New Bounds for the Sine Function and Tangent Function

: Using the power series expansion technique, this paper established two new inequalities for the sine function and tangent function bounded by the functions x 2 ( sin ( λ x ) / ( λ x )) α and x 2 ( tan ( µ x ) / ( µ x )) β . These results are better than the ones in the previous literature.


Introduction
Because of the fact the functions cos x and (sin x)/x are less than 1 for x ∈ (0, π/2), in order to determine this relationship (sin x)/x and the weighted geometric mean of cos x and 1, we examine the Taylor expansion of the following function: When choosing β = 1/3 in above formula we can obtain the following fact sin x x − (cos x) 1/3 = 1 45 x 4 + 19 5670 which will motivate us to prove the following inequality sin x x > (cos x) 1/3 (1) holds for 0 < x < π/2. The above inequality was confirmed by Mitrinović and Adamović in [1], so we call it Mitrinović-Adamović inequality. On the other hand, the relationship between (sin x)/x and the weighted arithmetic mean of cos x and 1 has been discussed in Zhu [2] just published, described as the following inequality similarly: In 1451, using a geometrical method Nicolaus De Cusa (1401-1464) discovered (3), and in 1664 when considering the estimation of π Christian Huygens (1629-1695) confirmed (2). In view of the above historical facts (see [3][4][5][6][7][8][9][10]), we call the inequality (2) Cusa-Huygens inequality.
This paper focuses on some new bounds for the functions sin(x)/x and tan(x)/x and wants to improve the following inequalities: Recently, Wu and Bercu [63] thought of Fourier series technology to approximate these two functions. They considered the power series expansion of the following function To obtain a slightly higher precision approximation, they let This technique can be used to deal with the approximation problem of another function tan(x)/x − 1, and then they obtain the following results.
In this paper, we want to obtain an approximation with appropriate accuracy about these two functions. We examine the power series expansion of function in the following form We can obtain that In the same way, we obtain With the above foreshadowing, we can now announce the main work of this paper which established two inequalities of exponential type for the functions 1 − (sin x)/x and (tan x)/x − 1 bounded by the function x 2 (sin(λx)/(λx)) α and x 2 (tan(µx)/(µx)) β as follows. Then the double inequality holds with the best constants φ and ϕ.
holds with the best constant 1/3.

Proof.
Since are positive for n ≥ 3. In order to prove Lemma 1, it suffices to prove that for n ≥ 3, To note the fact we only need to prove By mathematical induction, we can prove the inequality (23). First, the inequality (23) is obviously true for n = 3. Assume that (23) holds for n = m ≥ 3, that is, holds. In the following, we shall prove that (23) holds for n = m + 1. Since we can complete the proof of (23) when showing that In fact, holds for all m ≥ 3. This completes the proof of Lemma 1.
It is not difficult to prove the following conclusion in the similar way.

Comparison of New and Old Results
When letting φ = 1 in (19)  x   42 5 .
By using the similar proof method of Theorem 1 it is not difficult to prove the following results: hold for all x ∈ (0, π/2). So new results (19) and (20) are better that the old ones (16) and (17), respectively. In addition, there are deeper conclusions: x 4 + 1 3780 x 6 (34)
Funding: This paper is supported by the Natural Science Foundation of China grants No. 61772025.