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.


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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