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A class of Ruscheweyh-type harmonic univalent functions with varying arguments. (English) Zbl 1050.30010
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 $$V_{\overline H}(n,\alpha)$$.

##### 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
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