Univalent Functions in the Möbius Invariant QK Space
It is shown that a univalent function f belongs to QK if and only if sup a∈𝔻∫01M∞2(r,f∘φa-f(a))K′(log (1/r))dr<∞, where φa(z)=(a-z)/(1-a¯z), provided K satisfies certain regularity conditions. It is also shown that under these conditions QK contains all univalent Bloch functions if and only if ∫0...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2011-01-01
|
| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/259796 |
| Summary: | It is shown that a univalent function f belongs to QK if and only if sup a∈𝔻∫01M∞2(r,f∘φa-f(a))K′(log (1/r))dr<∞, where φa(z)=(a-z)/(1-a¯z), provided K satisfies certain regularity conditions. It is also shown that under these conditions QK contains all univalent Bloch functions if and only if ∫01(log ((1+r)/(1-r)))2K′(log (1/r))dr<∞. |
|---|---|
| ISSN: | 1085-3375 1687-0409 |