Fractional inclusions of the Hermite–Hadamard type for m-polynomial convex interval-valued functions

Abstract

Abstract The notion of m-polynomial convex interval-valued function Ψ = [ ψ − , ψ + ] $\Psi =[\psi ^{-}, \psi ^{+}]$ is hereby proposed. We point out a relationship that exists between Ψ and its component real-valued functions ψ − $\psi ^{-}$ and ψ + $\psi ^{+}$ . For this class of functions, we establish loads of new set inclusions of the Hermite–Hadamard type involving the ρ-Riemann–Liouville fractional integral operators. In particular, we prove, among other things, that if a set-valued function Ψ defined on a convex set S is m-polynomial convex, ρ , ϵ > 0 $\rho,\epsilon >0$ and ζ , η ∈ S $\zeta,\eta \in {\mathbf{S}}$ , then m m + 2 − m − 1 Ψ ( ζ + η 2 ) ⊇ Γ ρ ( ϵ + ρ ) ( η − ζ ) ϵ ρ [ ρ J ζ + ϵ Ψ ( η ) + ρ J η − ϵ Ψ ( ζ ) ] ⊇ Ψ ( ζ ) + Ψ ( η ) m ∑ p = 1 m S p ( ϵ ; ρ ) , $$\begin{aligned} \frac{m}{m+2^{-m}-1}\Psi \biggl(\frac{\zeta +\eta }{2} \biggr)& \supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr] \\ & \supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{m}\sum_{p=1}^{m}S_{p}( \epsilon;\rho ), \end{aligned}$$ where Ψ is Lebesgue integrable on [ ζ , η ] $[\zeta,\eta ]$ , S p ( ϵ ; ρ ) = 2 − ϵ ϵ + ρ p − ϵ ρ B ( ϵ ρ , p + 1 ) $S_{p}(\epsilon;\rho )=2-\frac{\epsilon }{\epsilon +\rho p}- \frac{\epsilon }{\rho }\mathcal{B} (\frac{\epsilon }{\rho }, p+1 )$ and B $\mathcal{B}$ is the beta function. We extend, generalize, and complement existing results in the literature. By taking m ≥ 2 $m\geq 2$ , we derive loads of new and interesting inclusions. We hope that the idea and results obtained herein will be a catalyst towards further investigation.

Keywords

• Hermite–Hadamard
• m-polynomial convex
• Interval-valued function
• ρ-Riemann–Liouville