New inequalities of Huygens-type involving tangent and sine functions

Abstract

Using the estimations of the even-indexed Bernoulli number and Euler number this paper established some new inequalities for the three functions $2\left( \sin x\right) /x+\left( \tan x\right) /x$, $\left( \sin x\right) /x+2\left( \tan (x/2)\right) /\left( x/2\right) $ and $2x/\sin x+x/\tan x$ bounded by the powers of tangent function.