A class of superordination-preserving convex integral operator

If H(U) denotes the space of analytic functions in the unit disk U, for the integral operator Aα,β,γ,δ : K → H(U), with K ⊂ H(U), defined by Aα,β,γ,δ[f ](z) = [ β + γ zγ ∫ z 0 f(t)h(t)t d t ]1/β , ( α, β, γ, δ ∈ C and h ∈ H(U) ) , we will determine sufficient conditions on g1, g2, α, β and γ such that zh(z) [ g1(z) z ]α ≺ zh(z) [ f(z) z ]α ≺ zh(z) [ g2(z) z ]α