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

Abstract

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 $\bigcap_{f\in X^{(n)}} f(\Delta)$, as well as the covering domain, i.e. the set $\bigcup_{f\in X^{(n)}} f(\Delta)$. The results depend on the parity of $n\in N$. 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 $f\in X^{(n)}$ there is $m\prec f$, and $\bigwedge_{f\in X^{(n)}}\ [k\prec f \Rightarrow k\prec m]$, 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 $f\in X^{(n)}$ there is $f\prec M$, and $\bigwedge_{f\in X^{(n)}}\ [f\prec k \Rightarrow M\prec k]$, 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.