Open Access
2018 On the Growth of Real Functions and their Derivatives
Jürgen Grahl, Shahar Nevo
Real Anal. Exchange 43(2): 333-346 (2018). DOI: 10.14321/realanalexch.43.2.0333


We show that for any \(k\)-times differentiable function \(f:[a,\infty)\to\mathbb{R}\), any integer \(q\ge 0\) and any \(\alpha>1\) the inequality \[ \liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\ldots\cdot \log_q x \cdot |f^{(k)}(x)|}{1+|f(x)|^\alpha}= 0 \] holds and that this result is the best possible in the sense that \(\log_q x\) cannot be replaced by \((\log_q x)^\beta\) with any \(\beta>1\).


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Jürgen Grahl. Shahar Nevo. "On the Growth of Real Functions and their Derivatives." Real Anal. Exchange 43 (2) 333 - 346, 2018.


Published: 2018
First available in Project Euclid: 27 June 2018

zbMATH: 06924893
MathSciNet: MR3942582
Digital Object Identifier: 10.14321/realanalexch.43.2.0333

Primary: 26A12 , 26D10

Keywords: differential inequalities , growth of real-valued functions

Rights: Copyright © 2018 Michigan State University Press

Vol.43 • No. 2 • 2018
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