hyperbolic functions

By using some tools of analysis, we establish some analytical properties such as monotonicity and inequalities involving the hyperbolic sine integral function. As applications of some of the established properties, we obtain some rational bounds for the hyperbolic tangent function.

By using L'Hospital's rule for monotonicity, we provide an alternative proof of a monotonicity property of a certain function involving the exponential function. This new approach is very concise.

In this paper‎, ‎we introduce the hyperbolic tan-X family of distributions as a new statistical distribution defined by a trigonometric function‎. ‎Some properties and applications of the new family are investigated in some detail‎. ‎A characterization theorem extending the new class of statistical distributions is also considered‎. ‎Avenues for further research are also presented‎.