Vol. 17, No. 3, 1966

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ISSN: 0030-8730
Functions analytic in a finite disk and having asymptotically prescribed characteristic

Daniel Francis Shea, Jr.

Vol. 17 (1966), No. 3, 549–560

Let f(z) be analytic in the region |z| < R (R +). Then in the interval 0 r < R, Nevanlinna’s characteristic

         1--∫ 2π +    i𝜃
T (r,f) = 2π  0 log|f(re )|d𝜃

is known to be nonnegative, nondecreasing and convex in log r; however, it is not known whether these properties characterize completely T(r,f).

Recently, A. Edrei and W. H. J. Fuchs have investigated one aspect of this question; they have shown that if Λ(r) is an arbitrary convex function of log r defined for r0 r < +and such that log r = o(Λ(r)) as r +, then it is possible to find an entire function f(z) such that

T(r,f) ∽ Λ (r) (r → +∞ ),                  (A )

except possibly for values of r belonging to an exceptional set of finite measure. In this note I establish an analogue of this result for the case of functions regular in a disk of finite radius R.

Mathematical Subject Classification
Primary: 30.66
Received: 18 February 1965
Published: 1 June 1966
Daniel Francis Shea, Jr.