Isoperimetric type inequalities for harmonic functions

Abstract

For 0 < p < + ∞ let h p be the harmonic Hardy space and let b p be the harmonic Bergman space of harmonic functions on the open unit disk U . Given 1 ⩽ p < + ∞ , denote by ‖ ⋅ ‖ b p and ‖ ⋅ ‖ h p the norms in the spaces b p and h p , respectively. In this paper, we establish the harmonic h p -analogue of the known isoperimetric type inequality ‖ f ‖ b 2 p ⩽ ‖ f ‖ h p , where f is an arbitrary holomorphic function in the classical Hardy space H p . We prove that for arbitrary p > 1 , every function f ∈ h p satisfies the inequality ‖ f ‖ b 2 p ⩽ a p ‖ f ‖ h p , where a p > 1 is a suitable constant depending only on p. Furthermore, by using the Carleman inequality in the form ‖ f ‖ b 4 ⩽ ‖ f ‖ h 2 with f ∈ H 2 , we prove the following refinement of the above inequality for p = 2 ‖ f ‖ b 4 ⩽ 1.5 + 2 4 ‖ f ‖ h 2 .