On Young's inequality

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:469

Article

Alzer, Horst¹  Kwong, Man Kam² 

[1] Morsbacher Str 10, D-51545 Waldbrol, Germany

[2] Hong Kong Polytech Univ, Dept Appl Math, Hunghom, Hong Kong, Peoples R China

关键词: Young's inequality;    Trigonometric sums;    Turin-type inequalities;    Harmonic functions;    Absolutely monotonic;   

DOI  :  10.1016/j.jmaa.2018.06.061

来源: Elsevier

【 摘 要 】

We present some inequalities for trigonometric sums. Among others, we prove the following refinements of the classical Young inequality. (1) Let m >= 3 be an odd integer, then for all n >= m - 1, Sigma(n)(k=1) cos(k theta)/k >= Sigma(m)(k=1)(-1)(k)/k. The sign of equality holds if and only if n = m and theta = pi. The special case m = 3 is due to Brown and Koumandos (1997). (2) For all even integers n >= 2 and real numbers r is an element of (0,1] and theta is an element of [0, pi] we have Sigma(n)(k=1) cos(k theta)/k r(k) >= -5/48 (5+root 5) = -0.75375.... The sign of equality holds if and only if n = 4, r = 1 and 0 = 4 pi/5. We apply this result to prove the absolute monotonicity of a function which is defined in terms of the log-function. (C) 2018 Published by Elsevier Inc.

【 授权许可】

Free