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The notion of m-polynomial convex interval-valued function $\Psi =[\psi ^{-}, \psi ^{+}]$ is hereby proposed. We point out a relationship that exists between Ψ and its component real-valued functions $\psi ^{-}$ and $\psi ^{+}$ . 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, $\rho,\epsilon >0$ and $\zeta,\eta \in {\mathbf{S}}$ , then $$\begin{aligned} \frac{m}{m+2^{-m}-1}\Psi \biggl(\frac{\zeta +\eta }{2} \biggr)& \supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr] \\ & \supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{m}\sum_{p=1}^{m}S_{p}( \epsilon;\rho ), \end{aligned}$$ where Ψ is Lebesgue integrable on $[\zeta,\eta ]$ , $S_{p}(\epsilon;\rho )=2-\frac{\epsilon }{\epsilon +\rho p}- \frac{\epsilon }{\rho }\mathcal{B} (\frac{\epsilon }{\rho }, p+1 )$ and $\mathcal{B}$ is the beta function. We extend, generalize, and complement existing results in the literature. By taking $m\geq 2$ , we derive loads of new and interesting inclusions. We hope that the idea and results obtained herein will be a catalyst towards further investigation.

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