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The principal inverse of the gamma function

Author: Mitsuru Uchiyama
Journal: Proc. Amer. Math. Soc. 140 (2012), 1343-1348
MSC (2010): Primary 33B15; Secondary 26A48, 47A63
Published electronically: August 3, 2011
MathSciNet review: 2869117
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Abstract: Let $ \Gamma(x)$ be the gamma function in the real axis and $ \alpha$ the maximal zero of $ \Gamma'(x)$. We call the inverse function of $ \Gamma (x)\vert _{(\alpha, \infty)}$ the principal inverse and denote it by $ \Gamma ^{-1}(x)$. We show that $ \Gamma ^{-1}(x)$ has the holomorphic extension $ \Gamma ^{-1}(z)$ to $ {\bf C}\setminus (-\infty, \Gamma(\alpha)]$, which maps the upper half-plane into itself, namely a Pick function, and that $ \Gamma (\Gamma^{-1}(z))= z$ on $ {\bf C}\setminus (-\infty, \Gamma(\alpha)].$

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  • 1. H. Alzer and C. Berg, Some classes of completely monotonic functions II, Ramanujan J. 11(2006)225-248. MR 2267677 (2007k:33001)
  • 2. C. Berg and H. L. Pedersen, Pick functions related to the gamma function, Rocky Mountain J. of Math. 32(2002)507-525. MR 1934903 (2004h:33007)
  • 3. C. Berg, On powers of Stieltjes moment sequences, I, J. Theoretical Probability 18(2005)871-889. MR 2289936 (2008i:44008)
  • 4. R. Bhatia, Matrix Analysis, Springer, 1996. MR 1477662 (98i:15003)
  • 5. R. Bahtia and T. Sano, Loewner matrices and operator convexity, Math. Ann. 344(2009)703-716. MR 2501306 (2010g:47037)
  • 6. A. Koranyi, On a theorem of Löwner and its connections with resolvents of selfadjoint transformations, Acta Sci. Math. 17(1956)63-70. MR 0082656 (18:588c)
  • 7. K. Löwner, Über monotone Matrixfunctionen, Math. Z. 38(1934)177-216. MR 1545446
  • 8. W.F.Donoghue, Monotone Matrix Functions and Analytic Continuation, Springer-Verlag, 1974. MR 0486556 (58:6279)
  • 9. C. H. Fitzgerald, On analytic continuation to a Schlicht function, Proc. Amer. Math. Soc. 18(1967)788-792. MR 0219712 (36:2791)
  • 10. R. A. Horn, Schlicht mapping and infinitely divisible kernels, Pacific J. of Math. 38(1971)423-430. MR 0310208 (46:9310)
  • 11. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991. MR 1091716 (92e:15003)
  • 12. M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford Univ. Press, 1985. MR 822228 (87e:47001)
  • 13. M. Uchiyama, Operator monotone functions, positive definite kernels and majorization, Proc. Amer. Math. Soc. 138(2010)3985-3996. MR 2679620

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Additional Information

Mitsuru Uchiyama
Affiliation: Department of Mathematics, Interdisciplinary Faculty of Science and Engineering, Shimane University, Matsue City, Shimane 690-8504, Japan

Keywords: Gamma function, Pick function, Nevanlinna function, Loewner kernel function
Received by editor(s): November 18, 2010
Received by editor(s) in revised form: January 4, 2011
Published electronically: August 3, 2011
Additional Notes: The author was supported in part by (JSPS) KAKENHI 21540181
Communicated by: Richard Rochberg
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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