Constants related to double Nörlund und Cesàro means (Q1330815)

Let \(q = (q(n))\) and \(r = (r(n))\), \(n = 0,1,2, \dots,\) be positive increasing sequences. Then, for a double sequence \(x = (x(i,j))\) of real or complex numbers, the double Nörlund mean \(t(m,n,q,r,x)\) is defined by \[ t(m,n,q,r,x) : = {1 \over Q(m)R(n)} \sum^ m_{i = 0} \sum^ n_{j = 0} q(m - i)r(n - j)x(i,j), \] where \(Q(m) : = \sum^ m_{i = 0} q(i)\) and \(R(n) : = \sum^ n_{j = 0} r(j)\). Moreover, for \(q,r\) and \(x\) as above let \[ \| x \|_{q,r} : = \sup \biggl\{ \bigl | t(m,n,q,r,x) \bigr | : m \geq 0,\;n \geq 0 \biggr\}. \] The paper is concerned with inequalities between \(\| x \|_{q,r}\) and \(\| x \|_{1,1}\) \((: = \| x \|_{(1,1, \dots), (1,1, \dots)})\). While the inequality \(\| x \|_{q,r} \leq \| x \|_{1,1}\) holds in any case (Theorem 1), inequalities in the reverse direction are more delicate and hold only under restrictions on \(q,r\) and \(x\).