AN IDENTITY RELATED TO JORDAN ’ S INEQUALITY

The main purpose of this note is to establish an 
identity which states that the function sinx/x is a power series of (π2−4x2) with positive coefficients for all x≠0. This enable us to obtain a much stronger Jordan's inequality 
than that obtained before.


Introduction
The well-known Jordan's inequality states that with equality holds if and only if x = π/2 (see [5]).It plays an important role in many areas of pure and applied mathematics.The inequality (1.1) is first extended to the following: and then, it is further extended to the following: with equality holds if and only if x = π/2 (see [2,4,6]).The inequality (1.3) is slightly stronger than the inequality (1.2) and is sharp in the sense that 1/π 3 cannot be replaced by a larger constant.More recently, the monotone form of L'Hopital's rule (see [1,Lemma 5.1]) has been successfully used by Zhu [9,10], Wu and Debnath [7,8] in the sharpening Jordan's inequality.For example, it has been shown that if 0 < x ≤ π/2, then hold with equality if and only if x = π/2.Furthermore, the constants 1/π 3 and (π − 2)/π 3 in (1.4) as well as the constants (12 − π 2 )/(16π 5 ) and (π − 3)/π 5 in (1.5) are the best.Also, in the process of sharpening Jordan's inequality, one can use the same method as did in [7] to introduce a parameter θ (0 < θ ≤ π) to replace the value π/2.Unfortunately, the preceding method will become cumbersome to execute in the further generalization of Jordan's inequality.
In this note we establish an identity which states that the function sinx/x is a power series of (π 2 − 4x 2 ) with positive coefficients for all x = 0.This enables us to obtain a much better inequality than (1.4)

Main result
The main result relating to Jordan's inequality is contained in the following.
The above established inequality (2.14) is much stronger than the left-hand side of inequality (1.5).Also one can add more positive terms to the right-hand side of inequality (2.14) to get higher accuracy.
Finally, it should be pointed out that, in order to give the right-hand side of inequality (1.4) or (1.5), the following Taylor expansion for x/ sinx will play an important role as the above established identity (2.1) or (2.13).
Taylor expansion of x/ sinx.
Recall that the Bernoulli numbers B n and the functions B n (x) are defined by It is familiar that they have the following properties (see [3, Section I-13]): Also one can add more positive terms to the right-hand side of inequality (2.20) to get higher accuracy.