Integral inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate the convergence and stability of the methods. Unfortunately, mathematical methods are useless if the method is not convergent or stable. Thus, there is a present day need for accurate inequalities in proving the existence and uniqueness of the mathematical methods. Convexity play a concrete role in the field of inequalities due to the behaviour of its definition. There is a strong relationship between convexity and symmetry. Which ever one we work on, we can apply to the other one due to the strong correlation produced between them especially in recent few years. In this article, we first introduced the notion of $\lambda$ -incomplete gamma function. Using the new notation, we established a few inequalities of the Hermite–Hadamard (HH) type involved the tempered fractional integrals for the convex functions which cover the previously published result such as Riemann integrals, Riemann–Liouville fractional integrals. Finally, three example are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.

In this paper, we firstly obtain two identities for multiplicative differentiable functions. Then by using these identities, we establish Ostrowski and Simpson type inequalities for multiplicative integrals. At the end we give detail applications of our main results.

In this note, we obtain new some inequalities of Simpson’s type based on convexity. Some applications for special means of real numbers are also given.

In the present note, we proved Hermite-Hadamard-Mercer inequalities for fractional integrals and we established some new fractional inequalities connected with the right and left-sides of Hermite-Hadamard-Mercer type inequalities for differentiable mappings whose derivatives in absolute value are convex.

In this study, the family F and F-convex function are given with its properties. In view of this, we establish some new inequalities of Hermite–Hadamard type for differentiable function. Moreover, we establish some trapezoid type inequalities for functions whose second derivatives in absolute values are F-convex. We also show that through the notion of F-convex we can find some new Hermite–Hadamard type and trapezoid type inequalities for the Riemann–Liouville fractional integrals and classical integrals.