Slowly varying functions in the complex plane

Let f be analytic and have no zeros in $|\arg z| < \alpha \leqslant \pi$; f is called slowly varying if, for every $\lambda > 0,f(\lambda z)/f(z) \to 1$ uniformly in $|\arg z| \leqslant \beta < \alpha$, when $|z| \to \infty$. One shows that f is slowly varying if and only if $zf’(z)/f(z) \to 0$ uniformly in $|\arg z| \leqslant \beta < \alpha$, when $|z| \to \infty$.