Isoperimetric inequalities for conformal moments of plane domains
The first step of this paper is to prove the inequality \[ \left |\sum^n_{k=0} a_kb_{n-k} \right|^2\leq B(\alpha,\beta) \sum^n_{k=0}{k!(n-k)!|a_k b_{n-k} |^2\over n!B(k+\alpha, n-k+\beta)} \tag{1} \] where \((a_k)\) and \((b_k)\) are the sequences of arbitrary complex numbers and \(\Gamma(x)\) and \(B(x,y)\) are Euler's functions, and \(\alpha>0\) and \(\beta>0\). The authors are using the inequality (1) to prove the inequality \[ \|fg|_{\alpha+ \beta}\leq\|f\|_\alpha \|g\|_\beta \tag{2} \] where \(\alpha\geq 1\) and \(\beta\geq 1\) and \(B(\Omega)\), \(B(\Omega)\) are the weighted Bergman space with the norm \[ \|f\|_\alpha= \left({\alpha-1 \over\pi}\iint_\Omega \bigl|f(z)\bigr |^2 R^{\alpha-2}(z,\Omega) dx dy\right)^{\frac 12} \] and \(f(z)\) is an analytic function in \(\Omega\) and \(R(z,\Omega)\) is the conformal radius of \(\Omega\) at the point \(z,\Omega\) is a simply connected domain in the \(z\) plane. The inequality (2) is used to derive isoperimetric inequality for the geometric functionals which are closely related to the torsional rigidity of a simply connected domain. (Corollary 1.1, corollary 1.2 and corollary 1.2).
scientific article; zbMATH DE number 1820064