New Inequalities of Cusa–Huygens Type

Using the power series expansions of the functions cotx,1/sinx and 1/sin2x, and the estimate of the ratio of two adjacent even-indexed Bernoulli numbers, we improve Cusa–Huygens inequality in two directions on 0,π/2. Our results are much better than those in the existing literature.


Introduction
For x ∈ (0, π/2), we know that the functions cos x and (sin x)/x are less than 1. In order to confirm the relationship between (sin x)/x and the weighted arithmetic mean of cos x and 1, we can examine the Taylor expansion of the following function: Obviously, when choosing β = 1/3 we can get the following fact: which inspires us to prove that for 0 < x < π/2, The existing mathematical historical data (see [1][2][3][4][5][6][7][8]) show that the above inequality (2) was discovered by Nicolaus De Cusa (1401-1464) using a geometrical method in 1451 and was later in 1664 confirmed by CHRISTIAN HUYGENS (1629-1695) when considering the estimation of π. Because of the contribution of Nicolaus De Cusa and Christian Huygens to this inequality (1), we call it Cusa-Huygens inequality. Recently, Zhu [9] provided two improvements of (2) as follows.
In this paper, we use the power series expansions of two functions cot x and 1/ sin x and their derivative functions to prove the main conclusions. We know that the Taylor coefficients of these power series expansions are closely related to the Bernoulli number, which is related to the Riemann zeta function through the following identity: The latest research information on the Riemann zeta function can be found in Milovanović and Rassias [46].

Lemma 2 ([52-54])
. Let B 2n be the even-indexed Bernoulli numbers, n = 1, 2, · · ·. Then To prove our results, we also need the monotone form of the L'Hospital rule shown in [55][56][57] and the criterion for the monotonicity of the quotient of power series shown in [58].
) are also increasing (or decreasing) on (a, b).

Proof.
Let Then is increasing for n ≥ 1, by Lemma 4 we have that a (x)/b (x) is increasing on (0, π/2). By Lemma 3 we get that the function increasing on (0, π/2). At the same time, we find that This completes the proof of Lemma 5.

Remarks
In this section, we compare new conclusions (12) and (13) with (9)-(11). Remark 1. The inequality (12) is better than the one (9) because − 1 180 The last inequality follows from Lemma 5 due to Remark 3. We can find that the inequality (12) is better than the right-hand side one of (13) on (0, 1.4117) while (13) is better than the one (12) on (1.4117, π/2). So the inequality (12) has the advantage on the left-hand side of the interval (0, π/2) and the advantage of the inequality (13) lies near this point π/2.

Conclusions
In this paper, using the power series expansions of the functions cot x, 1/ sin x, and 1/ sin 2 x, and the estimate of the ratio of two adjacent even-indexed Bernoulli numbers, we obtained some new Cusa-Huygens type inequalities, which greatly improve known results.
Funding: This paper will be supported by the natural science foundation of China (61772025).

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Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable.