Improvement of isoperimetric type inequality for harmonic functions in the case [formula omitted].

Let b p and h p ( 0 < p < ∞ ) be the harmonic Bergman space and harmonic Hardy space of harmonic functions on the open unit disk U , respectively. Given 1 ≤ p < ∞ , denote by ‖ ⋅ ‖ b p the norm in the space b p and by ‖ ⋅ ‖ h p the norm in the space h p . In this paper we establish some improvements of the constant a p appearing in the inequality ‖ f ‖ b 2 p ≤ a p ‖ f ‖ h p , given on Kalaj and Meštrović (2011) [Theorem 1.1], for p = 4 and f real harmonic, as ‖ f ‖ b 8 ≤ 28 + 35 2 + 24 2 ( 2 + 2 ) + 4 4 ( 2 + 2 ) 64 8 ‖ f ‖ h 4 . [ABSTRACT FROM AUTHOR]