Weighted integrals of holomorphic functions in the unit polydisc

• Main Author: Stević Stevo
• Format: Article
• Language: English
• Published: SpringerOpen 2005-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2005/463631
• ISSN: 1025-5834; 1029-242X

<p/> <p>Let <inline-formula><graphic file="1029-242X-2005-463631-i1.gif"/></inline-formula> be a measurable function defined on the unit polydisc <inline-formula><graphic file="1029-242X-2005-463631-i2.gif"/></inline-formula> in <inline-formula><graphic file="1029-242X-2005-463631-i3.gif"/></inline-formula> and let <inline-formula><graphic file="1029-242X-2005-463631-i4.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2005-463631-i5.gif"/></inline-formula>, be admissible weights on the unit disk <inline-formula><graphic file="1029-242X-2005-463631-i6.gif"/></inline-formula>, with distortion functions <inline-formula><graphic file="1029-242X-2005-463631-i7.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2005-463631-i8.gif"/></inline-formula>, where <inline-formula><graphic file="1029-242X-2005-463631-i9.gif"/></inline-formula>, and <inline-formula><graphic file="1029-242X-2005-463631-i10.gif"/></inline-formula>. We prove the following result: if <inline-formula><graphic file="1029-242X-2005-463631-i11.gif"/></inline-formula> and for all <inline-formula><graphic file="1029-242X-2005-463631-i12.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2005-463631-i13.gif"/></inline-formula>, then <inline-formula><graphic file="1029-242X-2005-463631-i14.gif"/></inline-formula> and there is a positive constant <inline-formula><graphic file="1029-242X-2005-463631-i15.gif"/></inline-formula> such that <inline-formula><graphic file="1029-242X-2005-463631-i16.gif"/></inline-formula>.</p>