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When does the first derivative exceed the geometricmean of a function and its second derivative?
Ernie Tuck
2

Given a real function u = u(x) defined in some region of real x, the quantity w = u0 −uu00
is “often” (but by no means always) positive throughout that region. The purpose of thisnote is to suggest some restrictions on u that are sufficient to guarantee that w > 0. (Strictly0speaking, the verbalised title of the note is relevant only to the special case when√ u, u and00
0u are all separately positive, in which case w > 0 does indeed imply that u > uu00 ).
Since w/u2 is the derivative of −u0 /u, the condition w > 0 is equivalent to u0 /u being adecreasing function, or to − log u being a convex function. There is a relationship between
a decreasing property of u0 /u and reality of the zeros of u (see e.g. [1], p. 128), and mostfunctions u of interest here possess many real zeros.

1

Examples on the whole real line

The example u = sin x, which is such that w = 1 for all x, is perhaps too simple, thoughemphatic! The slightly more general example u = k + sin x for real constant k, which is suchthat w = 1 + k sin x, is instructive in that w > 0 for all x if and only if |k| < 1. Tellingly, if|k| < 1 then u has infinitely many real zeros but no complex zeros, whereas if |k| > 1 thenu has only complex zeros with non-zero imaginary part.
Simple polynomials like u = x2 −k with w = 2(x2 +k) again illustrate w > 0 for functions
u having only simple real zeros (k > 0), and w changing sign if u has complex zeros (k < 0).
On the other hand, real zeros are not essential for w > 0, as the example u = exp(−x2 )with w = 2 exp(−2x2 ) shows; but it is of interest that this function u has two real inflexion
points.
These examples and some others are considered further later.

2

Preliminary results
2(1) Directly from its definition, w = u0 − uu00 > 0 in any region where uu00 < 0.
(2) Since w0 = u0 u00 − uu000 , similarly w0 > 0 in any region where u0 u00 > 0 and uu000 < 0. If
this is so in x > a, then also w > 0 in x > a, providing w(a) ≥ 0.
2(3) Since w00 = u00 − uu0000 , then w00 > 0 in any region where uu0000 < 0. If this is so in x > a,then also w0 > 0 in x > a providing w0 (a) ≥ 0, and then also w > 0 providing w(a) ≥ 0.
(4) Also directly from the definition, in order that w > 0, any turning point (u0 = 0) musthave uu00 < 0, so it must be either a positive (u > 0) maximum (u00 < 0), or a negative(u < 0) minimum (u00 > 0).
It is straightforward using these four results to analyse fully the region between any twoconsecutive simple zeros, paying particular attention to turning points and inflexion points.
Then, by considering a general segment as the union of such regions and zero-free end
regions, the following can be shown.

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Sufficient conditions

Let u = u(x) be a smooth real function defined in some segment (or all) of the real x-axis.
2Then u0 − uu00 > 0 if all of the following are true:– any zeros of u are simple, i.e. u0 does not vanish where u vanishes;– any turning points of u are either positive maxima or negative minima, i.e. uu00 < 0where u0 vanishes;– there is exactly one turning point and at most two inflexion points between any twoconsecutive zeros, and if there are two such inflexion points, the turning point lies between
them;– either uu00 < 0, or else (in a region with uu00 > 0 between a zero and an inflexion point)
u000 changes sign at most once and uu0000 < 0 in any part of that region where u0 u000 > 0.

4Remarks

The conditions stated are sufficient, but it is not clear how many of them are necessary. Inparticular, the involvement of the fourth derivative in the last condition seems at first sighta little unlikely; nevertheless it may well be necessary. I note via direct computation thatthe real-valued Riemann function u = ξ(1/2 + ix) on its critical line ([1], p. 16) satisfies allof the above, at least for 6 < x < 108 . In particular, it always has uu0000 < 0 in the relatively
uncommon regions where both uu00 > 0 and u0 u000 > 0.
A good example of the necessity of some of the other conditions is u(x) = (4 − x2 )(2 + x2 )on [−2, 2], which has three turning points, and suffers w(0) = −32. Similarly, u(x) =(4 − x2 )(3 + 2x + x2 ) has only one turning point, at x = 1, but two inflexion points in(−2, 1), and suffers w(−1/2) = −99/16. On the other hand u(x) = (4 − x2 )(4 + 2x + x2 )succeeds in having w(x) > 0 in all of [−2, 2], in spite of the fact that its two inflexion points
are also both to the left of its (single) turning point.
The above sufficient conditions are mostly of interest for functions with many real zeros.
In particular they say little about the everywhere-positive example u = exp(−x2 ), except
for showing that w > 0 between the inflexion points at x2 = 1/2, although we actually knowthat w > 0 everywhere for this function.
The “too-simple” example u = sin x with infinitely many zeros has the property that its
inflexion points and zeros coincide. Functions like u = ξ(1/2 + ix) with many more-or-lessevenly spaced zeros often have inflexion points that are in a sense “close” to their zeros. Thatis, there is one (and only one) turning point about midway between every pair of zeros, andthen one (and only one) inflexion point about midway between every pair of turning points.
Only in very special cases like u = sin x does this rough idea become exact enough to cause
the inflexion points to coincide with the zeros, but in many other cases (e.g. u = k + sin x
with |k| < 1) there is not more than one inflexion point between a zero and the nearest
turning point, which is consistent with the above sufficient conditions.

Acknowledgements
I thank John Harper, John van der Hoek and Jim Hill for comments on this work.

References[1] H.M. Edwards, Riemann’s Zeta Function (Academic Press 1974).
Department of Applied Mathematics, The University of Adelaide, SA 5005
Received 16 February 2005, accepted 11 May 2005.