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

The notion of m -polynomial convex interval-valued function Ψ =[ψ ⁻, ψ ⁺] Ψ = [ ψ − , ψ + ] is hereby proposed. We point out a relationship that exists between Ψ and its component real-valued functions ψ ⁻ ψ − and ψ ⁺ ψ + . 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 ρ , ϵ > 0 and ζ,η ∈ S ζ , η ∈ S , then $${aligned} {m}{m+2⁻ᵐ-1}Ψ (ζ +η /2 )& ⊇ {Γ ρ(ε +ρ )}{(η -ζ )ε /ρ} [{ρ{J}}_{ζ ⁺}ε Ψ (η )+ρ{J}_{η ⁻}εΨ (ζ ) ] \\ & ⊇ Ψ (ζ )+Ψ (η )/m∑ₚ₌₁ᵐSₚ( ε;ρ ), {aligned}$$ 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 ) , where Ψ is Lebesgue integrable on $[ζ,η ]$ [ ζ , η ] , $Sₚ(ε;ρ )=2-ε /ε +ρ p- ε /ρ B (ε /ρ , p+1 )$ S p (