On the growth of real functions and their derivatives

We show that for any $k$-times continuously differentiable function $f:[a,\mathrm{\infty })⟶\mathbb{R}$, any integer $q\ge 0$ and any $\alpha >1$ the inequality $$\underset{x\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{x^k\cdot \log x\cdot {\log }_{2}x\cdot \cdots \cdot {\log }_{q}x\cdot f^{(k)}(x)}{1+\mathrm{\mid }f(x){\mathrm{\mid }}^{\alpha }}\le 0$$ holds.