On functions convex in the direction of the real axis with a fixed second coefficient

In the paper we consider the class Gamma of analytic and univalent functions f in the unit disk Delta, normalized by f(0) = f(0) - 1 = 0, having real coefficients and such that f(Delta) is convex in the direction of the real axis. We are especially interested in some subclasses of Gamma. The most important of them is Gamma(c) consisting of those functions which have the second coefficients of the Taylor expansion fixed and equal to c. We obtain the Koebe set for this class as well as for the classes Gamma(+)(c) and Gamma (c) of functions which are in some sense convex in the direction of positive and negative axes respectively. (C) 2011 Elsevier Inc. All rights reserved.