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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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