A function $f:\Bbb{R}^n\to \Bbb{C}$ is said to satisfy Mean Value property on $\Bbb{R}^n$ if for all $x$ we have $f(x)=\frac{1}{B(x,r)}\int\limits_{B(x,r)}f(y)\ dy$ for all $r>0$.
I have proved that there does not exists any $L^p$ non-zero function which satisfies the above property using Holder's inequality where $1\le p<\infty$. How to check the $L^\infty$?
Can anyone help me in this regard? Thanks for help in advance.