Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

# Simple Search

Query:
Enter a query and click »Search«...
Format:
Display: entries per page entries
Zbl 0890.33004
Balasubramanian, R.; Ponnusamy, S.; Vuorinen, M.
Functional inequalities for the quotients of hypergeometric functions.
(English)
[J] J. Math. Anal. Appl. 218, No.1, 256-268 (1998). ISSN 0022-247X

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 $b \in (0,2-a)$, $${F(a,b;a+b;r^2) \over F(a,b;a+b;1-r^2)} + {F(a,b;a+b;s^2) \over F(a,b;a+b;1-s^2)} \geq 2{F(a,b;a+b;rs) \over F(a,b;a+b;1-rs)}. .$$
[D.M.Bressoud (St.Paul)]
MSC 2000:
*33C20 Generalized hypergeometric series
33E05 Elliptic functions and integrals

Citations: Zbl 0692.33001

Cited in: Zbl 0946.33003

Highlights
Master Server