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

The notion of m-polynomial convex interval-valued function Psi=[psi-,psi+] is hereby proposed. We point out a relationship that exists between Psi 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 rho -Riemann-Liouville fractional integral operators. In particular, we prove, among other things, that if a set-valued function Psi defined on a convex set S is m-polynomial convex, rho ,E>0 and zeta,eta is an element of S, then mm+2-m-1Psi(zeta+eta 2)superset of Gamma rho (E+rho)(eta-zeta)E rho[rho J zeta +EPsi(eta)+rho J eta -EPsi(zeta)]superset of Psi(zeta)+Psi(eta )mSigma p=1mSp(E;rho), where Psi is Lebesgue integrable on [zeta,eta], Sp(E;rho)=2-EE+rho p-E rhoB(E rho,p+1) and B is the beta function. We extend, generalize, and complement existing results in the literature. By taking m >= 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.