Transfinite diameter of Bernstein sets in <inline-formula><graphic file="1029-242X-2002-680202-i1.gif"/></inline-formula>

• Main Authors: Bialas-Cie&#380; Leokadia, Jedrzejowski Mieczys&#322;aw
• Format: Article
• Language: English
• Published: SpringerOpen 2002-01-01
• Series: Journal of Inequalities and Applications
• Subjects: Bernstein and Markov inequality; Transfinite diameter; Pluripolar sets
• Online Access: http://www.journalofinequalitiesandapplications.com/content/7/680202
• ISSN: 1025-5834; 1029-242X

<p/> <p>Let <inline-formula><graphic file="1029-242X-2002-680202-i2.gif"/></inline-formula> be a compact set in <inline-formula><graphic file="1029-242X-2002-680202-i3.gif"/></inline-formula> satisfying the following generalized Bernstein inequality: for each <inline-formula><graphic file="1029-242X-2002-680202-i4.gif"/></inline-formula> such that <inline-formula><graphic file="1029-242X-2002-680202-i5.gif"/></inline-formula>, for each polynomial <inline-formula><graphic file="1029-242X-2002-680202-i6.gif"/></inline-formula> of degree <inline-formula><graphic file="1029-242X-2002-680202-i7.gif"/></inline-formula> <inline-formula><graphic file="1029-242X-2002-680202-i8.gif"/></inline-formula> where <inline-formula><graphic file="1029-242X-2002-680202-i9.gif"/></inline-formula> is a constant independent of <inline-formula><graphic file="1029-242X-2002-680202-i10.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2002-680202-i11.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2002-680202-i12.gif"/></inline-formula> is an infinite set of natural numbers that is also independent of <inline-formula><graphic file="1029-242X-2002-680202-i13.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2002-680202-i14.gif"/></inline-formula>. We give an estimate for the transfinite diameter <inline-formula><graphic file="1029-242X-2002-680202-i15.gif"/></inline-formula> of the set <inline-formula><graphic file="1029-242X-2002-680202-i16.gif"/></inline-formula>: <inline-formula><graphic file="1029-242X-2002-680202-i17.gif"/></inline-formula> For <inline-formula><graphic file="1029-242X-2002-680202-i18.gif"/></inline-formula> satisfying the usual Bernstein inequality (i.e., <inline-formula><graphic file="1029-242X-2002-680202-i19.gif"/></inline-formula>), we prove that <inline-formula><graphic file="1029-242X-2002-680202-i20.gif"/></inline-formula></p>