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Harmonic univalent functions with varying arguments. (English) Zbl 1026.30016
Let \({\mathcal H}^*\) denote the known class of harmonic sense-preserving univalent functions \(f\) of the form \[ f=h+\overline g,\;h(z)=z+ \sum^{+ \infty}_{n=2}a_nz^n,\;g(z)=b_1z+\sum^{+\infty}_{n=2}b_nz^n,\;0\leq b_1 <1,\tag{1} \] \(z\in\Delta: =\{z\in\mathbb{C}: |z|<1\}\) and such that \({ \partial \over\partial \theta}\arg(f(re^{i \theta}))\geq 0\), \(z=re^{i\theta} \in \Delta\). The authors consider the class \({\mathcal V}_{\mathcal H}\) of functions \(f\) of the form (1) such that there exists \(\varphi\) so that, mod \(2\pi\), \[ \alpha_n = (n-1)\varphi\equiv \pi,\quad \beta_n+(n-1)\varphi \equiv 0,\quad n\geq 2,\tag{2} \] where \(\alpha_n=\arg a_n\), \(\beta_n=\arg b_n\) and the class \({\mathcal V}_{{\mathcal H}^*}={\mathcal H}^* \cap {\mathcal V}_{\mathcal H}\). In this paper they obtained necessary and sufficient coefficient conditions, distortion theorems and extreme points for the class \({\mathcal V}_{{\mathcal H}^*}\). In the last section the authors extended these results to the functions with varying arguments that are starlike or convex of positive order. For example in Theorem 2.1 we have: If \(f=h+\overline g\in {\mathcal V}_{\mathcal H}\), then \(f\in{\mathcal V}_{{\mathcal H}^*}\) if and only if \(\sum^{+ \infty}_{n=2} n(|a_n|+ |b_n|)\leq 1-b_1\).

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions