,wfori=1,2,where(33)ℳ1Nf,w=∑α=1nfwα−nf11/n∑α=1n1/wα,ℳ2Nf,w=nfw1+fwn−∑α=1nfwα−nf11/w1+1/wn−1/n∑α=1n1/wα.Proof.
Letbmin∈ℬn0 be a constant n-tuple, i.e.,(34)bmin=min1≤k≤nbα,…,min1≤k≤nbα.
Then, for anyb∈ℬn0, we have b≥bmin. So, by applying Theorem 6, we have(35)ℳif,w,b≥ℳif,w,bmin.
On the contrary,(36)ℳ1f,w,bmin=min1≤k≤nbα∑α=1nfwα−nf11/n∑α=1n1/wα,ℳ1f,w,bmin=min1≤k≤nbαnfw1+fwn−∑α=1nfwα−nf11/w1+1/wn−1/n∑α=1n1/wα,i.e., ℳ1f,w,bmin=min1≤k≤nbαℳiNf,w. 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:Jℝ≔{b=bαk∈ℕ:bα∈ℝ are such that BK≔∑α∈Ibα≠0, for all K∈Pfℕ}F+C,ℝ≔{f∈FC,ℝ: fx>0, for all x∈C}J+ℝ≔{b∈Jℝ: bα≥0, for all k∈ℕ}J∗ℝ≔{w=wαα∈ℕ: wα∈C, for all k∈ℕ}Pfℕ≔{K⊂ℕ: K is finite}HConvC,ℝ≔the cone of all HCFs on CFC,ℝ≔ the linear space of all real functions on C
Now, we consider more general functionals:(37)D1f,K,b,w;Ψ≔BKΨ1BK∑α∈Kbαfwα−f11/BK∑α∈Kbα/wα,D2f,K,b,w;Ψ≔BKΨfW1+fX21BK∑α∈Kbαfwα−f11/w1+1/x2−1/BK∑α∈Kbα/wα,D3f,K,b,w;Ψ≔Ψ∑α∈KhbαBKfwα−f11/BK∑α∈Kbα/wα,D4f,K,b,w;Ψ≔ΨfW1+fX21BK∑α∈KhbαBKfwα−f11/w1+1/x2−1/BK∑α∈Kbα/wα,where f∈HConvC,ℝ, K∈Pfℕ, b∈J+ℝ, w∈J∗C, and Ψ:0,∞⟶ℝ is a convex function whose properties will determine the behavior of functional Di,i=1,2,3,4, as follows. Obviously, for Ψt=t, we recapture, from functional Di, the functional ℳi considered in Theorem 5.
First of all, we observe that, by Jensen-type inequality, the functionalDi is well defined and positive homogenous in the third variable, that is,(38)Dif,K,mb,w;Ψ=mDif,K,b,w;Ψ,for any m>0 and b∈J+ℝ.
The following result concerning the superadditivity and the monotonicity of the functionalDi,i=1,2, as function of weights holds.Theorem 8.
Letf∈HConvC,ℝ, K∈Pfℕ, w∈J∗C, and Ψ:0,∞⟶ℝ be monotonic nondecreasing and concave function.(i)
Ifb,c∈J+ℝ, then(39)Dif,K,b+c,w;Ψ≥Dif,K,b,w;Ψ+Dif,K,c,w;Ψ.That is,Di is superadditive as a function of weights.(ii)
Ifb,c∈J+ℝ, with b≥c, meaning that bi≥ci, for each i∈ℕ and Ψ:0,∞⟶0,∞, then(40)Dif,K,b,w;Ψ≥Dif,K,c,w;Ψ≥0.
That is,Di is monotonic nondecreasing as function of weights.Proof.
.(i)
Letb,c∈J+ℝ; then, by the harmonic convexity of f on C,(41)1BK+CK∑α∈Kbα+cαfwα−f11/BK+BK∑α∈Kbα+cα/wα=BK1/BK∑α∈Kbαfwα+CK1/CK∑α∈KcαfwαBK+CK−f1BK/BK+CK.1/BK∑α∈Kbα/wα+CK/BK+CK.1/CK∑α∈Kcα/wα≥BK1/BK∑α∈Kbαfwα+CK1/CK∑α∈KcαfwαBK+CK−BKf1/1/BK∑α∈Kbα/wα+CKf1/1/CK∑α∈Kcα/wαBK+CK=BK1/BK∑α∈Kbαfwα−f1/1/BK∑α∈Kbα/wαBK+CK+CK1/CK∑α∈Kcαfwα−f1/1/CK∑α∈Kcα/wαBK+CK.SinceΨ is monotonically nondecreasing and concave, then, by (40),(42)Ψ1BK+CK∑α∈Kbα+cαfwα−f11/BK+BK∑α∈Kbα+cα/wα≥BKΨ1/BK∑α∈Kbαfwα−f1/1/BK∑α∈Kbα/wαBK+CK+CKΨ1/CK∑α∈Kcαfwα−f1/1/CK∑α∈Kcα/wαBK+CK,which, by multiplication withBK+CK>0, produces the desired result (39) for i=1.Similarly, we can easily verify result (39), for i=2.(ii)
Ifb≥c, then by (i),(43)Dif,K,b,w;Ψ=Dif,K,b−c+c,w;Ψ≥Dif,K,b−c,w;Ψ+Dif,K,c,w;Ψ≥Dif,K,c,w;Ψ,since Dif,K,b−c,w;Ψ≥0.Corollary 2.
Letf∈HConvC,ℝ, K∈Pfℕ, w∈J∗C, and Ψ:0,∞⟶ℝ be a monotonically nondecreasing and concave function.If there exist the numbersm and M with M≥m≥0 such that Mc≥b≥mc, then(44)MCKΨ1CK∑α∈Kcαfwα−f11/CK∑α∈Kcα/wα≥BKΨ1BK∑α∈Kbαfwα−f11/BK∑α∈Kbα/wα≥mCKΨ1CK∑α∈Kcαfwα−f11/CK∑α∈Kcα/wα.In particular,(45)MmΨ1CK∑α∈Kcαfwα−f11/CK∑α∈Kcα/wα≥Ψ1BK∑α∈Kbαfwα−f11/BK∑α∈Kbα/wα≥mMΨ1CK∑α∈Kcαfwα−f11/CK∑α∈Kcα/wα.For a functionΦ:0,∞⟶0,∞, we now consider the functionals(46)D1f,K,b,w;Ψ,Φ≔∑α∈KΦbαΨ1∑α∈KΦbα∑α∈KΦbαfwα−f11/∑α∈KΦbα∑α∈KΦbα/wα,D2f,K,b,w;Ψ,Φ≔∑α∈KΦbαΨfw1+fwn−1/∑α∈KΦbα∑α∈KΦbαfwα−f11/w1+1/wn−1/∑α∈KΦbα∑α∈KΦbα/wα,where f∈HConvC,ℝ, K∈Pfℕ, w∈J∗C, and b∈J+ℝ. Now, if we denote by Φb the sequence Φbαk∈ℕ, then we observe that, for i=1,2,(47)Dif,K,b,w;Ψ,Φ=Dif,K,Φb,w;Ψ.The following result may be stated.Corollary 3.
Letf∈HConvC,ℝ, K∈Pfℕ, w∈J∗C, and Ψ:0,∞⟶0,∞ be monotonically nondecreasing and concave function. If Φ:0,∞⟶0,∞ is concave, then Dif,K,.,w;Ψ,Φ is also concave on J+ℝ, for i=1,2.Proof.
Utilizing the properties of monotonicity, superadditivity, and positive homogeneity of functionalDif,K,.,w;Ψ, we have(48)Dif,K,tb+1−tc,w;Ψ,Φ=Dif,K,Φtb+1−tc,w;Ψ≥Dif,K,tΦb+1−tΦc,w;Ψ≥Dif,K,tΦb,w;Ψ+Dif,K,1−tΦc,w;Ψ=tDif,K,Φb,w;Ψ+1−tDif,K,Φc,w;Ψ=tDif,K,b,w;Ψ,Φ+1−tDif,K,c,w;Ψ,Φ,for any b,c∈J+ℝ and t∈0,1, which proves the statement.
The following result concerning the superadditivity and monotonicity of the functionalDif,K,b,w;Ψ, for i=1,2, as an index set function holds.Theorem 9.
Letf∈HConvC,ℝ, K∈Pfℕ, w∈J∗C, and b∈J+ℝ. Assume that Ψ:0,∞⟶0,∞ is a monotonically nondecreasing and concave function.(i)
IfK,L∈Pfℕ with K∩L≠ϕ,(49)Dif,K∪L,b,w;Ψ≥Dif,K,b,w;Ψ+Dif,L,b,w;Ψ.That is,Dif,K,.,w;Ψ is superadditive as an index set function on Pfℕ.(ii)
IfK,L∈Pfℕ with L⊂K and Ψ:0,∞⟶0,∞, then(50)Dif,K,b,w;Ψ≥Dif,L,b,w;Ψ≥0.That is,Dif,K,.,w;Ψ is monotonically nondecreasing as an index set function on Pfℕ.Proof.
.(i)
LetK,L∈Pfℕ with K∩L≠ϕ. By the harmonic convexity of f on C, we have(51)1PK∪L∑j∈K∪Lbαfwα−f11/PK∪L∑j∈K∪Lbα/wα=BK1/BK∑α∈Kbαfwα+PL1/PL∑l∈LqlfxlBK+PL−f1BK/BK+PL.1/BK∑α∈Kbα/wα+PL/BK+PL.1/PL∑l∈Lql/xl≥BK1/BK∑α∈Kbαfwα+PL1/PL∑l∈LqlfxlBK+PL−BKf1/1/BK∑α∈Kbα/wα+PLf1/1/PL∑l∈Lql/xlBK+PL=BK1/BK∑α∈Kbαfwα−f1/1/BK∑α∈Kbα/wαBK+PL+PL1/PL∑l∈Lqlfxl−f1/1/PL∑l∈Lql/xlBK+PL.SinceΨ is monotonically nondecreasing and concave, then, by (50),(52)Ψ1PK∪L∑j∈K∪Lbαfwα−f11/PK∪L∑j∈K∪Lbα/wα≥BKΨ1/BK∑α∈Kbαfwα−f1/1/BK∑α∈Kbα/wαBK+PL+PLΨ1/PL∑l∈Lqlfxl−f1/1/PL∑l∈LqlfxlBK+PL,which, by multiplication withBK+PL>0, produces the desired result (48) for i=1.Similarly, we can easily verify result (48), for i=2.(ii)
LetK,L∈Pfℕ with L⊂K; then,(53)Dif,K,b,w;Ψ=Dif,K\L∪L,b,w;Ψ≥Dif,K\L,b,w;Ψ+Dif,L,b,w;Ψ≥Dif,L,b,w;Ψ≥0,since Dif,K\L,b,w;Ψ≥0.


## 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.



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