On new proofs of Wilker’s inequalities involving trigonometric functions. (English) Zbl 1040.26006

Guo, Bai-Ni; Qiao, Bao-Min; Qi, Feng; Li, Wei

Math. Inequal. Appl. 6, No. 1, 19-22 (2003).

By using some properties of Bernoulli’s numbers and techniques of calculus, the best possible constants \(\alpha=\frac8{45}\) and \(\beta=\left(\frac2\pi\right)^4\) in the inequality \[ 2+\alpha x^3\tan x>\left(\frac{\sin x}x\right)^2+\frac{\tan x}x>2+\beta x^3\tan x \] are confirmed.

Reviewer: Feng Qi (Henan)

MSC:

26D05	Inequalities for trigonometric functions and polynomials
11B68	Bernoulli and Euler numbers and polynomials
33B10	Exponential and trigonometric functions
42A05	Trigonometric polynomials, inequalities, extremal problems

Keywords:

inequality; sine function; tangent function; Bernoulli number; Wilker’s inequality