Covering Problems for Functions $$n$$ n -Fold Symmetric and Convex in the Direction of the Real Axis II

Abstract

Let F denote the class of all functions univalent in the unit disk ≡ {ζ ∈ C : |ζ | < 1} and convex in the direction of the real axis. The paper deals with the subclass F (n) of these functions f which satisfy the property f (εz) = ε f (z) for all z ∈ , where ε = e 2πi/n . The functions of this subclass are called n-fold symmetric. For F (n) , where n is odd positive integer, the following sets, f ∈F (n) f ( )-the Koebe set and f ∈F (n) f ( )-the covering set, are discussed. As corollaries, we derive the Koebe and the covering constants for F (n) .

Koebe sets for the class of functions convex in two directions

“…If α ∈ [α 1 , α 2 ] then the curves given by (12) and ( 13) have one common point in the closed first quarter of the complex plane.…”

“…For α = 0 and α = 1 the set K α reduces to two well-known families: K 0 = K(1) consisting of functions convex in the direction of the real axis and K 1 = K(i) consisting of functions convex in the direction of the imaginary axis. These classes were discussed, among others, by Hengartner and Schober, Goodman and Saff, Ciozda, Brown, and Prokhorov (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). The set K α , α ∈ (0, 1) has not been discussed yet.…”

Some extremal problems for trinomials with fold symmetry

“…This radius is called the Koebe radius. In [9][10][11]13], some examples of finding the Koebe radius for different classes of functions are given.…”