Geodesic
Normalized incomplete beta function: log-concavity in parameters and other properties
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 30, Tome 440 (2015), pp. 138-161

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Logarithmic concavity/convexity in parameters of the normalized incomplete beta function has been demonstrated by Finner and Roters in 1997 as a corollary of a rather difficult result based on generalized reproductive property of certain distributions. In the first part of this paper we give a direct analytic proof of the logarithmic concavity/convexity mentioned above. In the second part, we strengthen these results by proving that power series coefficients of the generalized Turán determinants formed by the parameter shifts of the normalized incomplete beta function have constant sign under some additional restrictions. Our method also leads to various other new facts which may be of independent interest. In particular, we establish linearization formulas and two-sided bounds for the above mentioned Turán determinants. Further, we find two identities of combinatorial type which we believe to be new. 
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     author = {D. B. Karp},
     title = {Normalized incomplete beta function: log-concavity in parameters and other properties},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {138--161},
     publisher = {mathdoc},
     volume = {440},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_440_a9/}
}
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D. B. Karp. Normalized incomplete beta function: log-concavity in parameters and other properties. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 30, Tome 440 (2015), pp. 138-161. http://geodesic.mathdoc.fr/item/ZNSL_2015_440_a9/