On the Growth of Real Functions and their Derivatives

Abstract

We show that for any \(k\)-times differentiable function \(f:[a,\infty)\to\mathbb{R}\), any integer \(q\ge 0\) and any \(\alpha>1\) the inequality \[ \liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\ldots\cdot \log_q x \cdot |f^{(k)}(x)|}{1+|f(x)|^\alpha}= 0 \] holds and that this result is the best possible in the sense that \(\log_q x\) cannot be replaced by \((\log_q x)^\beta\) with any \(\beta>1\).