Computational Methods and Function Theory 1 (2001), No. 1, 81--88
Copyright Heldermann Verlag 2001
Roger W. Barnard
barnard@math.ttu.edu, Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409-1042, U.S.A.
Kendall C. Richards
richardk@southwestern.edu, Department of Mathematics, Southwestern University, Georgetown, Texas 78626, U.S.A.
Recent efforts to obtain bounds for the complete elliptic integral $$ \frac{\pi}{2}\cdot\,_{2}F_{1}\left(-\frac{1}{2},\frac{1}{2};1;r^2\right) $$ in terms of power means and other related means have precipitated the search for similar bounds for the more general $\,_{2}F_{1}(\alpha,\beta;\gamma;r)$. In an early paper, B. C. Carlson considered the approximation of the {\em hypergeometric mean values} $(\,_{2}F_{1}(-a,b;b+c;r))^{1/a}$ in terms of {\em means of order} $t$, given by $M_t(s,r):= \{(1-s) +s(1-r)^t\}^{1/t}$. In this note, a refinement of one such approximation is established by first proving a general positivity result involving $\,{}_3F_2$.
Keywords: Hypergeometric function, generalized hypergeometric function, means of order $t$.
MSC 2000: 33C, 41A.
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