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On the growth of real functions and their derivatives. (English) Zbl 1400.26007

Let \(q\) be a nonnegative integer and assume that \(\alpha>1\) is a real number. The main result of this paper establishes that if \(f:[1,\infty)\rightarrow{\mathbb R}\) is \(k\)-times continuously differentiable, then \[ \liminf_{x\rightarrow +\infty}\frac{x^k\cdot\log x\cdot \log_2 x\cdot\dots\cdot\log_q x\cdot f^{(k)}(x)}{1+|f(x)|^\alpha}\leq 0, \] where \(\log_p x\) denotes the \(p\)-times iterated natural logarithm. It is optimal in the sense that \(\log_qx\) cannot be replaced by \((\log_qx)^\beta\) with any \(\beta>1\).

MSC:

26A12 Rate of growth of functions, orders of infinity, slowly varying functions
26D10 Inequalities involving derivatives and differential and integral operators
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