Dirichlet integral and star-function inequalities

Abstract:

Let $\lambda (z)$ be a smooth function in an annulus, $\tilde \lambda (z)$ its symmetric rearrangement and ${\lambda ^ * }(z)$ its star-function. A formula is proved relating $\Delta {\lambda ^ * },\Delta \lambda$, and the Dirichlet integrands of $\lambda$ and $\tilde \lambda$. The formula shows the relationship between Dirichlet integral inequalities and the subharmonicity of ${\lambda ^ * }$ for subharmonic $\lambda$, and gives an explicit formula for $\Delta {\lambda ^ * }$.