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Alzer, Horst; Qiu, Songliang

Monotonicity theorems and inequalities for the complete elliptic integrals. (English) Zbl 1059.33029

J. Comput. Appl. Math. 172, No. 2, 289-312 (2004).
Summary: We prove monotonicity properties of certain combinations of complete elliptic integrals of the first and second kind, \(\mathcal K\) and \(\mathcal E\). These results lead to sharp symmetrical bounds for \(\mathcal K\) and \(\mathcal E \), which improve recently discovered inequalities.

Cited in 113 Documents

MSC:

33E05 Elliptic functions and integrals
26D15 Inequalities for sums, series and integrals
33C05 Classical hypergeometric functions, \({}_2F_1\)

Keywords:

Complete elliptic integrals; Monotonicity; Inequalities; Mean values; Arc length of an ellipse
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\textit{H. Alzer} and \textit{S. Qiu}, J. Comput. Appl. Math. 172, No. 2, 289--312 (2004; Zbl 1059.33029)
Full Text: DOI

Digital Library of Mathematical Functions:

§19.9(i) Complete Integrals ‣ §19.9 Inequalities ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals

References:

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