## The principal inverse of the gamma function

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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 $\Gamma (x)|_{(\alpha , \infty )}$ the principal inverse and denote it by $\Gamma ^{-1}(x)$. We show that $\Gamma ^{-1}(x)$ has the holomorphic extension $\Gamma ^{-1}(z)$ to $\textbf {C}\setminus (-\infty , \Gamma (\alpha )]$, which maps the upper half-plane into itself, namely a Pick function, and that $\Gamma (\Gamma ^{-1}(z))= z$ on $\textbf {C}\setminus (-\infty , \Gamma (\alpha )].$## References

- Horst Alzer and Christian Berg,
*Some classes of completely monotonic functions. II*, Ramanujan J.**11**(2006), no. 2, 225–248. MR**2267677**, DOI 10.1007/s11139-006-6510-5 - Christian Berg and Henrik L. Pedersen,
*Pick functions related to the gamma function*, Rocky Mountain J. Math.**32**(2002), no. 2, 507–525. Conference on Special Functions (Tempe, AZ, 2000). MR**1934903**, DOI 10.1216/rmjm/1030539684 - Christian Berg,
*On powers of Stieltjes moment sequences. I*, J. Theoret. Probab.**18**(2005), no. 4, 871–889. MR**2289936**, DOI 10.1007/s10959-005-7530-6 - Rajendra Bhatia,
*Matrix analysis*, Graduate Texts in Mathematics, vol. 169, Springer-Verlag, New York, 1997. MR**1477662**, DOI 10.1007/978-1-4612-0653-8 - Rajendra Bhatia and Takashi Sano,
*Loewner matrices and operator convexity*, Math. Ann.**344**(2009), no. 3, 703–716. MR**2501306**, DOI 10.1007/s00208-008-0323-3 - A. Korányi,
*On a theorem of Löwner and its connections with resolvents of selfadjoint transformations*, Acta Sci. Math. (Szeged)**17**(1956), 63–70. MR**82656** - Karl Löwner,
*Über monotone Matrixfunktionen*, Math. Z.**38**(1934), no. 1, 177–216 (German). MR**1545446**, DOI 10.1007/BF01170633 - William F. Donoghue Jr.,
*Monotone matrix functions and analytic continuation*, Die Grundlehren der mathematischen Wissenschaften, Band 207, Springer-Verlag, New York-Heidelberg, 1974. MR**0486556** - Carl H. Fitzgerald,
*On analytic continuation to a schlicht function*, Proc. Amer. Math. Soc.**18**(1967), 788–792. MR**219712**, DOI 10.1090/S0002-9939-1967-0219712-9 - Roger A. Horn,
*Schlicht mappings and infinitely divisible kernels*, Pacific J. Math.**38**(1971), 423–430. MR**310208** - Roger A. Horn and Charles R. Johnson,
*Topics in matrix analysis*, Cambridge University Press, Cambridge, 1991. MR**1091716**, DOI 10.1017/CBO9780511840371 - Marvin Rosenblum and James Rovnyak,
*Hardy classes and operator theory*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. Oxford Science Publications. MR**822228** - Mitsuru Uchiyama,
*Operator monotone functions, positive definite kernel and majorization*, Proc. Amer. Math. Soc.**138**(2010), no. 11, 3985–3996. MR**2679620**, DOI 10.1090/S0002-9939-10-10386-4

## Additional Information

**Mitsuru Uchiyama**- Affiliation: Department of Mathematics, Interdisciplinary Faculty of Science and Engineering, Shimane University, Matsue City, Shimane 690-8504, Japan
- MR Author ID: 198919
- Email: uchiyama@riko.shimane-u.ac.jp
- 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
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**140**(2012), 1343-1348 - MSC (2010): Primary 33B15; Secondary 26A48, 47A63
- DOI: https://doi.org/10.1090/S0002-9939-2011-11023-2
- MathSciNet review: 2869117