Properties on a subclass of univalent functions defined by using a multiplier transformation and Ruscheweyh derivative

Abstract

Abstract In this paper we have introduced and studied the subclass ℛ𝒥 (d, α, β) of univalent functions defined by the linear operator RI n , λ , l γ f ( z ) $RI_{n,\lambda ,l}^\gamma f(z)$ defined by using the Ruscheweyh derivative Rnf(z) and multiplier transformation I (n, λ, l) f(z), as RI n , λ , l γ : 𝒜 → 𝒜 $RI_{n,\lambda ,l}^\gamma :{\cal A} \to {\cal A}$ , RI n , λ , l γ f ( z ) = ( 1 − γ ) R n f ( z ) + γ I ( n , λ , l ) f ( z ) $RI_{n,\lambda ,l}^\gamma f(z) = (1 - \gamma )R^n f(z) + \gamma I(n,\lambda ,l)f(z)$ , z ∈ U, where 𝒜 n ={f ∈ ℋ(U) : f(z) = z + an +1 zn +1 + . . . , z ∈ U}is the class of normalized analytic functions with 𝒜1 = 𝒜. The main object is to investigate several properties such as coefficient estimates, distortion theorems, closure theorems, neighborhoods and the radii of starlikeness, convexity and close-to-convexity of functions belonging to the class ℛ𝒥(d, α, β).