Vol. 104, No. 1, 1983

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Zeros of successive iterates of multiplier-sequence operators

Carl L. Prather and Ken Shaw

Vol. 104 (1983), No. 1, 205–218

The authors study the zero free regions of iterates of multiplier-sequence operators for (i) functions analytic in the disc |z| < R and (ii) functions analytic in |z| > T. Integral representations for the iterates of each class of functions are given. As a consequence the authors give a generalization of the Post-Widder inversion formula. Other applications include an investigation of the zero free regions for iterates of fractional integrals as well as connections between results obtained here and recent final set results for iterates of operators on balanced sums.

Mathematical Subject Classification 2000
Primary: 30C99
Secondary: 30D99, 44A15
Received: 2 June 1980
Published: 1 January 1983
Carl L. Prather
Ken Shaw