Gaussian integral means in the Fock space

We explore the relation between Hadamard products of two entire functions in the weighted Fock space F γ p and their integral Gaussian means. By introducing the key auxiliary function κ γ , r , we show that the growth of the Gaussian means of the Hadamard product f ∗ g ∗ κ γ , r is controlled by the growth of Gaussian means of f and g . Among several consequences of this observation, in particular, we establish the sub-multiplicative property f ∗ g ∗ κ F 1 ≤ ‖ f ‖ F 1 ‖ g ‖ F 1 for functions in the classical Fock space F 1 .