Subclasses of harmonic starlike functions (Q1611520)

Let us consider functions \(f\) harmonic in \(\Delta=\{z \in\mathbb{C}: |z|<1\}\) of the form \[ f=h+ \overline g,\;h(z)=z+\sum^\infty_{n=2} a_n z^n,\;g(z)= \sum^\infty_{n=1} b_nz^n,\;z\in\Delta,\;|b_1|<1. \tag{1} \] In this paper the authors consider the classes \(HS_p(\alpha)\) and \(HUCV (\alpha) \), \(0\leq \alpha<1\), of functions \(f\) of the form (1) respectively satisfying the conditions \[ \sum^\infty_{n=2} (2n-1-\alpha)/ (1-\alpha) \bigl[ |a_n |+|b_n|\bigr]\leq 1-|b_1|, \tag{2} \] \[ \sum^\infty_{n=2} n (2n-1-\alpha) /(1-\alpha) \bigl[|a_n|+|b_n|\bigr]\leq 1-|b_1|. \tag{3} \] In particular, they obtain: Theorem 1. For \(0\leq\alpha <1 \), we have \(HS_p(\alpha) \subset HS^*\) and \(HUCV(\alpha)\subset HC\) where \(HS^*\) denotes the class of harmonic univalent starlike functions and \(HC\)-harmonic convex functions. Theorem 2. Each function in the class \(HS_p(\alpha)\) maps the disk \(|z|=r <\frac 12\) onto a convex domain. The constant 1/2 is the best possible. Theorem 4. Let \(t_n=(1-\alpha)/ (2n-1-\alpha)\). For \(b_1\) fixed, the extreme points for \(HS_p(\alpha)\) are \(\{z+t_nxz^n+ \overline {b_1z}\} \cup\{z+ \overline{b_ +t_nxz^n}\}\) where \(n\geq 2\), and \(|x|=1-|b_1|\). One can see that the paper belongs to a series of many publications concerning conditions like (2)--(3), which guarantee appropriate properties of some classes of functions holomorphic or harmonic in the disk \(\Delta\).