On Fejér's inequalities for the Legendre polynomials

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 ] .