On Hedenmalm-Shimorin type inequalities (Q2091647)

The authors directly prove an Hedenmalm-Shimorin inequality for short antidiagonals. The main results are given in the following theorems. Theorem 1.1. For any infinite complex-valued matrix \(M=\{m_{j,k}\}_{j,k=1}^{\infty}\), we have \[ \sum_{l=2}^{\infty}s^l\left|\sum_{j+k=l}\frac{m_{j,k}}{\sqrt{jk}}\right|^2\leq(\|M\|_{1\to2}^2+\|M\|_{2\to\infty}^2)s\log\frac{1}{1-s},\;\; 0\leq s\leq1, \] provided that the two quantities defined as follows \[ \|M\|_{1\to2}^2=\sup_{k\geq1}\sum_{j=1}^{\infty}|m_{j,k}|^2\;\;\text{and}\;\;\|M\|_{2\to\infty}^2= \sup_{j\geq1}\sum_{k=1}^{\infty}|m_{j,k}|^2 \] are both finite. Theorem 1.3. Let \(\{m_{i,j,k}\}_{i,j,k=1}^{\infty}\) be a sequence of complex numbers such that \[ \sup_{j,k\geq1}\sum_{i=1}^{\infty}|m_{i,j,k}|^2+\sup_{i,k\geq1}\sum_{j=1}^{\infty}|m_{i,j,k}|^2+ \sup_{i,j\geq1}\sum_{k=1}^{\infty}|m_{i,j,k}|^2\leq1. \] Then \[ \sum_{l=3}^{\infty}\frac{s^l}{l+1}\left|\sum_{i+j+k=l}\frac{m_{i,j,k}}{\sqrt{ijk}}\right|^2\leq\frac{s}{2}\left(\log\frac{1}{1-s}\right)^2, \;\;0\leq s\leq1. \]