Estimates of Some Coefficient Functionals for Close-to-Convex Functions

For a given starlike function F alpha=z1-alpha z2, alpha is an element of[-1,1], the class C0(F alpha) is defined as follows: an analytic normalized function f belongs to C0(F alpha) if it satisfies Rezf '(z)F alpha(z)>0 in the open unit disk triangle. The condition defining this class can be rewritten in the following equivalent form Re{(1-alpha z2)f '(z)}>0, z is an element of triangle. The family C0(F alpha) is a subclass of the class of close-to-convex functions. The main aim of this paper is to maximize the modulus of a functional which is a linear combination with coefficients symmetric with respect to zero and is defined on the subfamily of C0(F alpha) of functions with a fixed second coefficient in its Taylor series expansion.