On the Modulus of Power Series of a Certain Type

Abstract

In a recent publication B. Kuttner [4] has proven that the functions QO f K (z) which are defined for | z | < l by £ (n-\-l) K z n have the property n=0 f K (z) ^ 0 for | z | < 1, 1 < K < 2. It is the purpose of this paper to give a new proof of this result. We obtain it as a special case of a theorem on the modulus of power series where the coefficients are totally monotone. This theorem also yields a generahzation of Kakeya's theorem [3] under the same assumption about the coefficients.1. We remark that the power series which are considered in what follows (with one exception in §3) have unique analytic extensions into the whole z-plane with the possible exception of z > 1 [and for/(z) = 2 a n z n , a n^Q , we will write /(l) = oo if 2 a n diverges]. A sequence {q n } is totally monotone if A k q n^Q for k, n = 0, 1, ... (Ag n = g n -q n+1 , A & q n = A*" 1 q n -A*" 1 q n+1 , k = 2,3,...), and by a theorem of Hausdorff [2] this is equivalent with a representation Jo g(t) monotonic increasing for 0 <£ < 1 (g(l) < oo) . THEOREM. J / / ( z ) = tt o + £ q n -i zn > a o rea^ an d {Qn} totally monotone, n = l then \f(re i0 )\^f(-r)pf(-l) for 0 <r < 1, 0 < Proof. We have for z = re ie f Jo cos 9-rt |o I-2rt cos d+{rt) 2 A straightforward calculation shows that cos 6-rt _ 1^-1 -r-1-2rtcosd+(rtf 1+rt 1+t' and the theorem follows immediately from this inequality.

Completely monotone sequences and harmonic mappings

“…In particular, if a + f (−1) ≥ 0, then we conclude that a + f (z) is non-vanishing on |z| < 1. If c = 0, this theorem reduces to the main result of [19].…”

Completely monotone sequences and harmonic mappings

“…In particular, if a + lim r→1 − f (−r) ≥ 0 and if f is non-constant, then we conclude that a + f (z) is non-vanishing on |z| < 1. If c = 0, this theorem reduces to the main result of [16].…”

On the Modulus of Power Series with Totally Monotone Coefficients

“…is positive in -1 < x < 1 not only for the special functions y(t), but also for any arbitrary increasing function s(t) with s(l) < oo. For this purpose we replace y(t) by s(t) in / 1} where s(t) is independent of A-, and regard I y as a function of A-, say I t (x But /i(i)*H(i-0 2 (<+' 2 )**(0) M(i-t) 2…”

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In a recent paper A. Peyerimhoff [2] showed that if/(z) = tfo + Sn^itfn-i 2 " where a 0 is real and {q n } a totally monotonet sequence then | z | = r^ 1 implies that |/(z)| ^ f ( -r). [This result is only significant if a 0 ^ 0, since otherwise/(-r) < 0.]

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