ON UNIVALENCE, STARLIKENESS AND CONVEXITY OF CERTAIN ANALYTIC-FUNCTIONS

Let t = (t0, t1,...,t(k) is-an-element-of (-infinity infinity)k+1. By R(t) we denote the class of functions f(z) = z + a2z2 + ... that are analytic in the unit disk U = {z : \z\ < 1} and satisfy the condition [GRAPHICS] It is known that (i) if f, g is-an-element-of R = R(0, 1, 1) then f * g is-an-element-of R, Re [f(z)/z] > 1/2, Ref(z) > -1 + 2 log 2, and f * g is convex, (ii) if f is-an-element-of R (0, 4/3, 4/3) then f is startlike. In the present paper we solve problems similar to the above in a large number of classes R(t). Some known inequalities are improved and some known results are proved under weaker conditions.