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Generalization of quantum calculus and corresponding Hermite-Hadamard inequalities
(Springer Basel Ag, 2024) Akbar, Saira Bano; Abbas, Mujahid; Budak, Hueseyin
The aim of this paper is first to introduce generalizations of quantum integrals and derivatives which are called (phi-h) integrals and (phi-h) derivatives, respectively. Then we investigate some implicit integral inequalities for (phi-h) integrals. Different classes of convex functions are used to prove these inequalities for symmetric functions. Under certain assumptions, Hermite-Hadamard-type inequalities for q-integrals are deduced. The results presented herein are applicable to convex, m-convex, and & hstrok;-convex functions defined on the non-negative part of the real line.
New versions of the Hermite-Hadamard inequality for (φ-h)-integrals
(Springer, 2024) Akbar, Saira Bano; Abbas, Mujahid; Nazeer, Waqas; Budak, Hueseyin
In this paper, we prove some new versions of the Hermite-Hadamard inequality for (phi-h)-integrals. For this aim, we use the tangent and secant lines at the same special points. Moreover, we investigate the relations between the newly obtained results and earlier published papers. We also present some new results by special choices of phi and h.