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2011d.00686 Tuck, Ernie When does the first derivative exceed the geometric mean of a function and its second derivative? Aust. Math. Soc. Gaz. 32, No. 4, 267-268 (2005). 2005 The Australian Mathematical Society, Melbourne EN I25 I45 Zbl 0315.10035 Zbl 1113.11303

http://www.austms.org.au/Gazette/2005/Sep05/Tuck.pdf

Introduction: Given a real function $u = u(x)$ defined in some region of real $x$, the quantity $w = u'{}^2-uu''$ is ``often'' (but by no means always) positive throughout that region. The purpose of this note is to suggest some restrictions on $u$ that are sufficient to guarantee that $w > 0$. (Strictly speaking, the verbalised title of the note is relevant only to the special case when $u$, $u'$ and $u''$ are all separately positive, in which case $w > 0$ does indeed imply that $u' > \sqrt{uu''}$). Since $w/u^2$ is the derivative of $u'/u$, the condition $w > 0$ is equivalent to $u'/u$ being a decreasing function, or to $-\log u$ being a convex function. There is a relationship between a decreasing property of $u'/u$ and reality of the zeros of $u$ [see e.g. , p. 128 in {\it H. M. Edwards}, Riemann's Zeta Function. Academic Press (1974; Zbl 0315.10035), reprint Dover (2001; Zbl 1113.11303)], and most functions $u$ of interest here possess many real zeros.