A sufficient condition fork′(z)(k(z)∈H ^q,q≥1) to be ofH ¹

Abstract

We proved ifk(z)∈H ^q (q≥1),g(z) is analytic on |z|=1,$\left\| {g(e^{i\theta } ) + k(e^{i\theta } )} \right\|q = \mathop {\min }\limits_{h \in H^q } \left\| {g(e^{i\theta } ) + h(e^{i\theta } )} \right\|$, thenk′(z)∈H ¹, especially, ifq=1, thenk(z) is an analytic function on the closed unit disk |z|≤1.