# A property of hyperbolic general family of function spaces with Hadamard gap series

## Abstract

In this article, we obtained results which characterized the hyperbolic general family functions $$F^{*}{(p,q,s)}$$ and $$F^{*}_0{(p,q,s)}$$ by the coefficients of certain lacunary series expansions in the unit disk. Moreover, we obtain a sufficient and necessary condition for the hyperbolic function $$f^*$$ with Hadamard gaps, that is, for $$f(z)= \sum_{k=1}^{\infty} a_{k}z^{n_k}$$ satisfying $$\frac{n_{k+1}}{n_k}\geq \lambda > 1$$ for all $$k$$, to belong to $$F^{*}{(p,q,s)}$$ and $$F^{*}_{0}{(p,q,s)}$$ on the unite disk $$\mathbb{D}$$.

## Article Details

How to Cite
Kamal, A., & yassen, T. (2018). A property of hyperbolic general family of function spaces with Hadamard gap series. Journal of Advanced Studies in Topology, 9(1), 34–42. https://doi.org/10.20454/jast.2018.1324
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