Univalent Functions in the Möbius Invariant ð ‘„ ð ¾ Space

Fernando Pérez-González and Jouni Rättyä

Abstract and Applied Analysis, 2011, vol. 2011, 1-11

Abstract:

It is shown that a univalent function ð ‘“ belongs to ð ‘„ ð ¾ if and only if s u p ð ‘Ž ∈ ð ”» ∫ 1 0 ð ‘€ 2 ∞ ( ð ‘Ÿ , ð ‘“ ∘ 𠜑 ð ‘Ž − ð ‘“ ( ð ‘Ž ) ) ð ¾ â€² ( l o g ( 1 / ð ‘Ÿ ) ) ð ‘‘ ð ‘Ÿ < ∞ , where 𠜑 ð ‘Ž ( 𠑧 ) = ( ð ‘Ž − 𠑧 ) / ( 1 − ð ‘Ž 𠑧 ) , provided ð ¾ satisfies certain regularity conditions. It is also shown that under these conditions ð ‘„ ð ¾ contains all univalent Bloch functions if and only if ∫ 1 0 ( l o g ( ( 1 + ð ‘Ÿ ) / ( 1 − ð ‘Ÿ ) ) ) 2 ð ¾ â€² ( l o g ( 1 / ð ‘Ÿ ) ) ð ‘‘ ð ‘Ÿ < ∞ .

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:259796

DOI: 10.1155/2011/259796

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