A function f in Hp on the unit disc U of the complex plane has the uniform growth
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00001966/resource/name/S0008414X00001966_eq1.gif?pub-status=live)
We consider in this paper a subspace
of Hp with better uniform growth
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00001966/resource/name/S0008414X00001966_eq2.gif?pub-status=live)
For the previous results on
see [5, 6, 7]. We start with proving an inequality on Hp which is related to the Hardy-Stein identity (Theorem 2.1) in Section 2. This is applied in the subsequent section to prove some space imbedding theorems related to
(Theorems 3.1 and 3.5). These theorems have some known theorems as their corollaries. Finally we prove some coefficient relations on
in the last section.
The authors wish to thank Professor Patrick Ahern for the helpful conversations during his visit to Korea. Actually he suggested to the first author the possibility of Theorem 2.1 some years ago.