I know that a bound for the Bessel function of the second kind of the first order is given by $$K_{1}(x)<\frac{1}{x}.$$ But it also holds that $$K_{1}(x)\sim\sqrt{\frac{\pi}{2x}}e^{-x}$$ for $x\to+\infty$.
The first bound is very loose when approaching infinity, so I was wondering if there is a bound valid on the entire $(0, +\infty)$ but which decays exponentially at infinity.