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Turan's Inequalities for Trigonometric Polynomials
Author(s) -
Bojanov Borislav
Publication year - 1996
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/53.3.539
Subject(s) - mathematics , trigonometry , corollary , trigonometric polynomial , trigonometric functions , order (exchange) , combinatorics , constant (computer programming) , regular polygon , convex function , function (biology) , inequality , set (abstract data type) , pure mathematics , algebra over a field , mathematical analysis , geometry , computer science , finance , evolutionary biology , economics , biology , programming language
We present here a technique for establishing inequalities of the form c ‖ f ‖ ⩽ ∫ 0 2 πφ ( |f ( k )( t ) | ) d t ⩽ M ‖ f ‖ in the set of all trigonometric polynomials of order n which have only real zeros. The function φ is assumed to be convex and increasing on [0, ∞). As a corollary of the main result we get Turan's inequalities ‖ f ( k )‖ q ⩾ c ( n , k , q ) ‖ f ‖with the exact constant c ( n , k , q ) for each 1 ⩽ q ⩽ ∞, n and k .