Sharp inequalities related to the Adamovic-Mitrinovic, Cusa, Wilker and Huygens results

Abstract

In this paper, we establish sharp inequalities for trigonometric functions. For example, we consider the Wilker inequality and prove that for 0 < x < ?/2 and n ? 1, 2 + (?n?1 j=2 dj+1x2j+ ?nx2n) x3 tan x < (sin x/x)2 + tan x/x < 2 + (?n?1 j=3 dj+1x2j+ Dnx2n) x3 tan x with the best possible constants ?n = dn and Dn = 2?6 ? 168?4 + 15120/945?4 (2/?) 2n ? ?n?1 j=2 dj+1 (2/?/)2n?2j , where dk = 22k+2 ((4k + 6) |B2k+2| + (?1)k+1)/(2k + 3)! and Bk are the Bernoulli numbers (k ? N0 := N? {0}). This improves and generalizes the results given by Mortici, Nenezic and Malesevic.