On functions convex in the direction of the real axis with real coefficients

The paper is concerned with the class X((n))) consisting of all functions, which are n-fold symmetric, convex in the direction of the real axis and have real coefficients. For this class we determine the Koebe domain, i.e. the set boolean AND(f is an element of X(n)) f(Delta), as well as the covering domain, i.e. the set boolean OR(f is an element of X(n)) f(A). The results depend on the parity of n is an element of N. We also obtain the minorant and the majorant for this class. These functions are defined as follows. If there exists an analytic, univalent function m satisfying the following conditions: m'(0) > 0, for every f is an element of x((n)) there is m (sic) f, and Lambda(f is an element of X(n)) [k (sic) f double right arrow k (sic) m], then this function is called the minorant of X((n)). Similarly, if there exists an analytic, univalent function M such that M'(0) > 0, for every f is an element of X((n)) there is f (sic) M, and Lambda(f is an element of X(n)) [f (sic) k double right arrow M (sic) k], then this function is called the majorant of X((n)). If these functions exist, then m (Delta) and M(Delta) coincide with the Koebe domain and the covering domain for X((n)), respectively.