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Convex subclass of harmonic starlike functions. (English) Zbl 1066.31001
A complex valued harmonic function $$f$$ defined in a simply connected domain $$\Omega$$ can be represented as $$f = h + \overline{g}$$, where $$h$$ and $$g$$ are holomorphic in $$\Omega$$. Such an $$f$$ is locally univalent and sense preserving in $$\Omega$$ if and only if $$|h'(z)| > |g'(z)|$$ in $$\Omega$$. Let $$S_H$$ the class of functions $$f = h + \overline{g}$$ that are harmonic and sense preserving in the unit disk $$\mathbb D$$ for which $$f(0) = f_z(0) - 1 = 0$$. For $$f = h + \overline{g} \in S_H$$ the functions $$f$$ and $$g$$ can be expressed as $h(z) = z + \sum_{n=2}^\infty a_nz^n, \quad g(z) = \sum_{n=1}^\infty b_nz^n, \quad |b_1| < 1. \tag{1}$ The class $$S_H$$ was introduced by J. Clunie and T. Sheil-Small [Ann. Acad. Sci. Fenn., Ser. A I Math. 9, 3–25 (1984; Zbl 0506.30007)] who also investigated its geometric subclasses and obtained some coefficient bounds. J. M. Jahangiri [J. Math. Anal. Appl. 235, 470–477 (1999; Zbl 0940.30003)] defined the class $${\mathcal F}_H(\alpha)$$ consisting of functions $$f = h + \overline{g}$$ such that $$h$$ and $$g$$ are of the form $h(z) = z - \sum_{n=2}^\infty a_nz^n, \quad g(z) = \sum_{n=1}^\infty |b_n|z^n, \tag{2}$ which satisfy the condition $\operatorname{Re} \left(\frac{z h'(z) - \overline{zg'(z)}}{h(z) + g(z)} \right) > \alpha, \quad 0 \leq \alpha < 1.$ Jahangiri proved that if $$f = h + \overline{g}$$ is given by (1) and if $\sum_{n=1}^{\infty} \left(\frac{n - \alpha}{1 - \alpha}|a_n| + \frac{n + \alpha}{1 - \alpha}|b_n| \right) \leq 2, \quad 0 \leq \alpha < 1, \quad a_1 = 1,$ then $$f$$ is harmonic, univalent, and starlike of order $$\alpha$$ in $$\mathbb D$$. This condition is also necessary if $$f \in {\mathcal F}_H(\alpha)$$. In the paper the authors consider the subclass $$S_H^{*}(\lambda, \alpha)$$ of $$S_H$$ of those functions $$f = h + \overline{g} \in S_H$$ that satisfy the condition $\operatorname{Re} \left(\frac{zh'(z) - \overline{zg'(z)}}{\lambda (zh'(z) - \overline{zg'(z)})+ (1 - \lambda)(h(z) + \overline{g(z)}} \right) > \alpha, \tag{3}$ for some $$\alpha$$, $$(0 \leq \alpha < 1)$$, $$\lambda$$, $$(0\leq \lambda < 1)$$, and $$z \in \mathbb D$$, and the the subclass $$TS_H^{*}(\lambda, \alpha)$$ of $$S_H^{*}(\lambda, \alpha)$$ consisting of functions $$f = h + \overline{g}$$ such that $$h$$ and $$g$$ satisfy (2). In the note it is shown that if $$f = h + \overline{g}$$ satisfy (1) and $\sum_{n=1}^{\infty}\left(\frac{n - \alpha - \alpha \lambda(n - 1)}{1 - \alpha}|a_n| + \frac{n + \alpha - \alpha \lambda (n + 1)}{1 - \alpha}|b_n| \right) \leq 2 \tag{4}$ where $$a_1 = 1$$, $$0 \leq \lambda < 1, 0 \leq \alpha < 1$$, then $$f$$ is harmonic univalent in $$\mathbb D$$, and, for $$\lambda \leq (1 - \alpha)/(1 + \alpha)$$, $$f \in S_H^{*}(\lambda, \alpha)$$. Moreover, if $$f = h + \overline{g}$$ satisfy (2), then (4) is a necessary and sufficient condition for $$f \in TS_H^{*}(\lambda, \alpha)$$ whenever the parameters satisfy the same restrictions as above. Other properties for the subclass $$TS_H^{*}(\lambda, \alpha)$$ are proved; for instance, $$TS_H^{*}(\lambda, \alpha)$$ is closed under convex combinations.

##### MSC:
 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions 30D30 Meromorphic functions of one complex variable, general theory
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##### References:
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