Functional inequalities for the quotients of hypergeometric functions. (English) Zbl 0890.33004

It was proven in [SIAM J. Math. Anal. 21, No. 2, 536-549 (1990; Zbl 0692.33001)] that for the complete elliptic integral of the first kind, \[ \int_0^{\pi/2} {d\theta \over \sqrt{1-r^2\sin^2\theta}} = {\pi \over 2} F\left( {1\over 2}, {1 \over 2}; 1; r^2 \right), \] if we define \[ \mu(r) = { F(1/2,1/2;1;1-r^2) \over F(1/2,1/2;1;r^2) }, \] then \(\mu(r) + \mu(s) \leq 2 \mu(\sqrt{rs})\) for all \(r,s \in (0,1)\). In this paper, the authors prove that for the function \[ m(r) = { F(a,1-a,1;1-r^2) \over F(a,1-a,1;r^2) },\quad 0<a<1, \] the same inequality, \(m(r) + m(s) \leq 2 m(\sqrt{rs})\), holds for all \(r,s \in (0,1)\). They also prove that for \(a \in (0,2)\) and \(b \in (0,2-a)\), \[ {F(a,b;a+b;r^2) \over F(a,b;a+b;1-r^2)} + {F(a,b;a+b;s^2) \over F(a,b;a+b;1-s^2)} \geq 2{F(a,b;a+b;rs) \over F(a,b;a+b;1-rs)}. . \]


33C20 Generalized hypergeometric series, \({}_pF_q\)
33E05 Elliptic functions and integrals


Zbl 0692.33001
Full Text: DOI


[1] Anderson, G. D.; Barnard, R. W.; Richards, K. C.; Vamanamurthy, M. K.; Vuorinen, M., Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc., 347, 1713-1723 (1995) · Zbl 0826.33003
[2] Anderson, G. D.; Vamanamurthy, M. K.; Vuorinen, M., Functional inequalities for complete elliptic integrals and their ratios, SIAM J. Math. Anal., 21, 536-549 (1990) · Zbl 0692.33001
[3] Anderson, G. D.; Vamanamurthy, M. K.; Vuorinen, M., Hypergeometric functions and elliptic integrals, (Srivastava, H. M.; Owa, S., Current Topics in Analytic Function Theory (1992), World Scientific: World Scientific Singapore/London), 48-85 · Zbl 0984.33502
[4] Askey, R., S. Ramanujan and hypergeometric and basic hypergeometric series, Uspekhi Mat. Nauk, 45, 33-76 (1990)
[5] Bateman, H., (Erdelyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions (1953), McGraw-Hill: McGraw-Hill New York)
[6] Berndt, B. C., Ramanujan’s Notebooks (1987), Springer-Verlag: Springer-Verlag New York · Zbl 0389.10002
[7] Berndt, B. C.; Bhargava, S.; Garvan, F. G., Ramanujan’s theories of elliptic functions to alternative bases, Trans. Amer. Math. Soc., 347, 4163-4244 (1995) · Zbl 0843.33012
[8] Borwein, J. M.; Borwein, P. B., Pi and the AGM (1987), Wiley · Zbl 0699.10044
[9] Lehto, O.; Virtanen, K. I., Quasiconformal Mappings in the Plane. Quasiconformal Mappings in the Plane, Grundlehren Math. Wiss., 126 (1973), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0267.30016
[10] Ponnusamy, S.; Vuorinen, M., Asymptotic expansions and inequalities for hypergeometric functions, Mathematika, 44, 43-64 (1997) · Zbl 0897.33001
[11] Whittaker, E. T.; Watson, G. N., A Course of Modern Analysis (1958), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0108.26903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.