Smoothly Varying Functions and Perfect Proximate Orders

It is shown in this paper that $h(r)$ is a smoothly varying function of order $\rho$ if and only if the function $\rho(r)=(\ln h(r))/\ln r$ is a perfect proximate order, i.e., an infinitely differentiable (in a neighborhood of $+\infty$) function for which the conditions $\lim_{r\to+\infty}\rho(r)=\rho$, $\rho\in\mathbb R$, and $\lim_{r\to+\infty}r^n\ln r\rho^{(n)}(r)=0$ for all $n\in\mathbb N$ are satisfied. Consequences of the result indicated above are also obtained in this paper.