Abstract
Trapezoidal inequalities for functions of divers natures are useful in numerical computations. The authors have proved an identity for a generalized integral operator via twice differentiable preinvex function. By applying the established identity, the generalized trapezoidal type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in recent past decades. Various special cases have been identified. Some applications of presented results to special means have been analyzed. The ideas and techniques of this paper may stimulate further research.
Highlights
The following inequality, named Hermite– Hadamard inequality, is one of the most famous inequalities in the literature for convex functions.Theorem 1
It is important to summarize the study of fractional integrals
Let S ⊆ Rn be an invex set with respect to η : S × S −→ Rn
Summary
The following inequality, named Hermite– Hadamard inequality, is one of the most famous inequalities in the literature for convex functions. The aim of this paper is to establish trapezoidal type generalized integral inequalities for preinvex. Let S ⊆ Rn be an invex set with respect to η : S × S −→ Rn. A function f : S −→ [0, +∞) is said to be preinvex with respect to η, if for every x, y ∈ S and t ∈ [0, 1], f x + tη(y, x) ≤ (1 − t)f (x) + tf (y). Let define a function φ : [0, +∞) −→ [0, +∞) satisfying the following conditions:. Motivated by the above literatures, the main objective of this paper is to discover, an interesting identity in order to study some new bounds regarding general trapezoidal type integral inequalities. By using the established identity as an auxiliary result, some new estimates for trapezoidal type integral inequalities via generalized integrals are obtained. The ideas and techniques of this paper may stimulate further research in the field of integral inequalities
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