On natural approaches related to classical trigonometric inequalities
Résumé
In this paper, new refinements and improvements of Mitrinovic-Cusa and related inequalities are presented.
We prove in particular for \\ $0 < x < \frac{\pi}{2}$\ and any \ $n \geq 5$ $$ P_n(x)\ <\ (\sin x)^2- x^3\cot x < P_{n-1}(x) + \left[\left(\frac{2}{\pi}\right)^{2n} - \sum_{k=3}^{n-1} a_k \left(\frac{2}{\pi}\right)^{2n-2k}\right] x^{2n} $$ where $P_n(x) = \frac{x^6}{15}-\frac{x^8}{945} +\sum_{k=5}^n a_k x^{2k}$ is a $n$-polynomial, with positive coefficients \quad $a_{{k}}=\frac{{2}^{2\,k-2}}{\ \left( 2\,k-2 \right) ! } \left( \left| {B}_{ 2\,k-2} \right| +{\frac { \left( -1\right) ^{k+1}}{ \left( 2\,k-1 \right) k}} \right),$\
$ B_{2k} $\ are Bernoulli numbers. We also prove for any \ $n \geq 5$ $$\left(\frac{1+\cos x}{2}\right)^2 < \cos x + \frac{P_n(x)}{x^3} < \left(\frac{\sin(x)}{x}\right)^3.$$
The analysis results show that our bounds are tighter than previous methods.
Domaines
Analyse classique [math.CA]
Origine : Fichiers produits par l'(les) auteur(s)