In this article, some new generalized fractional integral inequalities of midpoint and trapezoid type for twice differentiable convex functions are obtained. In view of this, we obtain new integral inequalities of midpoint and trapezoid type for twice differentiable convex functions in a form classical integral and Riemann-Liouville fractional integrals. Finally, we apply our new inequalities to construct inequalities involving moments of a continuous random variable.
We consider the Hermite-Hadamard inequality and related results on integral inequalities, in the context of fractional integrals and derivatives defined using Mittag-Leffler kernels, specifically the Atangana-Baleanu and Prabhakar models of fractional calculus.
Integral inequalities play a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods. Thus, the present days need to seek accurate inequalities for proving the existence and uniqueness of the mathematical methods. The concept of convexity plays a strong role in the field of inequalities due to the behavior of its definition. There is a strong relationship between convexity and symmetry. Whichever one we work on, we can apply it to the other one due the strong correlation produced between them, especially in the past few years. In this article, we firstly point out the known Hermite–Hadamard (HH) type inequalities which are related to our main findings. In view of these, we obtain a new inequality of Hermite–Hadamard type for Riemann–Liouville fractional integrals. In addition, we establish a few inequalities of Hermite–Hadamard type for the Riemann integrals and Riemann–Liouville fractional integrals. Finally, three examples are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.
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 λ -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 article, we have established new Hermite–Hadamard's type inequalities for Riemann–Liouville fractional integrals of convex functions with respect to increasing functions. Our obtained inequalities generalize some recent obtained inequalities in the literature involving classical integrals and Riemann–Liouville fractional integrals. Finally, applications of our work are demonstrated via the known special functions of real numbers.
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