A NOTE ON FUNCTIONS OF MEAN BLOCH TYPES

Abstract

Abstract. A characterization of the holomorphic function spaces of me-an Bloch type on the unit disc is deduced in terms of the induced distance. 1. IntroductionWe introduce basic definitions, previous results, and the goal of this paperthat we will involve.Let D = {z ∈ C: |z| 0, let B α ( D ) be the space of holomorphic functions on D satisfyingsup z∈D i1 −|z| 2 ¢ α |f 0 ( z ) | 0 and 1 ≤ p < ∞ , let B pα ( D ) be the space holomorphic functionssatisfyingsup 0 <r< 1 (1 −r 2 ) α µZ T |f 0 ( rζ ) | p dσ ( ζ )¶ 1 p < ∞. The spaces B α ( D ) and B p ( D ) occurred in the literature in connection withthe Lipschitz space Lip α ( D ) and the mean Lipschitz space Lip p ( D ) which, for0 < α < 1 and 1 ≤ p < ∞ , are defined to consist of f holomorphic in D suchthat |f ( z ) −f ( w ) | ≤ C|z −w| α , z,w ∈ D, and of f ∈ H p ( D ) such thatµZ T |f