Vol. 27, No. 3, 1968

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Pick’s conditions and analyticity

Alan Carleton Hindmarsh

Vol. 27 (1968), No. 3, 527–531

Let w(z) be a function in the open upper half plane (UHP) with values in UHP, and let Pn = (dij) be the n × n matrix of difference quotients

dij =   zi − zj

formed from any n points z1,z2,,zn UHP. It was shown by G. Pick that if w(z) is also analytic in UHP, then the Pn are all nonnegative definite Hermitian matrices (denoted Pn 0). In what follows, two converse results are derived.

(1) If D is a domain in UHP, w(z) is continuous in D and has values in UHP, and P3 0 for all choices of the z1,z2,z3 D, then w(z) is analytic in D. It is well known that the condition P2 0 does not imply anything of this sort, but corresponds only to a distance-shrinking property of w(z) in the noneuclidean geometry of UHP.

(2) If w is as before, but Pn 0 for all n and all z1,,zn D, i.e., {w(z) w(ζ)}(z ζ) is a nonnegative definite kernel in D, then w(z) is analytic in D and has an analytic extension to UHP whose values are in UHP.

Mathematical Subject Classification
Primary: 30.28
Received: 6 March 1968
Published: 1 December 1968
Alan Carleton Hindmarsh