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Metric property of a real polynomial
Author(s) -
Komarov Mikhail A.
Publication year - 2021
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12421
Subject(s) - mathematics , degree (music) , polynomial , rational function , combinatorics , corollary , measure (data warehouse) , function (biology) , metric (unit) , discrete mathematics , class (philosophy) , pure mathematics , mathematical analysis , operations management , physics , database , evolutionary biology , artificial intelligence , computer science , acoustics , economics , biology
Abstract For x ∈ R , let M ( P , δ ) be the measure m { x : | P ′ ( x ) / ( n P ( x ) ) | ⩾ δ } ( δ > 0 ), where P is an arbitrary polynomial of positive degree n . We prove thatsup Q M ( Q , δ ) = π / δ in the class of all real polynomials Q . As a corollary, we obtain the estimate of the derivative of a rational function, improving the known results of Gonchar and Dolzhenko, and prove that the inequality m { x : r ′ ( x ) / r ( x ) ⩾ n } ⩽ 2 π ( r is any real rational function of degree n ), conjectured by Borwein, Rakhmanov and Saff, is valid, at the least, for even and for odd functions r .