STABILITY OF THE INTEGRAL CONVOLUTION OF k-UNIFORMLY CONVEX AND k-STARLIKE FUNCTIONS

Abstract. For a constant k ∈ [0,∞) a normalized function f, an-alytic in the unit disk, is said to be k-uniformly convex if Re (1 + zf ′′(z)/f ′(z))> k|zf ′′(z)/f ′(z) | at any point in the unit disk. The class of k-uniformly convex functions is denoted k-UCV (cf. [4]). The function g is said to be k-starlike if g(z) = zf ′(z) and f ∈ k-UCV. For analytic functions f, g, where f(z) = z + a2z2 + · · · and g(z) = z + b2z2 + · · · , the integral convolution is defined as follows: (f ⊗ g)(z) = z + n=2 anbn n zn. In this note a problem of stability of the integral convolution of k-uniformly convex and k-starlike functions is investigated. 2000 Mathematics Subject Classification. 30C45, 30C50, 30C55. Key words and phrases. k-uniformly convex functions, k-starlike functions, Hadamar