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

In this paper we have introduced and studied the subclass ℛ𝒥 (d, α, β) of univalent functions defined by the linear operator RIn,λ,lγf(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 RIn,λ,lγ:𝒜→𝒜$RI_{n,\lambda ,l}^\gamma :{\cal A} \to {\cal A}$, RIn,λ,lγf(z)=(1−γ)Rnf(z)+γI(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 + a_n+1zⁿ⁺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).