Extremal problems in subclasses of univalent functions.(Russian)Zbl 0679.30010

Let $$E=\{| z| <1\}$$ denote the unit disk. For $$\alpha\geq 0$$, $$| \beta | <\pi /2$$, and $$0\leq \gamma <1$$, let M($$\alpha$$,$$\beta$$,$$\gamma)$$ denote the class of functions $$F(z)=z+..$$. that are analytic and univalent on E and satisfy the conditions $$F(z)F'(z)/z\neq 0$$ and Re$$I(F,\alpha,\beta)> \gamma\cos\beta$$, where $I(F,\alpha,\beta)=(\delta -\alpha)(zF'(z)/F(z))+\alpha (1+(zF''(z)/F'(z)))$ and $$\delta =e^{i\beta}$$. The class $$\tilde M$$ is defined similarly, except that $$\delta =\cos \beta$$. Clearly, $$M(0,0,\gamma)$$ is the class of starlike functions of order $$\gamma$$, $$M(0,\beta,\gamma)$$ is the class of $$\beta$$-spirallike functions of order $$\gamma$$, and $$\tilde M(\alpha\cos\beta,\beta,\gamma)= M(\alpha,0,\gamma)$$. The classes M and $$\tilde M$$ can also be defined as the solution sets of certain differential equations.
The author examines several quantities related to the classes M and $$\tilde M.$$ For example, the order of $$\beta$$-starlikeness of functions in $$\tilde M$$ is determined, and bounds are found for the real and imaginary parts of the function $$F(z)/(\delta zF'(z))$$ on $$| z| =r$$.
Reviewer: Renate McLaughlin

MSC:

 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C70 Extremal problems for conformal and quasiconformal mappings, variational methods
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