Abstract
The main goal of this research is to introduce a new form of generalized Hermite–Hadamard and Simpson type inequalities utilizing Riemann–Liouville fractional integral by a new class of preinvex functions which is known as strongly generalized ( phi,h,s )-preinvex functions in the second sense. It is observed that the derived inequalities are generalizations of the inequalities obtained by W. Liu, W. Wen (Filomat 30(2):333–342, 2016).
Highlights
Convexity plays a focal and major part in mathematical finance, economics, engineering, management sciences, and optimization theory
A huge speculation of convex functions is that of invex functions presented in [2]
It is realized that the preinvex functions and invex sets may not be convex functions and convex sets, respectively
Summary
Convexity plays a focal and major part in mathematical finance, economics, engineering, management sciences, and optimization theory. We present new generalized inequalities using the Riemann–Liouville fractional integral by the class of strongly generalized (φ, h, s)-preinvex functions in the second sense. Definition 3 ([5]) The function f on the invex set Kφξ is said to be sφ-preinvex with respect to ξ and φ if f x + zeiφξ (y, x) ≤ (1 – z)sf (x) + zsf (y), ∀x, y ∈ Kφξ , z ∈ [0, 1], s ∈ Definition 4 The function f on the invex set K is said to be strongly generalized (φ, h, s)preinvex in the second sense with modulus c > 0 if it is nonnegative, and for all u, v ∈ K and z × s ∈ (0, 1) × Lemma 1 Let Kφξ ⊆ R be a φ-invex subset with respect to φ(·) and ξ : Kφξ × Kφξ ⊆ R with u < u + eiφξ (v, u) and 0≤φ π 2
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