# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Inequalities for the perimeter of an ellipse. (English) Zbl 0985.26009

The authors describe a method to study whether an algebraic approximation to the perimeter of an ellipse is from above or below. By the representation of the perimeter in terms of hypergeometric functions the problem boils down to establishing the sign of the error

$E\left(x\right)=F\left(1/2,-1/2;1;x\right)-A\left(x\right),$

where $A\left(x\right)$ is an algebraic function (depending on the approximation chosen) of the parameter $x\in \left(0,1\right)$ related to the eccentricity of the ellipse. This problem can be tackled analyzing the sign of a series whose entries are all $>0$ starting from a sufficiently large index. Thus, the question is reduced to the sign of a polynomial given by the sum of a finite number of terms of the series. In the situation described its coefficients are integers, and we can apply a Sturm sequence argument with the aid of a computer algebra system performing integer arithmetics.

In this way, the authors show that several classical formulas approximate the elliptical perimeter from below, proving in particular a conjecture by Vuorinen on a Muir’s formula.

##### MSC:
 26D07 Inequalities involving other types of real functions 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$ 33C75 Elliptic integrals as hypergeometric functions 41A30 Approximation by other special function classes