Abstract
In this paper, certain Hermite–Hadamard–Mercer-type inequalities are proved via Riemann–-Liouville fractional integral operators. We established several new variants of Hermite–Hadamard’s inequalities for Riemann–Liouville fractional integral operators by utilizing Jensen–Mercer inequality for differentiable mappingϒwhose derivatives in the absolute values are convex. Moreover, we construct new lemmas for differentiable functionsϒ′,ϒ″, andϒ‴and formulate related inequalities for these differentiable functions using variants of Hölder’s inequality.
Highlights
Jensen inequality in the literature states that if Υ is a convex function on the interval [θ1, θ2], n n
Notable contributions have been made on Jensen–Mercer’s type inequality
Anjidani and Changalvaiy worked on reverse Jensen–Mercer-type operator inequalities and Jensen–Mercer operator inequalities for superquadratic functions
Summary
By using Jensen–Mercer inequalities, Hermite–Hadamardtype inequalities can be expressed in Riemann–Liouville fractional integral form as follows: Theorem 2. By change of variables u ((1 + λ)/2)z1 + ((1 − λ)/2)z2 and v ((1 − λ)/2)z1 + ((1 + λ)/2)z2, λ ∈ [0, 1], we obtain Both sides by λα− 1 and integrate the resulting inequality with respect to λ over [0, 1]. For the proof of second inequality of (5), we first note that if Υ is convex function, for λ ∈ [0, 1], it gives λz2. ≤ 2 Υ θ1 + Υ θ2 − Υ z1 + Υ z2 Multiplying both sides by λα− 1 and integrating the resulting inequality with respect to λ over [0, 1], we have. For the proof of second inequality of (6), we first note that if Υ is convex function, for λ ∈ [0, 1], Υz1.
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