Total Positivity of Mean Values and Hypergeometric Functions

@article{Carlson1983TotalPO,
  title={Total Positivity of Mean Values and Hypergeometric Functions},
  author={Bille C. Carlson and John L. Gustafson},
  journal={Siam Journal on Mathematical Analysis},
  year={1983},
  volume={14},
  pages={389-395}
}
The weighted power mean of two positive variables is strictly totally positive (STP) if its order t satisfies $ - \infty 0$, which is equivalent to ${}_2 F_1 $ with argument ${{1 - x} / y}$, is STP if $\alpha ,\beta \beta '$, and $\beta + \beta ' - \alpha $ are positive. With weaker restrictions this function is represented in a new way as a convolution. Higher order positivity is discussed for some other hypergeometric functions, including incomplete elliptic integrals. 
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