Cosider the [K-Bessel function][1] $$K_\nu(x):= \frac\pi 2 \frac{I_{-\nu}(x)-I_\nu(x)}{\sin(\nu\pi)}.$$ 

See also <cite authors="Watson, G. N.">_Watson, G. N._, [**A treatise on the theory of Bessel functions.**](https://zbmath.org/?q=an:48.0412.02), Cambridge: University Press, Chapter III, p.78.</cite> 

This book contains many results on the asymptotic behaviour of functions of this kind. However, I am interested in upper bound estimates against simpler terms. Something similar is used in a another paper to show resolvent estimates by estimating K-Bessel functions against exponential and polynomial terms. At the moment I try to reproduce this estimates as a part of my research. However, no explanation on these kinds of estimated is provided and it falls short in providing the literature so I am looking for a reference on this topic. 


[1]: https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions:_I%CE%B1,_K%CE%B1