Inequalities for the integral means of holomorphic functions in the strongly pseudoconvex domain
Commun. Korean Math. Soc. 2005 Vol. 20, No. 2, 339-350
Printed June 1, 2005
Hong Rae Cho, Jinkee Lee
Pusan National University, Pusan National University
Abstract : We obtain the following two inequalities on a strongly pseudoconvex domain $\Omega$ in ${\mathbb C}^n$ : for $f\in\mathcal O(\Omega)$ \begin{align*} \int_0^{\delta_0} t^{a|\alpha|+b} M_p^a (t, D^\alpha f) \,dt &\lesssim\int_0^{\delta_0} t^{b} M_p^a (t, f) \,dt\\ \int_0^{\delta_0} t^{b} M_p^a (t, f)\,dt &\lesssim \sum_{j=0}^m\int_0^{\delta_0} t^{am+b} M_p^a\Big(t,\mathcal N^j f\Big) \,dt. \end{align*} In \cite{S}, Shi proved these results for the unit ball in ${\Bbb C}^n$. These are generalizations of some classical results of Hardy and Littlewood.
Keywords : strongly pseudoconvex domain, integral means, Levi polynomial
MSC numbers : 32A10, 32A99
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