Fejér–Pachpatte–Mercer-Type Inequalities for Harmonically Convex Functions Involving Exponential Function in Kernel

Abstract

In the present study, fractional variants of Hermite–Hadamard, Hermite–Hadamard–Fejér, and Pachpatte inequalities are studied by employing Mercer concept. Firstly, new Hermite–Hadamard–Mercer-type inequalities are presented for harmonically convex functions involving fractional integral operators with exponential kernel. Then, weighted Hadamard–Fejér–Mercer-type inequalities involving exponential function as kernel are proved. Finally, Pachpatte–Mercer-type inequalities for products of harmonically convex functions via fractional integral operators with exponential kernel are constructed.

Estimations of the Jensen Gap and Their Applications Based on 6-Convexity

“…From inequalities (15) and (16), we conclude that the estimates for the Jensen gap given in (11) provide better estimates than the estimates mentioned in inequality (2) in [40].…”

“…Since, the function Φ(δ) = δ κ τ is 6-convex on(0, ∞) for the mentioned values of τ and κ, we assume Φ(δ) = δ κ τ , δ ς = a ς , and ς = b τ ς in (11), and we obtain (25). (iii)-(iv) By utilizing (11) for δ ς = a ς , ς = b τ ς , and Φ(δ) = δ κ τ , we deduce the converse of (25).…”

“…Convex functions have many attractive and important properties, and due to these properties and their characteristics, convex functions play a leading role in the solutions to many complicated problems [7][8][9]. Moreover, convex functions are also popular because they deal with problems very smoothly [10][11][12]. Due to this, convex functions have attracted the attention of many researchers [13][14][15].…”

Some integral inequalities via new generalized harmonically convexity

“…In order to estimate and improve the error bounds for some well‐known integral inequalities, including the trapezoidal, midpoint, and Ostrowski‐type inequalities, inequality () has been established and generalized in numerous ways for various classes of convex functions [3–28]. Dragomir and Agarwal [11] established some inequalities of the trapezoidal type for differentiable convex functions by taking into consideration the above inequality.…”

Integral Inequalities of Integer and Fractional Orders for n–Polynomial Harmonically tgs–Convex Functions and Their Applications

“…Gürbüz et al [23] worked on the Caputo-Fabrizio operator, Mohammed et al [24] used tempered fractional integrals, Sahoo et al [25] used k− Riemann-Liouville fractional integrals, and Khan et al [26] established a new version of Hermite-Hadamard inequality employing generalized conformable fractional integrals. Butt and his collaborators, for example, worked on Jensen-Mercer type inequalities using novel fractional operators (see [27][28][29]), while Set et al [30] and Fernandez et al [31] presented the Hermite-Hadamard inequality using the Atangana-Baleanu fractional operator. For some recent generalizations of the Hermite-Hadamard inequality, we suggest interested readers to see [32][33][34][35][36] and the references therein.…”