On Young's inequality

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.