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For $F(x)={}_2F_1 (a,b;c;x)$, with $c=a+b$, $a>0$, $b>0$, it has been proved in [1] that $\log F(x)$ is convex on $(0,1)$.

I numerically checked that with a variety of $a,\ b$ values, $\log F(x)$ is not only convex, but also has a Taylor series in x consisting of strictly positive coefficients. Can this be proved?

[1] Generalized convexity and inequalities, Anderson, Vamanamurthy, Vuorinen, Journal of Mathematical Analysis and Applications, Volume 335, Issue 2,

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Please note the paper, may be it will be useful:

D. Karp, S.M. Sitnik, Log-convexity and log-concavity of hypergeometric-like functions, Journal of Mathematical Analysis and Applications, Volume 364, Issue 2, P. 384-394.

There is some general result in this paper on positive Taylor coefficients.

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