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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
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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

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