On the growth of real functions and their derivatives

TL;DR

This paper establishes a limit inferior inequality relating the growth of a k-times differentiable function and its derivatives, involving iterated logarithms, as x approaches infinity.

Contribution

It proves a new inequality connecting the growth of functions and their derivatives using iterated logarithms, extending previous understanding.

Findings

The inequality holds for all functions with specified smoothness and growth conditions.

It generalizes known results by involving multiple iterated logarithms.

The result applies to functions on unbounded intervals with specific differentiability.

Abstract

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.