Let \(f\) be a function which is analytic in the open unit disk. Then \(f\) is called uniformly normal if there is a constant \(K\) such that \(| f'(z)|\leq{K\over 1-| z|^ 2}\) for \(| z|<1\). (Such functions are also called Bloch functions.) A series \(\sum^ \infty_{k=0} a_ k z^{n_ k}\) convergent for \(| z|<1\) is said to have Hadamard gaps provided that there is a number \(\lambda>1\) such that \(n_{k+1}/n_ k\geq\lambda\) \((k\in\mathbb{N}_ 0)\).
The authors proved that for a series \(f\) having Hadamard gaps then \(f\) is uniformly normal if and only if the sequence \(\{a_ k\}\) is bounded. Let \(F(S)\) denote the set of functions given by the Hadamard gap series above where \(S=\{n_ k\}\) is fixed and \(\{a_ k\}\) varies over all sequences for which the power series converges for \(| z|<1\). The authors prove that \(F(S)\) is a normal family if and only if the set of sequences \(\{a_ k\}\) is uniformly bounded.