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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(x)=F(1/2,-1/2;1;x)-A(x),

where A(x) is an algebraic function (depending on the approximation chosen) of the parameter x(0,1) 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:
26D07Inequalities involving other types of real functions
33C05Classical hypergeometric functions, 2 F 1
33C75Elliptic integrals as hypergeometric functions
41A30Approximation by other special function classes