Extension of Inverses of Entire Functions of Genus 1 and 2 to the Upper Half Plane

Henrik Laurberg Pedersen

doi:10.1007/s40315-013-0020-4

Extension of Inverses of Entire Functions of Genus 1 and 2 to the Upper Half Plane

Henrik Laurberg Pedersen

Institut for Matematiske Fag

1 Citationer (Scopus)

Abstract

Any entire function of genus 1 which is positive on the positive real axis and which has only negative zeros decreases on some unbounded interval of the positive axis. The inverse of its reciprocal is shown to have an extension from that interval to a Pick-function in the upper half plane. A similar result holds for a class of entire functions of genus 2 and in particular the inverse function of Barnes’ double gamma function on a certain interval of the positive axis can be extended to a Pick-function. These results are proved using positive and negative definite kernels.

Originalsprog	Engelsk
Tidsskrift	Computational Methods and Function Theory
Vol/bind	13
Udgave nummer	2
Sider (fra-til)	263-275
ISSN	1617-9447
DOI	https://doi.org/10.1007/s40315-013-0020-4
Status	Udgivet - aug. 2013

Adgang til dokumentet

10.1007/s40315-013-0020-4

Citationsformater

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AB - Any entire function of genus 1 which is positive on the positive real axis and which has only negative zeros decreases on some unbounded interval of the positive axis. The inverse of its reciprocal is shown to have an extension from that interval to a Pick-function in the upper half plane. A similar result holds for a class of entire functions of genus 2 and in particular the inverse function of Barnes’ double gamma function on a certain interval of the positive axis can be extended to a Pick-function. These results are proved using positive and negative definite kernels.

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DO - 10.1007/s40315-013-0020-4

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