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On Fejér's inequalities for the Legendre polynomials
Author(s) -
Alzer Horst,
Kwong Man Kam
Publication year - 2017
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201600461
Subject(s) - legendre polynomials , mathematics , degree (music) , constant (computer programming) , inequality , polynomial , pure mathematics , associated legendre polynomials , combinatorics , mathematical analysis , classical orthogonal polynomials , orthogonal polynomials , gegenbauer polynomials , physics , computer science , acoustics , programming language
We present various inequalities for the sumS n ( t ) = ∑ k = 0 n P k ( t ) ,where P k denotes the Legendre polynomial of degree k . Among others we prove that the inequalities2 5 ( 1 + t ) ≤ S n ( t )and3 − 1 2 ( 1 − t 2 ) ≤ S n ( t )hold for all n ≥ 1 and t ∈ [ − 1 , 1 ] . The constant factors 2/5 and ( 3 − 1 ) / 2 are sharp. This refines a classical result of Fejér, who proved in 1908 thatS n ( t )is nonnegative for all n ≥ 1 and t ∈ [ − 1 , 1 ] .