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Let $u$ be a harmonic function and we define $$ q(r)=\int_{\partial B(0,r)}u^2(x)\,dx $$

The question is about to prove that $q(r)$ is log-convex, i.e., I want to show $\log q(r)$ is convex function of $\log r$

I compute $$ \frac{d^2}{(d\log r)^2} \log q(r) = r^2\frac{q''(r)}{q(r)}+r\frac{q'(r)}{q(r)}-\left(r\frac{q'(r)}{q(r)}\right)^2 $$ I was hoping to prove that $$ r\frac{q'(r)}{q(r)}\geq\left(r\frac{q'(r)}{q(r)}\right)^2 $$ i.e., $$r q'(r)\geq q(r)$$ However, unfortunately, I proved the opposite direction...

Any help is really welcome!

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  • $\begingroup$ This question is phrased as if it is an exercise. If it isn't an exercise, where did you come across this fact, and was there not a reference that you could look up? $\endgroup$
    – Yemon Choi
    Dec 6, 2014 at 22:07
  • $\begingroup$ Yes it is an exercise. above is what I tried... $\endgroup$
    – JumpJump
    Dec 6, 2014 at 22:19
  • $\begingroup$ If it is an exercise then it is not really on topic for this site (and you should be getting help from the person who assigned this exercise) $\endgroup$
    – Yemon Choi
    Dec 6, 2014 at 22:22
  • $\begingroup$ ok...I am prepare an exam so I just find some exercise online to practice. This problem I found on some prof's website but that prof is even in another country... $\endgroup$
    – JumpJump
    Dec 6, 2014 at 22:24
  • $\begingroup$ Are you sure of that definition? Usually "f(x) is log (or logarithmically) convex" just means: $\log f(x)$ is convex. $\endgroup$ Dec 7, 2014 at 0:16

1 Answer 1

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I think this is more or less an exercise, instead of a research problem, which is not suitable for posing on this site.

However, I provide an link for your problem,

http://www.math.caltech.edu/~2010-11/3term/ma110c/HarmonicFunctions.pdf

You could find more information by just google "subharmonic functions", here is an overview http://www.encyclopediaofmath.org/index.php/Subharmonic_function

There are also nice books in the London Mathematical Society Monographs on subharmonic functions by Hayman et al. , which contains all the desired properties of subharmonic functions.

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  • $\begingroup$ I'm so sorry to bother around with this problem. I tried it for a while and can not get it. So I post it here for some help. You mean I can find the answer for this problem in your link? $\endgroup$
    – JumpJump
    Dec 6, 2014 at 23:16
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    $\begingroup$ You can find both answers to your question in the pdf. This site is usually for real research problem or philosophy of some important results, or references request. $\endgroup$ Dec 7, 2014 at 0:05

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