Area-capacity-modulus inequality and area of the filled Julia set

Abstract

In this paper, we prove the general area-modulus inequality$\\operatorname{area}(K)\\leq\\operatorname{area}(D){\\rm~e}^{-4\\pi\\operatorname{mod}(A)},$where $K$ is a full regular compact set in the plane, $D$ is a bounded simply connected domain containing $K$ and $A=D\\backslash~K$.Here the equality holds if and only if $A$ is a round annulus.Then we establish the area-capacity-modulus inequality$\\operatorname{area}(K)\\leq\\pi\\operatorname{cap}(K)^2{\\rm~e}^{2L-4\\pi\\operatorname{mod}(\\Lambda)}$for full regular compact set $K$ in the plane, where $L$ is the maximal critical value of the Greenfunction induced by $K$ and $\\Lambda$ is the tree (in set theory) induced by $K$.Here the equality holds if and only if $K$ is a closed disk or $\\operatorname{mod}(\\Lambda)=\\infty$.Using this inequality, we obtain an upper bound for the area of the filledJulia set $K(f)$ for the monic complex polynomial $f$ of degree $d$. That is,$\\operatorname{area}(K(f))\\leq\\pi~{\\rm~e}^{-\\frac{2qL}{d-q-1}},$where $L=\\max\\{G_f(c)\\mid~f'(c)=0\\}$ and $q$ is the critical index for $f$ onthe set $G_f^{-1}(L)\\backslash~K(f)$.