×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Silverman, H, Harmonic univalent function with negative coefficients, J. math. anal. appl., 220, 283-289, (1998) · Zbl 0908.30013
[2] Silverman, H; Silvia, E.M, Subclasses of harmonic univalent functions, New zeal. J. math., 28, 275-284, (1999) · Zbl 0959.30003
[3] Clunie, J; Sheil-Small, T, Harmonic univalent functions, Ann. acad. sci. fenn. ser. A I math., 9, 3-25, (1984) · Zbl 0506.30007
[4] Jahangiri, J.M, Harmonic functions starlike in the unit disk, J. math. anal. appl., 235, 470-477, (1999) · Zbl 0940.30003
[5] Avcı, Y; Zlotkiewicz, E, On harmonic univalent mappings, Ann. univ. mariae cruie sklod. sec. A, 44, 1-7, (1990) · Zbl 0780.30013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.