An improved Vietoris sine inequality

We prove that if { a k } is a sequence of positive numbers, such that ( 2 j − 1 ) j + 1 2 j j a 2 j + 1 ≤ a 2 j ≤ 2 j − 1 2 j a 2 j − 1 for all j = 1 , 2 , … , then for all n = 1 , 2 , … , x ∈ [ 0 , π ] , ∑ k = 1 n a k sin ( k x ) ≥ 0 . An example is { a k } = { 1 , 1 2 , 1 2 , 3 4 2 , 1 3 , 5 6 3 ,...
Ausführliche Beschreibung

We prove that if { a k } is a sequence of positive numbers, such that ( 2 j − 1 ) j + 1 2 j j a 2 j + 1 ≤ a 2 j ≤ 2 j − 1 2 j a 2 j − 1 for all j = 1 , 2 , … , then for all n = 1 , 2 , … , x ∈ [ 0 , π ] , ∑ k = 1 n a k sin ( k x ) ≥ 0 . An example is { a k } = { 1 , 1 2 , 1 2 , 3 4 2 , 1 3 , 5 6 3 , … } = { 1 , 0.5 , 0.707 , 0.530 , 0.577 , 0.481 , … } where a k = 1 / ( k + 1 ) / 2 for odd k , and ( k − 1 ) / ( k k / 2 ) , for even k . This improves the well-known Vietoris sine inequality, by relaxing the requirement that { a n } has to be a nonincreasing sequence. Ausführliche Beschreibung