Inequalities for Harmonically Convex Mappings

: In this paper, we prove Hermite–Hadamard–Mercer inequalities, which is a new version of the Hermite–Hadamard inequalities for harmonically convex functions. We also prove Hermite– Hadamard–Mercer-type inequalities for functions whose ﬁrst derivatives in absolute value are harmonically convex. Finally, we discuss how special means can be used to address newly discovered inequalities.

In the theory of convex functions, the Hermite-Hadamard inequality is very important. It was independently discovered by C. Hermite and J. Hadamard (see also [2,3] (p. 137)): where f : I → R is a convex function over I, and κ 1 , κ 2 ∈ I, with κ 1 < κ 2 . In the case of concave mappings, the above inequality is satisfied in reverse order.
The following variant of the Jensen inequality, known as the Jensen-Mercer, was demonstrated by Mercer [4]: Theorem 1. If f is a convex function on [a, b], then the following inequality is true: where n ∑ j=1 λ j = 1, x j ∈ [a, b] and λ j ∈ [0, 1].
In [5], the idea of the Jensen-Mercer inequality was used by Kian and Moslehian, and the following Hermite-Hadamard-Mercer inequality was demonstrated: where f is a convex function on [κ 1 , κ 2 ]. For some recent studies linked to the Jensen-Mercer inequality, one can consult [6,7].

Harmonic Convexity and Related Inequalities
In this section, we will study the concepts of harmonically convex functions and the integral inequalities associated with them.

Definition 1.
A mapping such as f : I ⊂ R\{0} → R [8] is called harmonically convex if the following inequality holds for all x, y ∈ I and τ ∈ [0, 1] : When the inequality (6) is reversed, f is described as harmonically concave.

Main Results
For harmonically convex functions and differentiable harmonically convex functions, we will prove Hermite-Hadamard-Mercer-type inequalities in this section.
Theorem 7. For a harmonically convex mapping f : I ⊂ R\{0} → R with κ 1 , κ 2 ∈ I and κ 1 < κ 2 , the following inequality holds: Proof. Since the given mapping f is harmonically convex, we have By setting Integrating inequality (15) with respect to τ over an interval [0, 1], we have Thus, we obtain the first inequality of (13) because each integral on the right side of (16) is equal to To prove the second inequality in (13) through the harmonic convexity function of f , we have the following: By adding (17) and (18) and using inequality (10), we have Integrating inequality (19) with respect to τ over an interval [0, 1], we have Hence, we obtain the last inequality of (13).
The simple lemma below is needed to discover some new Hermite-Hadamard-Mercertype inequalities for functions whose first derivatives are harmonically convex.

Lemma 2.
For a differentiable mapping f : I ⊂ R\{0} → R on I • with κ 1 , κ 2 ∈ I and κ 1 < κ 2 , the following equality holds: Proof. Using the basic rules of integration, we have 1 0 Thus, we obtain the resultant equality (21) by multiplying the equality (22) with y−x 2xy .

Remark 2.
In Lemma 2, if we put x = κ 1 and y = κ 2 , then equality (21) becomes equality (9). Now, for the sake of brevity, we shall use the following notations: Theorem 8. The conditions of Lemma 2 are assumed to be true. The following inequality holds if the mapping | f | q , q ≥ 1 is harmonically convex on I: and Proof. We can deduce from Lemma 2 and the power mean inequality that: By inequality (10), we have the following: It is simple to verify this:
Theorem 9. The conditions of Lemma 2 are assumed to be true. The following inequality holds if the mapping | f | q , q > 1 is harmonically convex on I: where and Proof. We can deduce from Lemma 2 and Hölder's inequality that: By inequality (10), we have the following: It is simple to verify this: Proof. Inequality (13) in Theorem 7 for the mapping f : (0, ∞) → R, f (x) = x leads to this conclusion.

Conclusions
In this paper, we proved some new Hermite-Hadamard-Mercer inequalities for harmonically convex functions and differentiable harmonically convex functions. It was also demonstrated that the results of this paper generalize the findings ofİsçan in [8]. It is an interesting and challenging problem, and researchers may be able to obtain similar inequalities for various fractional operators in their future work.