D. C. Spencer

On finitely mean valent functions. II

for all R>0, where p is a positive number (not necessarily integral), we shall say that/(z) is p mean valent (p.m.v.)(1). This paper is a sequel to one of the same title to appear shortly in the Proceedings of the London Mathematical Society(2) in which I have shown that many of the known theorems concerning p-valent functions may be extended to the wider class of p.m.v. functions. I discuss here the behavior of p.m.v. functions on paths tending to points on the circumference | z| =1. The theorems which I discuss here remain true under hypotheses somewhat less restrictive than the one stated above. For example, the hypothesis that W(R) ^pwR2 only for R^RQ>0 would suffice (constants now depending on Ro as well as p). Furthermore, slightly less precise versions of the theorems (with p replaced by p + e) could be stated subject to the still weaker condition that W(R) lim sup-S p. ß->» irR2

On a Hardy-Littlewood problem of diophantine approximation

1. Let . When r is a positive integer, various writers have considered sums of the form where ω 1 and ω 2 are two positive numbers whose ratio θ = ω 1 /ω 2 is irrational and ξ is a real number satisfying 0 ≤ ξ 1 . In particular, Hardy and Littlewood ( 2,3,4 ), Ostrowski( 9 ), Hecke( 6 ), Behnke( 1 ), and Khintchine( 7 ) have given best possible approximations for sums of this type for various classes of irrational numbers. Most writers have confined themselves to the case r = 1, in which

On distortion in pseudo-conformal mapping

1.1. Suppose that the functions wk(zi, z2), k = \, 2(x), are regular in a four-dimensional domain Sßi of the complex variables Z\, z2. The transformation w of S81 into a domain 332 by a pair of functions Wk, for which d(wi, w2)/d(zi, z2) does not vanish identically, is called a PT (pseudo-conformal transformation). We are here concerned with general PTs in which the mapping is not necessarily one-one with respect to the schlicht space of the variables. Suppose that Si is a fixed schlicht domain in the space (21, s2), which contains the point (0, 0) in its interior. Let e = e(»i)