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

Balasubramanian, R.; Ponnusamy, S.; Vuorinen, M.

J. Math. Anal. Appl. 218, No. 1, 256-268 (1998).

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 \(b \in (0,2-a)\), \[ {F(a,b;a+b;r^2) \over F(a,b;a+b;1-r^2)} + {F(a,b;a+b;s^2) \over F(a,b;a+b;1-s^2)} \geq 2{F(a,b;a+b;rs) \over F(a,b;a+b;1-rs)}. . \]

Reviewer: D.M.Bressoud (St.Paul)

MSC:

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

Citations:

Zbl 0692.33001

References:

[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) · Zbl 0722.33009
[5]	Bateman, H., (Erdelyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions (1953), McGraw-Hill: McGraw-Hill New York) · Zbl 0052.29502
[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