Regularly varying functions and convolutions with real kernels

Abstract:

Let $\phi$ be a positive, measurable function and k a real-valued function on $(0,\infty ),k \in {L^1}(dt/t)$. We give conditions on $\phi$ and k sufficient to deduce the regular variation of $\phi$ from the assumption that \[ \alpha = \lim \limits _{x \to \infty } \frac {1}{{\phi (x)}}\int _0^\infty {\phi (t)k\left ( {\frac {x}{2}} \right )} \;\frac {{dt}}{t}\;{\text {exits}}\;(\alpha \ne 0,\infty ).\] The general theorems extend in certain ways results of other authors and yield a new theorem on the relation between the radial growth and zero-distribution of those entire functions which are canonical products of nonintegral order with negative zeros.