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SUMMARY:Inequalities of Duffin-Schaeffer type in the complex plane - Geno 
 Nikolov (Sofia)
DTSTART:20131121T150000Z
DTEND:20131121T160000Z
UID:TALK47418@talks.cam.ac.uk
CONTACT:34282
DESCRIPTION:The classical Markov inequality says that if a polynomial p of
  degree n is bounded by 1 on [-1\,1]\, then the maximal value of its k-th 
 derivative p(k) in the same interval is attained by  the Chebyshev polynom
 ial T_n(x) = cos n arccos x\, i.e.\, |p(k)(x) | <  T_n(k)(1). \n\nIn 1941\
 , Duffin and Schaeffer  found a refinement of this theorem to the effect t
 hat the conclusion still holds when the uniform boundedness |p(x)| < 1 is 
 replaced by a weaker conditions that |p| < 1 only at n+1 points at which |
 T_n| = 1. Moreover\, under such weaker assumption\, the Markov inequality 
 can be extended to the complex plane\, namely  |p(k)(x + iy)| < |T_n(k)(1 
 + iy)|. \n\nA crucial role in their proof is played by the so-called end-p
 oint domination property of T_n which is |T_n(x + iy)| < |T_n(1 + iy)|. We
  show that this property is featured more generally by the ultraspherical 
 polynomials P_n (which Chebyshev polynomials are a part of)\, and thus Duf
 fin-Schaeffer-type inequalities are valid for them too. The proof is based
  on an expansion formula for the squared modulus of an entire function fro
 m the Laguerre-Polya class\, due to Jensen\, and some further results\, In
  particular\, we prove a conjecture of Patrick from 1971 concerning the Ja
 cobi case.
LOCATION:MR 14\, CMS
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