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 \(\mathbb{D}\).