A superadditive property of Hadamard’s gamma function

Article

Abstract

Hadamard’s gamma function is defined by
$$H(x)=\frac{1}{\Gamma(1-x)}\frac{d}{dx}\log \frac{\Gamma(1/2-x/2)}{\Gamma(1-x/2)},$$
where Γ denotes the classical gamma function of Euler. H is an entire function, which satisfies H(n)=(n−1)! for all positive integers n. We prove the following superadditive property.
Let α be a real number. The inequality
$$H(x)+H(y)\leq H(x+y)$$
holds for all real numbers x,y with x,yα if and only if αα0=1.5031…. Here, α0 is the only solution of H(2t)=2H(t) in [1.5,∞).

Keywords

Hadamard’s and Euler’s gamma functions Psi function Superadditive Convex Inequalities 

Mathematics Subject Classification (2000)

33B15 39B62 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1965) Google Scholar
  2. 2.
    Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput. 66, 373–389 (1997) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alzer, H.: A power mean inequality for the gamma function. Monatsh. Math. 131, 179–188 (2000) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Alzer, H.: Mean-value inequalities for the polygamma functions. Aequ. Math. 61, 151–161 (2001) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Alzer, H., Ruscheweyh, S.: A subadditive property of the gamma function. J. Math. Anal. Appl. 285, 564–577 (2003) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Davis, P.J.: Leonhard Euler’s integral: a historical profile of the gamma function. Am. Math. Mon. 66, 849–869 (1959) MATHCrossRefGoogle Scholar
  7. 7.
    Gautschi, W.: The incomplete gamma function since Tricomi. In: Tricomi’s Ideas and Contemporary Applied Mathematics, Atti Convegni Lincei, vol. 147, pp. 203–237. Accad. Naz. Lincei, Rome (1998) Google Scholar
  8. 8.
    Luschny, P.: Is the gamma function misdefined? Or: Hadamard versus Euler—who found the better gamma function? http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html
  9. 9.
    Mijajlović, Ž., Malešević, B.: Differentially transcendental functions. arXiv:math.GM/0412354 v3, 9 Feb 2006
  10. 10.
    Newton, T.A.: Derivation of a factorial function by method of analogy. Am. Math. Mon. 68, 917–920 (1961) CrossRefGoogle Scholar
  11. 11.
    Sándor, J.: A bibliography on gamma functions: inequalities and applications. http://www.math.ubbcluj.ro/~jsandor/letolt/art42.pdf

Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer 2008

Authors and Affiliations

  1. 1.WaldbrölGermany

Personalised recommendations