Constants and characterization for certain classes of univalent harmonic mappings
Let \(\mathbb{D}=\{z\in\mathbb{C}: | z| <1\}\) be the open unit disk in the complex plane, and let \(\mathcal{H}\) be the class of complex valued harmonic functions \(f=h+\overline {g}\) in \(\mathbb D\), normalized by \(h(0)=g(0)=h^\prime (0)-1=0\) where \(h\) and \(g\) are analytic functions in the unit disk. A univalent function \(G\) is said to belong to \(\mathcal{K}(\alpha)\), the class of convex functions of order \(\alpha\), \(-1/2 \leq \alpha <1\), provided that \[ \mathrm{Re} \left (1+\frac{zG^{\prime\prime(z)}}{G^\prime(z)}\right)>\alpha,\qquad z\in\mathbb D. \] Similarly, one can define the class \(\mathcal{F}_{\mathcal{K}(\alpha)}\) as the class of functions \(f=h+\overline {g}\in \mathcal{H}\) such that for each \(z\in\mathbb D\) and some \(G\in\mathcal{K}(\alpha)\), \[ \left |\frac{h^\prime (z)}{G^\prime(z)}-1\right |<1-\left |\frac{h^\prime (z)}{G^\prime(z)}\right |. \] Finally, we define \[ \mathcal{F}^0_{\mathcal{K}(\alpha)}=\{f=h+\overline {g}\in\mathcal{F}_{\mathcal{K}(\alpha)}:g^\prime (0)=0\}. \] It is known that the functions in \(\mathcal{F}_{\mathcal{K}(\alpha)}\) are close-to-convex in \(\mathbb D\). In the paper under review, the authors discuss the \textsl{growth}, \textsl{covering}, \textsl{integral representation}, and \textsl{area} theorems for functions in \(\mathcal{F}^0_{\mathcal{K}(\alpha)}\). For instance, in Theorem 2.5, it is proved that for \(0\leq \alpha <1\), and \(f\in \mathcal{F}^0_{\mathcal{K}(\alpha)}\), the area of the image of \(\mathbb{D}_r=\{z\in \mathbb{C}: | z| <r\}\), for \(0<r<1\), satisfies the following estimates: \[ \int_0^{2\pi}\int_0^r\frac{\rho(1-\rho)^2}{(1+\rho)^{2(1-\alpha)}}d\rho\,d\theta \leq A(f(\mathbb{D}_r))\leq\int_0^{2\pi}\int_0^r\frac{\rho(1+\rho)^2}{(1-\rho)^{2(1-\alpha)}}d\rho\, d\theta. \] As another noticeable result established, among other things, by the authors, there is a sufficient condition which guarantees that the harmonic function \(f=h+\overline {g}\) is close-to-convex in \(\mathbb D\) provided that \(h\in\mathcal{K}(-\frac{1}{2})\) satisfies some more restrictions; see Theorem 4.1.
scientific article; zbMATH DE number 6478167