Extension of Vietoris’ Inequalities for Positivity of Trigonometric Polynomials

In this work, conditions on the coefficients {ak} are considered so that the corresponding sine sum n ∑ k=1 ak sin kθ and cosine sum a0 + n ∑ k=1 ak cos kθ are positive in the unit disc D. The monotonicity property of cosine sums is also discussed. Further a generalization of renowned Theorem of Vietoris’ for the positivity of cosine and sine sums is established. Various new results which follow from these inequalities include improved estimates for the location of the zeros of a class of trigonometric polynomials and new positive sums for Gegenbauer polynomials. 1. Preliminaries Positivity of trigonometric sums, which is key ingredient in Fourier Analysis, appears in various branches of mathematics and have many applications. In 1910, Fejer in connection with the study of Gibb’s phenomenon of Fourier series, conjectured that the partial sum of the series 1 2 (π − θ) = ∞ ∑ k=1 sin kθ k , 0 < θ ≤ π, are positive; i.e., n ∑ k=1 sin kθ k > 0, for all n ∈ N and θ ∈ (0, π). (1.1) This was proved by Jackson [17] in 1911 and later by Gronwall [15] in 1912. After this a number of different proofs appeared in the literature. Among these, a short ten line proof was provided by Landau [21]. See also [23, p.206] and [38, p.62]. Tomic [34] and HyltenCavallius [16] developed a geometric method of approaching such problems. Turan’s [35] striking proof shows that if n ∑ k=1 ak sin (2k − 1)θ ≥ 0, 0 < θ < π, 2010 Mathematics Subject Classification. 42A05, 42A32, 33C45, 26D05.