Abstract
In this paper, we introduce and study some properties for strong differential subordinations of analytic functions associated with Ruscheweyh derivative operator defined in the open unit disk and closed unit disk of the complex plane.
Highlights
Let U = {z ∈ C : z < 1} and U = {z ∈ C : z ≤ 1} denote the open unit disk and the closed unit disk of the complex plane, respectively
Let q(z, ζ) be a convex function in U × U for all ζ ∈ U and let h(z, ζ) = q(z, ζ) + nδzq′z (z, ζ), z ∈ U, ζ ∈ U, where δ > 0 and n is a positive integer
−1 zθ z 1 −1 t θ h(t, ζ)dt pp h(z, ζ)
Summary
Let H(U × U ) the class of all analytic functions in U × U. For n ∈ N = {1, 2, ...} and a ∈ C, let H∗[a, n, ζ] = { f ∈ H(U × U ) : f (z, ζ) = a + an(ζ) zn + an+1(ζ) zn+1 + L, z ∈ U , ζ ∈ U }, where ak (ζ) are holomorphic functions in U for k ≥ n. Let A∗n ζ = { f ∈ H(U × U ): f (z, ζ) = z + an+1(ζ) zn+1 + L, z ∈ U , ζ ∈ U }, where ak (ζ) are holomorphic functions in U for k ≥ n + 1. Denote the classes of starlike and convex functions in U × U by Sζ∗ and Kζ∗, respectively. K =2 which are analytic and univalent in U × U
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