On the growth of real functions and their derivatives

We show that for any $k$-times continuously differentiable function $f:[a,\infty)\longrightarrow{\mathbb R}$, any integer $q\ge 0$ and any $\alpha>1$ the inequality $$\liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\dots\cdot \log_q x \cdot f^{(k)}(x)}{1+|f(x)|^\alpha}\le 0 $$ holds.