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Sharp extensions of Bernstein's inequality to rational spaces
Author(s) -
Borwein Peter,
Erdélyi Tamás
Publication year - 1996
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300011876
Subject(s) - mathematics , rational function , unit circle , polynomial , bernstein polynomial , inequality , degree (music) , function (biology) , pure mathematics , combinatorics , unit (ring theory) , discrete mathematics , mathematical analysis , physics , mathematics education , evolutionary biology , acoustics , biology
Sharp extensions of some classical polynomial inequalities of Bernstein are established for rational function spaces on the unit circle, on K = r (mod 2 π), on [‐1, 1] and on ℝ. The key result is the establishment of the inequality| f ′ ( z 0 ) | ⩽ max {∑j = 1| a j | > 1n| a j | 2 − 1| a j − z 0 | 2, ∑j = 1| a j | < 1n1 −| a j | 2| a j − z 0 | 2}‖ f ‖∂ Dfor every rational function f = p n / q n , where p n is a polynomial of degree at most n with complex coefficients andq n ( z ) = ∏ j = 1 n( z − a j )with | a j | ≠ 1 for each j and for every z o ∈ δ D , where δ D ,= { z ∈ ℂ: | z | = l}. The above inequality is sharp at every z 0 ∈δ D .