Abstract
Simpson inequalities for differentiable convex functions and their fractional versions have been studied extensively. Simpson type inequalities for twice differentiable functions are also investigated. More precisely, Budak et al. established the first result on fractional Simpson inequality for twice differentiable functions. In the present article, we prove a new identity for twice differentiable functions. In addition to this, we establish several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. This paper is a new version of fractional Simpson type inequalities for twice differentiable functions.
Highlights
Simpson’s inequality plays a considerable role in several branches of mathematics
Since the convex theory is an effective way to solve a large number of problems from different branches of mathematics, many authors have studied the results of Simpson type for convex mapping
The remaining part of the paper proceeds as follows: In Sect. 2, after giving a general literature survey and the definition of Riemann–Liouville fractional integral operators, we prove an equality for twice differentiable functions
Summary
For four times continuously differentiable functions, the classical Simpson’s inequality is expressed as follows. Theorem 1 Suppose that F : [ρ1, ρ2] → R is a four times continuously differentiable mapping on (ρ1, ρ2), and let F (4) ∞ = sup |F (4)(κ)| < ∞. Some inequalities of Simpson type for s-convex functions are proved by using differentiable functions [4]. In the papers [34, 36], the new variants of Simpson type inequalities are established based on differentiable convex mapping.
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