On Fejér's inequalities for the Legendre polynomials

Horst Alzer and Man Kam Kwong

Mathematische Nachrichten, 2017, vol. 290, issue 17-18, 2740-2754

Abstract: We present various inequalities for the sum Sn(t)=∑k=0nPk(t),where Pk denotes the Legendre polynomial of degree k. Among others we prove that the inequalities 25(1+t)≤Sn(t)and3−12(1−t2)≤Sn(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 that Sn(t) is nonnegative for all n≥1 and t∈[−1,1].

Date: 2017
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