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

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.)