Browsing by Subject "Inequalities"
Now showing items 19 of 9

Article
An extension of Vietoris's inequalities
(2007)We establish a best possible extension of a famous Theorem of Vietoris about the positivity of a general class of cosine sums. Our result refines and sharpens several earlier generalizations of this Theorem, and settles ...

Article
Inequalities for two sine polynomials
(2006)We prove: (I) For all integers n ≥ 2 and real numbers x ∈ (0, π) we have (Formula Presented) with the best possible constant bounds (Formula Presented) (II) The inequality (Formula Presented) holds for all even integers n ...

Article
On a conjecture for trigonometric sums and starlike functions, II
(2010)We prove the case ρ=1/4 of the following conjecture of Koumandos and Ruscheweyh: let snμ(z)=Σk=0n(μ)k/ k!zk, and for ρε(0,1] let μ≤(ρ) be the unique solution of 0(ρ+1)πsin(tρπ)tμ1dt =0 in (0,1]. Then we have pipearg[(1 ...

Article
On a conjecture of Clark and Ismail
(2005)Let Φm (x) = xmψ(m) (x), where ψ denotes the logarithmic derivative of Euler's gamma function. Clark and Ismail prove in a recently published article that if m ∈ {1,2,..., 16}, then Φm(m) is completely monotonic on (0, ...

Article
On a conjectured inequality of Gautschi and Leopardi for Jacobi polynomials
(2007)Motivated by work on positive cubature formulae over the spherical surface, Gautschi and Leopardi conjectured that the inequality (equation presented) holds for α, β >  1 and n ≥ 1, θ∈ ∈(0, π), where Pn(α,β)(x) are the ...

Article
On a trigonometric sum of Vinogradov
(2004)The trigonometric sum f(m, n) = ∑ k=1 m1 sin(πkn/m) /sin(πk/m) (1 < m ∈ N, n ∈ N) has several applications in number theory. We prove that the mean value inequalities c1m(log m + γ  logπ/2) ≤ 1/m ∑ n=1 f(m, n) < c2m(log ...

Article
On the partial sums of a fourier series
(2008)We give sharp lower estimates for the partial sums of the Fourier series sin x + cos2x/2 + sin 3x/3 + cos 4x/4 + ..., with both an even and odd number of terms. Our results are obtained through a monotonicity property of ...

Article
A sharp bound for a sine polynomial
(2003)We prove that (Formula Presented) for all integers n ≥ 1 and real numbers x. The upper bound Si(π) is best possible. This result refines inequalities due to Fejér (1910) and Lenz (1951). © 2003, Instytut Matematyczny. All ...

Article
A Sharp Inequality for a Trigonometric Sum
(2013)We prove that the inequality, holds for all natural numbers n and real numbers x with x ∈ [0, Π]. The sign of equality is valid if and only if n = 1 and x = π /2. © 2012 Springer Basel AG.