# Circularly Symmetric Locally Univalent Functions

Bulletin of the Malaysian Mathematical Sciences Society, Feb 2016

Let $$D\subset \mathbb {C}$$ and $$0\in D$$. A set D is circularly symmetric if, for each $$\varrho \in \mathbb {R}^+$$, a set $$D\cap \{\zeta \in \mathbb {C}:|\zeta |=\varrho \}$$ is one of three forms: an empty set, a whole circle, a curve symmetric with respect to the real axis containing $$\varrho$$. A function f analytic in the unit disk $$\Delta \equiv \{\zeta \in \mathbb {C}:|\zeta |<1\}$$ and satisfying the normalization condition $$f(0)=f^{\prime }(0)-1=0$$ is circularly symmetric, if $$f(\Delta )$$ is a circularly symmetric set. The class of all such functions is denoted by X. In this paper, we focus on the subclass $$X^{\prime }$$ consisting of functions in X which are locally univalent. We obtain the results concerned with omitted values of $$f\in X^{\prime }$$ and some covering and distortion theorems. For functions in $$X^{\prime }$$ we also find the upper estimate of the n-th coefficient, as well as the region of variability of the second and the third coefficients. Furthermore, we derive the radii of starlikeness, convexity and univalence for $$X^{\prime }$$.

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Leopold Koczan, Paweł Zaprawa. Circularly Symmetric Locally Univalent Functions, Bulletin of the Malaysian Mathematical Sciences Society, 2016, 1615-1635, DOI: 10.1007/s40840-016-0329-z