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!



Prove a function, defined by integration of a harmonic function, is log-convex
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.


