Note on "Positive Cesaro Means of Numerical Series"

Abstract

We give a short proof of a result due to R. Askey. In this note we will give a short proof of Theorem 1 in the preceding paper by R. Askey. The proof utilizes a trick applied earlier by Askey and Pollard LI]. We restate Askey's result for convenience. Theorem. If y > a. > 1, and the (C, y) means of E a n are nonnegative then the (C, a) means of Ia rn are nonnegative for 0 Proof. We need to show that A n(r) > 0 for 0 (1w)-aIa rnwn= EA(r)wn. We may write (1 w)a1 a r-wn ) -var)1( + I(Y(rw) E a nrnwn The hypothesis gives that (I rw) 1E a rnwn has nonnegative power n series coefficients for r > 0. Now we need only show that h(w; r) = (1 w)-a-I(, rw)P+l has nonnegative power series coefficients for r in the given interval. Taking logs we get log h(w; r) = L(a + 1) (y + I)r n] _n and log h(w; r) then has nonnegative coefficients for 0 The same must be true of h(w; r) and so An(r) > 0 in this interval as claimed. BIBLIOGRAPHY 1. R. Askey and H. Pollard, Some absolutely monotonic and completely mono- tonic functions, SIAM. J. Math. Anal. 5 (1974). DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CINCINNATI, CINCINNATI, OHIO 45221 Received by the editors February 14, 1974. AMS (MOS) subject classifications (1970). Primary 40G05.