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

Department of Mathematical Sciences

1 Citation (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.

Original language	English
Journal	Computational Methods and Function Theory
Volume	13
Issue number	2
Pages (from-to)	263-275
ISSN	1617-9447
DOIs	https://doi.org/10.1007/s40315-013-0020-4
Publication status	Published - Aug 2013

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10.1007/s40315-013-0020-4

Cite this

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title = "Extension of Inverses of Entire Functions of Genus 1 and 2 to the Upper Half Plane",

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{\textquoteright} 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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T1 - Extension of Inverses of Entire Functions of Genus 1 and 2 to the Upper Half Plane

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N2 - 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.

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

M3 - Journal article

SN - 1617-9447

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EP - 275

JO - Computational Methods and Function Theory

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