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

In this paper, we establish sharp inequalities for trigonometric functions. For example, we consider the Wilker inequality and prove that for 0 < x < pi/2 and n >= 1, 2+(Sigma(j=2) (n-1) d(j+1)X(2j) + delta X-n(2n))X(3)tan x < (sin x/x)(2) + tan x/x < 2 + (Sigma(j=3) (n-1) d(j+1)X(2j) + DnX2n)X-3 tan x with the best possible constants delta(n) = d(n) and D-n = 2 pi(6) - 168 pi(4) + 15120 /945 pi 4 (2/pi)(2n) - Sigma(j=2) (n-1) d(j+1) (2/pi)(2n-2j) where d(k) = 2(2k+2)((4k + 6) |B2k+2| + (-1)(k+1))/(2k + 3)! and B-k are the Bernoulli numbers (k is an element of IN0 := IN boolean OR {0}). This improves and generalizes the results given by MORTICI, NENEZIC and MALESEVIC.

Automated summary

In this paper, sharp inequalities for trigonometric functions are established, including the Wilker inequality. The improved and generalized results prove that the inequalities hold for certain values of x and n, with the best possible constants determined. The formulas involve Bernoulli numbers and factorial calculations.