On generalized Hausdorff matrices (Q1109247)

Suppose \(\{\lambda_ n\}\) is a sequence with \(\lambda_ 0\geq 0\) and \(\lambda_ n>0\) for \(n>0\). Let \(\Omega\) be a simply connected region that contains every positive \(\lambda_ n\), and suppose that, for \(n=0,1,...,\Gamma_ n\) is a positively sensed Jordan contour lying in \(\Omega\) and enclosing every \(\lambda_ k\in \Omega\) with \(0\leq k\leq n\). Suppose that f is holomorphic in \(\Omega\) and that \(f(\lambda_ 0)\) is defined even when \(\lambda_ 0\not\in \Omega\). Define \[ (*)\quad \lambda_{n,k}=-\lambda_{k+1}....\lambda_ n\frac{1}{2\pi_ i}\int_{\Gamma_ n}\frac{f(z)dz}{(\lambda_ k-z)...(\lambda_ n- z)}+\delta_ k,\quad 0\leq k\leq n,\quad \lambda_{n,k}=0\quad for\quad k>n, \] where \(\delta_ k=f(\lambda_ 0)\) if if \(k=0\) and \(\lambda_ o\not\in \Omega\), and \(\delta_ k\) otherwise. Matrices whose entries are given by (*) one called generalized Hausdorff matrices. In this paper the authors prove two interesting theorems. In Theorem 1 they show that if \(\sum^{\infty}_{n=1}1/\lambda_ n=\infty\) and \(f(z)=\int^{1}_{0}t\quad zd\alpha (t),\) \(\alpha\in BV\), with \(\alpha (1)-\alpha (0)=1\) and \(\alpha (0+)=\alpha (0)\), then the matrix \(\lambda_{n,k}\) given by (*) is regular, while in Theorem 2 they establish the equivalence of generalized Hölder and Cesàro matrices.