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An internal geometric characterization of strongly starlike functions. (English) Zbl 0766.30008
A function $$f(z)=z+a_ 2 z^ 2+\dots$$, $$z\in D=\{z$$: $$| z|<1\}$$ is said to be strongly starlike of order $$\alpha$$, $$0<\alpha\leq 1$$, if $|\text{Arg}\{zf'(z)/f(z)\}|<\alpha\pi/2, \qquad z\in D.$ This class, denoted by $$S^*(\alpha)$$ was introduced by D. A. Brannan and W. E. Kirwan [J. Lond. Math. Soc., II. Ser. 1, 431-443 (1969; Zbl 0177.334)] and independently by J. Stankiewicz [Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 18, 143-146 (1970; Zbl 0195.363)] who called these functions $$\alpha$$-angularly starlike. He also presented an external geometric characterization of $$\alpha$$-angularly starlike functions: A normalized holomorphic and univalent function $$f$$ belongs to $$S^*(\alpha)$$ if and only if every point $$w\in\mathbb{C}\setminus f(D)$$ is a vertex of an angular sector with opening measure $$(1- \alpha)\pi$$ and bisected by the radius vector through $$w$$ is contained in $$\mathbb{C}\setminus f(D)$$. In this paper the authors presented an internal geometric characterization of the class $$S^*(\alpha)$$. By $$E(\alpha)$$ we denote the certain standard lens-shaped region which is the intersection of the two closed discs of radii $$(2\cos \alpha/2)^{-1}$$ both of which have 0 and 1 on their boundaries. We put also $$wE(\alpha)=\{\zeta=w\eta$$: $$\eta\in E(\alpha)\}$$.
Theorem. A normalized holomorphic and univalent function $$f$$ belongs to $$S^*(\alpha)$$ if and only if for every point $$w\in f(D)$$ the lens- shaped region $$wE(\alpha)$$ is contained in $$f(D)$$. The sharp estimates for $$| a_ 3-\alpha a_ 2^ 2|$$ and $$|\text{Arg}\{f(z)/z\}|$$ for the class $$S^*(\alpha)$$ are also obtained.

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C50 Coefficient problems for univalent and multivalent functions of one complex variable 30C75 Extremal problems for conformal and quasiconformal mappings, other methods