Some inequalities for entire functions
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Abstract:
For any entire functions $\varphi (z)$ and $\psi (z)$ on C with finite norm \[ {\left \{ {\frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|f(z){|^2}\exp ( - |z{|^2})dx\;dy} } } \right \}^{1/2}} < \infty ,\] we show that the inequality \[ \begin {array}{*{20}{c}} {\frac {2}{\pi }\int {\int \limits _{\mathbf {C}} {|\varphi (z)\psi (z){|^2}\exp ( - 2|z{|^2})\;dx\;dy} } } \hfill \\ { \leqslant \frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|\varphi (z){|^2}\exp ( - |z{|^2})\;dx\;dy\frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|\psi (z){|^2}\exp ( - |z{|^2})\;dx\;dy} } } } } \hfill \\ \end {array} \] 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.References
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Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 80 (1980), 254-258
- MSC: Primary 30D20; Secondary 15A69
- DOI: https://doi.org/10.1090/S0002-9939-1980-0577754-4
- MathSciNet review: 577754