New Inequalities of Cusa–Huygens Type

Ling Zhu

doi:10.3390/math9172101

Abstract

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.

Highlights

For x ∈ (0, π/2), we know that the functions cos x and/x are less than 1
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: ζ (2n) =
By substituting the power series expansions of all functions involved in Lemma 1 into f ( x ), we obtain that

Summary

Introduction

For x ∈ (0, π/2), we know that the functions cos x and (sin x )/x are less than 1. Bagul et al [48] drew two conclusions about the improvement of inequality (1): π sin x 2 + cos x. Inspired by inequalities (9)–(11), this paper intends to improve the famous inequality (1) from two different directions and to draw two results as follows. 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]

Lemmas

Proof of Theorem 1

Proof of Theorem 2

Remarks

Conclusions