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In this paper we have introduced and studied the subclass RD(d, α, ß) of univalent functions defined by the linear operator RDnλδ f(z) defined by using the Ruscheweyh derivative Rnf(z) and the generalized Salagean operator Dn955f(z), as RDnλ,α : A → A, RDnλδf (z) = (1-y) Rn f(z) + yDnλf(z), z ∈ U, where An = {f ∈ H(U) : f (z) = z + an+1zn+1 + . . . , z ∈ U} is the class of normalized analytic functions with AI = 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 RD(d, α, β). [ABSTRACT FROM AUTHOR]

Let A(p, n) denote the class of normalized analytic functions f(z) in the open unit disc f(z) = zp + ... ak zk, p, n ∈ ℕ := {1, 2, 3,...}. We consider in this paper the operator RIpγ (m, λ, l) f(z) := (1 - γ) Dm f(z) + γIp (m, λ, l) f(z) where Ip (m, λ, l) f(z) = zp + ∑k = p + n∞ [p + λ(k - p) + l/p + l]m ak zk and (m + 1) Dm + 1 f(z) = z(Dm f (z))' + m Dm f(z), m ∈ ℕ0, ℕ0 = ℕ ∪ {0}, λ ∈ ℝ, λ ≥ 0, l ≥ 0 is the Ruscheweyh operator. By making use of the above mentioned differential operator, a new subclass of p--valent functions in the open unit disc is introduced. The new subclass is denoted by ALpγ (m, n, µ, α, λ l). Parallel results, for some related classes including the class of starlike and convex functions respectively, are also obtained. [ABSTRACT FROM AUTHOR]

In the present paper we establish several strong differential subordinations regarding the new operator RIαm,λt, given by RIαm,λt, : A*nζ → A*nζ, RIαm,λtf(z, ζf) = (1 - α)Rmf(z, ζ), where Rmf(z, ζ) denote the Ruscheweyh derivative, I (m, λ, l) is the multiplier transformation and A*nζ = {f ∈ H(U x U̅), f(z, ζ) = z + an+1 (ζ) zn+1 + …, z ∈ U, ζ ∈ U̅} is the class normalized analytic functions. A certain subclass, denoted by RIm (δ, λ, l, α, ζ), of analytic functions is introduced by means of the new operator. [ABSTRACT FROM AUTHOR]

In the present paper we establish several strong differential subordinations regarding the new operator RDnλα, given by RD RDnλα : A*ς → A*ς, RDnλα f (z, ς) = (1 - α) Rn f(z, ς) + α Dn&lambda f(z, ς), where Rn f(z, ς) denote the Ruscheweyh derivative, Dnλ f(z, ς) is the generalized Sălăgean operator and A*nς = {f ∈ H(U x U̅), f(z, ς) = z + an+1 (ς) zn+1 + …, z ∈ U, ς U̅} is the class of normalized analytic functions with A*1ς = A*ς. A certain subclass, denoted by RDn (δ, λ, α), of analytic functions is introduced by means of the new operator. [ABSTRACT FROM AUTHOR]