Problem in harmonic analysis

Question

suppose $p$ be a fixed psitive real number and $f$ is an entire function with $$\lvert f(0) \rvert^p=\int_\mathbb{C}\ \lvert f(z)\exp(-\alpha\lvert z \rvert ^2) \rvert^p dA(z) $$ where $\alpha >0,$ $dA(z)=dxdy=\text{euclidean area measure}$ and $z=x+iy$. Does it follow that $f(z)$ is a constant function? If yes, how does one prove it?

Answer 1

You need $$ |f(0)|^p=\int_\mathbb{C}\ | f(z)|^p\exp(-p\alpha\lvert z \rvert ^2)\, dA(z) =g(\alpha). $$ Let $f$ be a polynomial (of even an $f$ with $|f(z)|\ll \exp(|z|^2)$), with $f(0)\ne 0$. Then $$ \lim_{a\to 0} g(\alpha)=\infty, $$ while $$ \lim_{a\to\infty} g(\alpha)=0. $$ Hence, there is a $\alpha_0>0$, such that $$ |f(0)|^p=\int_\mathbb{C}\ | f(z)|^p\exp(-p\alpha_0\lvert z \rvert ^2)\, dA(z) =g(\alpha_0). $$