A class of Ruscheweyh-type harmonic univalent functions with varying arguments. (English) Zbl 1050.30010

Murugusundaramoorthy, G.

Southwest J. Pure Appl. Math. 2003, No. 2, 90-95 (2003).

Denote by \(H\) the family of functions \(f= h+\overline g\) that are harmonic univalent and orientation preserving in the disc \(U= \{z:| z|< 1\}\) where \(h\) and \(g\) are of the form \[ h(z)= z+ \sum^\infty_{m=2} a_mz^m,\quad g(z)= b_1z+ \sum^\infty_{m=2} b_mz^m,\quad 0\leq b_1< 1.\tag{1} \] For \(f= h+\overline g\) given by (1) and \(n>-1\), we define the Rusheweyh derivative by \(D^nf(z)= D^n h(z)+ \overline{D^ng(z)}\), where \(D^n\) is the Ruschewey derivative of the holomorphic functions. Let \(\mathbb{R}_H(n,\alpha)\), \(n> -1\), \(0\leq\alpha< 1\) denote the class of harmonic functions \(f= h+\overline g\) of the form (1) such that \({\partial\over\partial\theta}(\text{arg\,}D^n f(z))\geq \alpha\), \(| z|= r< 1\) and let \(V_{\overline H}(n,\alpha):= \mathbb{R}_H(n,\alpha)\cap V_H\) where \(V_H\) [J. M. Jahangiri and H. Silverman, Int. J. Appl. Math. 8, No. 3, 267–275 (2002; Zbl 1026.30016)], is the class of functions \(f= h+\overline g\) of the form (1) and such that, \(\text{mod\,}2\pi\), \(\beta_m+ (m-1)\phi\equiv \pi\), \(\delta_m+ (m-1)\phi\equiv 0\), \(m\geq 2\), where \(\beta_m= \text{arg\,}a_m\) and \(\delta_m= \text{arg\,}b_m\).
In this paper the author gives the sufficient condition (of the Silverman’s kind) for \(f= h+\overline g\) given by (1) to be in the class \(\mathbb{R}_H(n,\alpha)\), and it is shown that this condition is also necessary for functions in the class \(V_{\overline H}(n,\alpha)\). The author obtains distortion theorems and characterizes the extreme points for \(V_{\overline H}(n,\alpha)\).

Reviewer: Zbigniew J. Jakubowski (Łódź)

MSC:

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

Keywords:

complex-valued harmonic univalent functions; Rusheweyh derivative; Hadamard product; starlike functions; coefficient bounds; distoration theorems