A New Comprehensive Class of Analytic Functions Defined by Ruscheweyh Derivative and Multiplier Transformations.

Let A(p, n) denote the class of normalized analytic functions f(z) in the open unit disc f(z) = z^p + ... a_k z^k, p, n ∈ ℕ := {1, 2, 3,...}. We consider in this paper the operator RI_p^γ (m, λ, l) f(z) := (1 - γ) D^m f(z) + γI_p (m, λ, l) f(z) where I_p (m, λ, l) f(z) = z^p + ∑_{k = p + n}^∞ [p + λ(k - p) + l/p + l]^m a_k z^k and (m + 1) D^{m + 1} f(z) = z(D^m f (z))' + m D^m 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 AL_p^γ (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]