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Landen inequalities for zero-balanced hypergeometric functions. (English) Zbl 1241.26021
Summary: For zero-balanced Gaussian hypergeometric functions $F(a, b; a + b; x)$, $a, b > 0$, we determine maximal regions of $ab$ plane where well-known Landen identities for the complete elliptic integral of the first kind turn on respective inequalities valid for each $x \in (0, 1)$. Thereby an exhausting answer is given to the open problem from the work by {\it G. D. Anderson, M. K. Vamanamurthy}, and {\it M. Vuorinen} [Conformal invariants, inequalities, and quasiconformal maps. Wiley (1997; Zbl 0885.30012), p. 79].

MSC:
26D15Inequalities for sums, series and integrals of real functions
33C05Classical hypergeometric functions, ${}_2F_1$
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References:
[1] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 3: More Special Functions, Gordon & Breach, New York, NY, USA, 1988. · Zbl 0733.00005
[2] M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, NY, USA, 1965.
[3] R. Askey, “Ramanujan and hypergeometric and basic hypergeometric series,” Russian Mathematical Surveys, vol. 45, no. 1, pp. 37-86, 1990. · Zbl 0722.33009 · doi:10.1070/RM1990v045n01ABEH002325
[4] G. D. Anderson, M. K. Vamanamurthy, and M. K. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1997. · Zbl 0885.30012
[5] F. Belzunce, E.-M. Ortega, and J. M. Ruiz, “On non-monotonic ageing properties from the Laplace transform, with actuarial applications,” Insurance: Mathematics & Economics, vol. 40, no. 1, pp. 1-14, 2007. · Zbl 1273.91231 · doi:10.1016/j.insmatheco.2006.01.010
[6] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, UK, 4th edition, 1958. · Zbl 45.0433.02
[7] A. Baricz, “Landen-type inequality for Bessel functions,” Computational Methods and Function Theory, vol. 5, no. 2, pp. 373-379, 2005. · Zbl 1104.33002 · doi:10.1007/BF03321104
[8] A. Baricz, Generalized Bessel Functions of the First Kind, vol. 1994 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010. · Zbl 1203.33001
[9] G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy, and M. Vuorinen, “Inequalities for zero-balanced hypergeometric functions,” Transactions of the American Mathematical Society, vol. 347, no. 5, pp. 1713-1723, 1995. · Zbl 0826.33003 · doi:10.2307/2154966
[10] S.-L. Qiu and M. Vuorinen, “Landen inequalities for hypergeometric functions,” Nagoya Mathematical Journal, vol. 154, pp. 31-56, 1999. · Zbl 0944.33004
[11] A. Baricz, Email to the second author, June 2005.
[12] S. Ponnusamy and M. Vuorinen, “Asymptotic expansions and inequalities for hypergeometric functions,” Mathematika, vol. 44, no. 2, pp. 278-301, 1997. · Zbl 0897.33001 · doi:10.1112/S0025579300012602