On Linear Combinations of n Analytic Functions

Let p ≥ 1 be an integer and let 0 ≤ α < p. An analytic function P(z) belongs to the class P(A,B,p,α) if P(z)=(p—α) P₁ (z) + α where P₁ (z) belongs to the class P (A,B) (-1 ≤ B < A ≤ 1) the well known Janowski class of analytic iunciions on |z| < l. Let f(z) = z^p + Σa_kz^k be analytic on | z | < 1 such that zf'(z)/f(z) = P(z) for some P(z) in the class P(A,B,p,α). We then say that f(z) belongs to the class S(A,B,p,α). In this note we consider linear combinations of analytic functions of the form Σ γ_j f_j(z) where f_j(z) {j=1,2.....,n) are in the class S(A,B,p,α) and γ_j n are complex numbers with Σ γ_j= 1 .We determine the radius of l n starlikeness and the radius of convexity of F(z) = Σ γ_jf_j(z).