## Region of variability of two subclasses of univalent functions.(English)Zbl 1114.30014

Let $$F_1$$ denote the class of functions $$f(z)=z+\dots$$, $$| z| <1$$, satisfying the condition $$\text{Re}(1+zf"(z)/f'(z))<3/2$$. Similarly, the class $$F_2$$ is defined by the condition $$\text{Re}(1+zf"(z)/f'(z))>-1/2$$. For $$| \lambda| \leq1$$ and $$| z_0| <1$$, define $C_1(\lambda)=\{f\in F_1: f"(0)=-\lambda\},\;\;\;C_2(\lambda)=\{f\in F_2: f"(0)=3\lambda\},$
$V_j(z_0,\lambda)=\{\log f'(z_0): f\in C_j(\lambda)\},\;\;j=1,2.$ The authors determine the regions $$C_1(\lambda)$$, $$C_2(\lambda)$$. Namely, for $$0\leq\lambda\leq1$$ and $$0<| z_0| <1$$, the boundary $$\partial V_1(z_0,\lambda)$$ is the Jordan curve $$\gamma(\theta)$$, $$\theta\in[-\pi,\pi]$$, given by $\gamma(\theta)=\int_0^{z_0}\frac{\delta(e^{i\theta}z,\lambda)}{z\delta(e^{i\theta}z,\lambda)-1}dz$ where
$\delta(z,\lambda)=\frac{z+\lambda}{1+\overline{\lambda}z}.$ The boundary $$\partial V_2(z_0,\lambda)$$ is described in the same way with obvious arithmetical changes in the rational kernel of the integral.

### MSC:

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

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