Some inequalities for entire functions

Abstract

For any entire functions φ ( z ) \varphi (z) and ψ ( z ) \psi (z) on C with finite norm \[ { 1 π ∫ ∫ C | f ( z ) | 2 exp ⁡ ( − | z | 2 ) d x d y } 1 / 2 > ∞ , {\left \{ {\frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|f(z){|^2}\exp ( - |z{|^2})dx\;dy} } } \right \}^{1/2}} > \infty , \] we show that the inequality \[ 2 π ∫ ∫ C | φ ( z ) ψ ( z ) | 2 exp ⁡ ( − 2 | z | 2 ) d x d y ⩽ 1 π ∫ ∫ C | φ ( z ) | 2 exp ⁡ ( − | z | 2 ) d x d y 1 π ∫ ∫ C | ψ ( z ) | 2 exp ⁡ ( − | z | 2 ) d x d y \begin {array}{*{20}{c}} {\frac {2}{\pi }\int {\int \limits _{\mathbf {C}} {|\varphi (z)\psi (z){|^2}\exp ( - 2|z{|^2})\;dx\;dy} } } \hfill \\ { \leqslant \frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|\varphi (z){|^2}\exp ( - |z{|^2})\;dx\;dy\frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|\psi (z){|^2}\exp ( - |z{|^2})\;dx\;dy} } } } } \hfill \\ \end {array} \] holds. This inequality is obtained as a special case of a general result. We also refer to some properties of a tensor product of spaces of entire functions.