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

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 
 
 h
 
 -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.


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][7][8][9] and references therein. e 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][11][12][13] and references therein. e 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.
holds, for all w 1 , w 2 ∈ I and t ∈ [0, 1]. If we reverse the above inequality, the function f becomes harmonic concave.
(i) e function f(w) � ln w is a HCF on the interval (0, ∞), but it is not a convex function. (ii) e function 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. convex, then f is HCF Varošanec, in [15], proposed the concept of h-convexity (also, see [16][17][18]) to unify numerous generalized aspects of convex functions. In a similar fashion, harmonic h-convexity unifies the various types of harmonic convexities.
for all w 1 , w 2 ∈ 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.
for all w 1 , w 2 ∈ 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].
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 (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][24][25][26][27][28][29][30]). In the sequel, the set of all nonnegative n-tuples b � (b 1 , . . . , b n ), such that B n ≔ n α�1 b α > 0, will be denoted with B 0 n . e difference between the right-hand and the left-hand side of inequalities (6)-(9) defines the following functionals: For a fixed function f and n-tuple w,

Main Results
In this section, we establish some properties of functionals related to Jensen-type inequalities for HCFs. (14) for i � 1, 2, 3, 4.
Proof. Take i � 1 in (28) and starting from definition, we have Journal of Mathematics while, after arranging and harmonic convexity of f, yields Finally, combining relation (15) and inequality (16), we obtain f, w, c). (17) Now, taking i � 2 in (28) and starting from the definition, we have while, after arranging and harmonic convexity of f, yields 4 Journal of Mathematics Finally, combining relation (18) and inequality (19), we obtain Taking i � 3 in (28) and starting from the definition, we have while, after arranging and harmonic h-convexity of f, yields Journal of Mathematics Finally, combining relation (21) and inequality (22), we obtain Similarly, it can be proved that Proof. e monotonicity property follows directly from superadditivity. Since b ≥ c, b can be represented as the sum of two n-tuples: b − c and c. Applying (28), we have Finally, M i (f, w, (b − c)) ≥ 0 by (10)- (12). So, we have that f, w, c), which proves the theorem.  � (b 1 , . . . , b n ) and c � (c 1 , . . . , c n ) be two n-tuples from B 0 n . Let m and M be real constants such that ≥ mM i (f, w, c). (29) Similarly, we obtain , w, c), (30) that is, where en, for any b ∈ B 0 n , we have b ≥ b min . So, by applying eorem 6, we have On the contrary, i.e., , w). So, it proves the right-hand side of inequality (35). e 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: where f ∈ HConv(C, R), K ∈ P f (N), b ∈ J + (R), w ∈ J * (C), and Ψ: (0, ∞) ⟶ R is a convex function whose properties will determine the behavior of functional D i , i � 1, 2, 3, 4, as follows. Obviously, for Ψ(t) � t, we recapture, from functional D i , the functional M i considered in eorem 5.
First of all, we observe that, by Jensen-type inequality, the functional D i is well defined and positive homogenous in the third variable, that is, for any m > 0 and b ∈ J + (R). e following result concerning the superadditivity and the monotonicity of the functional D i , i � 1, 2, as function of weights holds. □ Theorem 8. Let f ∈ HConv(C, R), K ∈ P f (N), w ∈ J * (C), and Ψ: (0, ∞) ⟶ R be monotonic nondecreasing and concave function.
at is, D i is superadditive as a function of weights.
at is, D i is monotonic nondecreasing as function of weights.
Proof (i) Let b, c ∈ J + (R); then, by the harmonic convexity of f on C, Since Ψ is monotonically nondecreasing and concave, then, by (40), which, by multiplication with B K + C K > 0, produces the desired result (39) for i � 1. Similarly, we can easily verify result (39), for i � 2.
If there exist the numbers m and M with M ≥ m ≥ 0 such that M c ≥ b ≥ mc, then In particular,
at is, D i (f, K, ., w; Ψ) is superadditive as an index set function on P f (N).
(ii) Let K, L ∈ P f (N) with L ⊂ K; then,

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