Prove that it is NOT true in general that:

1) Concave functions of concave functions are concave
2) Increasing functions of concave functions are concave
3) Quasi-Concave functions are concave

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Clarification of Question by roadapples-ga on 07 Dec 2002 23:40 PST

I was just looking over problems one and two here. I think these ARE
true. If I am right, please post a counter example proof showing that
they are true. Thanks in advance!

Hi, roadapple-ga:

Let's first define the terms which frame your questions.

[Concave and Quasi-Concave Functions of a Single Variable]
http://www.wilsonc.econ.nyu.edu/UMath/Handouts/ums02h03concaveandquasiconcavefunctionsonthereals.pdf

A function f:R -> R is (weakly) concave if and only if:

f(ta + (1-t)b) >= tf(a) + (1-t)f(b)

for any distinct a,b in R and any t in (0,1).

A function f:R -> R is (weakly) quasi-concave if and only if:

f(ta + (1-t)b) >= min(f(a),f(b))

for any distinct a,b in R and any t in (0,1).

If the weak inequalities >= above are replaced by strict inequalities
>, then the definitions become those respectively of strictly concave
and strictly quasi-concave.  Strictly concave implies weakly concave,
but not conversely, just as strictly increasing implies weakly
increasing and not conversely.

We now produce a set of examples that show in each part slightly more
than was asked for by this problem.

(1) Let f(x) = 1 - x^2, a strictly concave function.  Then:

f(f(x)) = 1 - (1 - x^2)^2 = 2x^2 - x^4

but 0 = (1/2)(-1) + (1/2)(1) and:

0 = f(f(0)) < (1/2)f(f(-1)) + (1/2)f(f(1)) = 1

Thus the composition of two strictly concave functions need not be
even weakly concave.

(2) Let f(x) = e^x, a strictly increasing function.

Let g(x) = -x^2, a strictly concave function.  Then:

f(g(x)) = e^-x^2

but 1 = (1/2)(0) + (1/2)(2) and:

1/e = f(g(1)) < (1/2)f(g(0)) + (1/2)f(g(2)) = (1 + e^-4)/2

Thus a strictly increasing function of a strictly concave function
need not be even weakly concave.

(3) Any strictly increasing function is strictly quasi-concave, since
if a < b:

f(ta + (1-t)b) > f(a) = min(f(a),f(b))

Let f(x) = e^x, a strictly increasing function and hence strictly
quasi-concave.

But f(x) = e^x is convex rather than concave, and in particular:

1 = f(0) < (1/2)f(-1) + (1/2)f(1) = ((1/e) + e)/2

Thus a strictly quasi-concave function need not be even weakly
concave.

regards, mathtalk-ga


Search Strategy

Keywords: "quasi-concave" function
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Clarification of Answer by mathtalk-ga on 08 Dec 2002 00:21 PST

Hi, roadapple-ga:

Thanks for taking time to rate my answer (and for the tip!).  The
feedback is encouraging to me, and it seems the timing worked for both
of us tonight!.

regards, mathtalk-ga

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Request for Answer Clarification by roadapples-ga on 08 Dec 2002 00:22 PST

No problem! I cant begin to tell you how long I have been stressing
over these. I actually found that site the first time around but it
also wasnt making much sense to me. I had the general idea of the
question but wasnt sure exactly how to prove it. Thanks again!

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Clarification of Answer by mathtalk-ga on 08 Dec 2002 04:37 PST

You're quite welcome.  I just included the site to be able to quote
the definitions for you without referring to a textbook.  Thanks again
for posting the interesting problem.

best wishes, mathtalk-ga