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 $0<x<\pi/2$ and $n\geq1$,
$$
2 + \left(\displaystyle\sum_{j=2}^{n-1}{d_{j+1}x^{2j} \!+ \delta_{n}x^{2n}}\!\!\right) x^{3}\tan x
<
\left(\nfrac{\sin x}{x}\right)^{\!2}+\nfrac{\tan x}{x}
<
2 + \left(\displaystyle\sum_{j=3}^{n-1}{d_{j+1}x^{2j} \!+ D_{n} x^{2n}}\!\!\right) x^{3}\tan x
$$
with the best possible constants
$$
\delta_{n} = d_{n}
\;\mbox{and}\;
D_{n}
=
\nfrac{2\pi^6-168\pi^4+15120}{945\pi^4} \left(\nfrac{2}{\pi}\right)^{\!2n}
\!-\,
\displaystyle\sum_{j=2}^{n-1}{d_{j+1}\left(\nnfrac{2}{\pi}\right)^{\!2n-2j}}\!,
$$
where
$
d_k
\!=\!
{2^{2k+2}{\big (}(4k+6) \, |B_{2k+2}|+(-1)^{k+1}{\big )}}/{(2k+3)!}
$
and
$
B_k
$
are the {\sc Bernoulli} numbers $\left(k \in \mathbb{N}_0\!:=\mathbb{N} \cup \{0\} \right)$.
This improves and generalizes the results given by {\sc Mortici}, {\sc Nenezi\'c} and {\sc Male\v{s}evi\'c}.