Some properties of subordination differential and superordination for univalent functions associated with the convolution operators

Abstract

In this paper, we define the Hadamard product (or convolution) (f * y([h])) (x)  of the analytic function in the unit disk µ = [ξ : |ξ| < 1, ξ ∈ Ȼ] with the non-zero parameter h, satisfactory to the relationship hη (f * ψ([η]))(ξ) = (hη - 1) (f * ψ([η + 1])) (ξ) + ξ(f * ψ([η + 1]))′ (ξ), to derive  subordination, superordination and sandwich results for functions of the form (f * ψ([η]))(ξ)  by using some properties of Subordination and Superordination concepts. Our results serve to generalize and improve some previous studies where the results of this research were applied to linear operator Bbd  band the multiplier operator ℑiℓ,m. This work can be generalized to other linear operators.