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Cesàro averaging operators. (English) Zbl 1024.47014
K. F. Andersen proved in [Proc. R. Soc. Edinb., Sect. A 126, 617-624 (1996; Zbl 0865.47020)] that the generalized Cesàro operator defined by \[ C^{\gamma} f(z)= \sum_{n=0}^\infty \bigg({1\over A_n^{\gamma+1}}\sum_{k=0}^n A_{n-k}^{\gamma}a_k\bigg) z^n, \] where \(f(z)=\sum_{n=0}^\infty a_n z^n\) is an analytic function on the unit disc \(U\) and \(A_k^\gamma\) is the \(k\)th coefficient of the series expansion satisfying of \((1-x)^{-1-\gamma}\), satisfies the following inequality \[ M_p( C^{\gamma} f, r)\leq C_{\gamma, p} M_p( f, r) \] for every \(0<r<1\) and Re \(\gamma>-1\), where \(M_p\) denotes the integral mean in \(L^p\). In the present paper, the above result is extended to analytic functions defined in the polydisk where the operator \(C^{\gamma}\) is substituted by the so-called generalized Cesàro operator \[ C^{\bar\gamma} f(z)= \sum_{|\alpha|=0} \Biggl({\sum_{\beta\leq\alpha} (\prod_{j=1}^n A_{\beta_j}^{\gamma_j}) a_{ \alpha-\beta } \over \prod_{j=1}^n A_{\alpha_j}^{\gamma_j+1}} \Biggr) z^\alpha, \] whenever \(f(z)=\sum_{|\alpha|=0}^\infty a_\alpha z^\alpha\) is an analytic function on \(U^n\).

MSC:
47B38 Linear operators on function spaces (general)
46E15 Banach spaces of continuous, differentiable or analytic functions
30H05 Spaces of bounded analytic functions of one complex variable
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