Inequalities for a symmetric elliptic integral

Carlson, B. C.

doi:10.1090/S0002-9939-1970-0257412-X

Inequalities for a symmetric elliptic integral
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by B. C. Carlson

Proc. Amer. Math. Soc. 25 (1970), 698-703

DOI: https://doi.org/10.1090/S0002-9939-1970-0257412-X

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Abstract:

Inequalities are found for an incomplete elliptic integral of the first kind which represents the reciprocal of the capacity of an ellipsoid with semiaxes $x,\;y,\;z$. One sequence of symmetric algebraic functions of $x,\;y,\;z$ converges to the value of the integral from below and two from above. Among the elements of these sequences are upper and lower approximations due to Pólya and Szegö.

References

G. Pólya and G. Szegö, Inequalities for the capacity of a condenser, Amer. J. Math. 67 (1945), 1–32. MR 11871, DOI 10.2307/2371912
B. C. Carlson, Normal elliptic integrals of the first and second kinds, Duke Math. J. 31 (1964), 405–419. MR 164067

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Sur un théorème de M. Liapounoff, relalif à l’équilibre d’une masse fluide

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D. G. Zill and B. C. Carlson, Symmetric elliptic integrals of the third kind, Math. Comp. 24 (1970), 199–214. MR 262553, DOI 10.1090/S0025-5718-1970-0262553-5

Inequality of mixed arithmetic and geometric means

B. C. Carlson, Some inequalities for hypergeometric functions, Proc. Amer. Math. Soc. 17 (1966), 32–39. MR 188497, DOI 10.1090/S0002-9939-1966-0188497-6
B. C. Carlson, Some series and bounds for incomplete elliptic integrals, J. Math. and Phys. 40 (1961), 125–134. MR 130398
B. C. Carlson, Hidden symmetries of special functions, SIAM Rev. 12 (1970), 332–345. MR 262554, DOI 10.1137/1012078

Inequalities

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