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

For a constant k ∈ [0,∞) a normalized function f , analytic 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 + a2z + · · · and g(z) = z + b2z + · · · , the integral convolution is defined as follows: (f ⊗ g)(z) = z + ∞ X n=2 anbn n z. In this note a problem of stability of the integral convolution of kuniformly convex and k-starlike functions is investigated. 2000 Mathematics Subject Classification. 30C45, 30C50, 30C55.