Normalized Incomplete Beta Function: Log-Concavity in Parameters and Other Properties
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The logarithmic concavity/convexity in parameters of the normalized incomplete beta function has been established 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, a direct analytic proof of the logarithmic concavity/convexity mentioned above is presented. These results are strengthened in the second part, where it is proved that the 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. The method of proof suggested also leads to various other new facts, which may be of independent interest. In particular, linearization formulas and two-sided bounds for the above-mentioned Turán determinants, and also two identities of combinatorial type, which we believe to be new, are established.
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- 3.G. E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press (1999).Google Scholar
- 8.A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis, 2nd edition, CRC Press (2003).Google Scholar
- 9.S. Das Gupta and S. K. Sarkar, “On TP 2 and log-concavity,” in: Inequalities in Statistics and Probability, Ed. by Y. L. Tong, IMS, Hayward, CA (1984), pp. 54–58.Google Scholar
- 11.S. I. Kalmykov and D. B. Karp, On the logarithmic concavity of series in gamma ratios, Izv. VUZov, Matematika, No. 6, 70–77 (2014).Google Scholar
- 13.D. S. Mitrinović, J. E. Pecarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers (1993).Google Scholar
- 14.J. E. Peˇcarić, F. Prosch, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press (1993).Google Scholar
- 15.A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Volume 3: More Special Functions, Gordon and Breach Science Publishers (1990).Google Scholar