Contraction property of differential operator on Fock space

Abstract

In the recent paper, \cite{tilli} Nicola and Tilli proved the Faber-Krahn inequality, which for $p=2$, states the following. If $f\in\mathcal{F}_α^2$ is an entire function from the corresponding Fock space, then $$\frac{1}π\int_Ω |f(z)|^2 e^{-π|z|^2} dx dy \le (1-e^{-|Ω|}) \|f\|^2_{2,π}.$$ Here $Ω$ is a domain in the complex plane and $|Ω|$ is its Lebesgue measure. This inequality is sharp and equality can be attained. We prove the following sharp inequality $$\int_Ω \frac{|f^{(n)}(z)|^2e^{-π|z|^2}}{π^n n ! L_n(-π|z|^2)}dxdy \le (1-e^{-(n+1)|Ω|})\|f\|^2_{2,π},$$ where $L_n$ is Laguerre polynomial, and $n\in\{0,1,2,3,4\} $. For $n=0$ it coincides with the result of Nicola and Tilli.