Inequalities for the Perimeter of an Ellipse

✍ Scribed by Roger W. Barnard; Kent Pearce; Lawrence Schovanec

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 96 KB
• Volume: 260
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2000.7128

✦ Synopsis

Functions in Geometric Function Theory" [6]

raised the question (Q1) Question. Is it true that A a b is an approximation to L a b / π a + b from below throughout the entire range of eccentricity e?

As is often the case for a mathematical conjecture, the insights gained in the course of its resolution and extended ramifications of the result supersede the original question. In this paper we develop a technique to answer Question (Q1) that readily extends to a broader class of inequalities, all of which are motivated by the problem of approximating the elliptical perimeter. The verification of each inequality is accomplished by showing the positivity of an infinite series. The proof of the positivity of the infinite series is achieved by utilizing a computer algebra system to execute a Sturm sequence argument.