Coefficient estimates for the inverses of starlike functions represented by symmetric gap series. (English) Zbl 1246.30032

Srivastava, H. M.; Mishra, A. K.; Kund, S. N.

Panam. Math. J. 21, No. 4, 105-123 (2011).

Let \(A_{p}\), \(p \in \mathbb{N}\), be the class of functions \(f\) analytic in the open unit disk \(\mathbb{U}\) having the following \(p\)-fold symmetric series expansion: \[ f(z)=z \left(1+\sum^{\infty}_{v=1}a_{vp}z^{vp} \right). \] The class of univalent functions in \(A_{p}\) is denoted here by \(S_{p}.\) A function \(f \in A_{p}\) is said to be in the class \(S^{*}_{p}(\alpha , \beta)\) (\(0 \leq \alpha<1; 0<\beta \leq 1\)) of univalent starlike functions of order \(\alpha\) and type \(\beta\) in \(\mathbb{U}\), if the following condition is satisfied for all \(z \in \mathbb{U}\): \[ \frac{|h(z)-1|}{|2\beta [h(z)-\alpha]-[h(z)-1]|}<1 \tag{2} \] where \(h(z)=zf^{\prime}(z)/f(z)\). Let \(\sum_{p}\) denote the class of analytic and univalent functions represented by \[ g(z)=z \left(1+\sum^{\infty}_{m=1}b_{mp}z^{-mp}\right), \] where \(z\in \mathbb{V}=\{z\in \mathbb{C}: 1<|z|< \infty \}\). We denote by \(\sum_{p}(\alpha , \beta)\) the class of univalent starlike functions of order \(\alpha\) and type \(\beta\) in \(\mathbb{V}.\) Thus, by definition, a function \(g \in \sum_{p}(\alpha , \beta)\) if and only if \(g \in \sum_{p}\) and the condition (2) is satisfied for \(z\in \mathbb{V}\) when \(h(z)\) is given by \(h(z)=zg^{\prime}(z)/g(z)\).
Let \(S^{-1}_{p}\) be the class of inverse functions \(f^{-1}\) of functions \(f\) in \(S_{p}\). Such functions have the following representation: \[ f^{-1}(w)=w \left(1+\sum^{\infty}_{m=1}A_{mp}w^{mp}\right) \] with \(|w|< \gamma(f)\), \(\gamma(f) \geq \frac{1}{4}\). Similarly, we let \(\sum^{-1}_{p}\) be the class of inverse functions \(g^{-1}\) of functions \(g\in \sum_{p}.\) These functions are represented by \[ g^{-1}(w)=w \left(1+\sum^{\infty}_{m=1}B_{mp}w^{-mp} \right) \] in some neighborhood of infinity. The function classes \(S^{*}_{p}(\alpha, \beta)^{-1}\) and \(\Sigma_{p}(\alpha, \beta)^{-1}\) are defined analogously.
In the present paper, the authors solve the coefficient estimate problem for the function classes \(S^{*}_{p}(\alpha, \beta)^{-1}\), \(\Sigma_{p}(\alpha, \beta)^{-1}\) and \(\Sigma_{p}(\alpha, \beta)\).

Reviewer: Mugur Acu (Sibiu)

MSC:

30C45

Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Keywords:

univalent functions; starlike functions; coefficients estimates