Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-27T12:37:03.579Z Has data issue: false hasContentIssue false
  • English
  • Français

Sharp inequalities for the complete elliptic integral of the first kind

Published online by Cambridge University Press:  01 September 1998

HORST ALZER
HORST ALZER
Affiliation:
Morsbacher Str. 10, 51545 Waldbröl, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

Let

formula here

be the complete elliptic integral of the first kind. Recently, S.-L. Qiu and M. K. Vamanamurthy have proved for all r∈(0, 1):

formula here

with r′=√1−r2. In this paper we establish that the converse inequality

formula here

holds for all r∈(0, 1). Moreover, we show that the constant factors ¼ and π/(4 log (2))−1 in (a 1) and (a 2) are best possible.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 124 , Issue 2 , September 1998 , pp. 309 - 314
Copyright
Cambridge Philosophical Society 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)