  In this paper, by investigating the monotonicity of a function composed of $%
\left( \sinh x\right) /x$ and $\cosh x$ with two parameters in $x$ on $% \left(
0,\infty \right) $, we prove serval theorems related to inequalities for
hyperbolic functions, which generalize known results and establish some new and
sharp inequalities. As applications, some new and sharp inequalities for
bivariate means are presented.
