Prove a function, defined by integration of a harmonic function, is log-convex [closed]

Question

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!

Answer 1

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.