RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Zap. Nauchn. Sem. POMI:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Zap. Nauchn. Sem. POMI, 2015, Volume 440, Pages 138–161 (Mi znsl6218)  

This article is cited in 1 scientific paper (total in 1 paper)

Normalized incomplete beta function: log-concavity in parameters and other properties

D. B. Karpab

a Far Eastern Federal University, 8 Sukhanova street, Vladivostok, 690950, Russia
b Institute of Applied Mathematics, FEBRAS, 7 Radio Street, Vladivostok, 690041, Russia

Abstract: 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.

Key words and phrases: incomplete beta function, Gauss hypergeometric function, log-concavity, combinatorial identity.

Full text: PDF file (215 kB)
References: PDF file   HTML file

English version:
Journal of Mathematical Sciences (New York), 2016, 217:1, 91–107

Bibliographic databases:

UDC: 517.58
Received: 21.09.2015
Language:

Citation: D. B. Karp, “Normalized incomplete beta function: log-concavity in parameters and other properties”, Analytical theory of numbers and theory of functions. Part 30, Zap. Nauchn. Sem. POMI, 440, POMI, St. Petersburg, 2015, 138–161; J. Math. Sci. (N. Y.), 217:1 (2016), 91–107

Citation in format AMSBIB
\Bibitem{Kar15}
\by D.~B.~Karp
\paper Normalized incomplete beta function: log-concavity in parameters and other properties
\inbook Analytical theory of numbers and theory of functions. Part~30
\serial Zap. Nauchn. Sem. POMI
\yr 2015
\vol 440
\pages 138--161
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6218}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3504464}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2016
\vol 217
\issue 1
\pages 91--107
\crossref{https://doi.org/10.1007/s10958-016-2958-z}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84978154010}


Linking options:
  • http://mi.mathnet.ru/eng/znsl6218
  • http://mi.mathnet.ru/eng/znsl/v440/p138

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. I. Kalmykov, D. B. Karp, “Inequalities for series in $q$-shifted factorials and $q$-gamma functions”, J. Math. Anal. Appl., 460:1 (2018), 332–351  crossref  mathscinet  zmath  isi  scopus
  • Записки научных семинаров ПОМИ
    Number of views:
    This page:95
    Full text:29
    References:19

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019