On the growth of real functions and their derivatives

Abstract

We show that for any $k$-times continuously differentiable function $f:[a,\infty )⟶\mathbb{R}$, any integer $q\ge 0$ and any $\alpha >1$ the inequality $$\underset{x\to \infty }{\mathrm{liminf}}\frac{x^k\cdot \log x\cdot {\log }_{2}x\cdot \cdots \cdot {\log }_{q}x\cdot f^{(k)}(x)}{1+|f(x){|}^{\alpha }}\le 0$$ holds.

10 pages