Directional Convexity and Characterizations of Beta and Gamma Functions

Martin Himmel; Janusz Matkowski

doi:10.68381/jca25060

Abstract

The logarithmic convexity of restrictions of the Beta function to rays parallel to the main diagonal and the functional equation

\varphi (x+1) = \frac{x(x+k)}{(2x+k+1)(2x+k)}\, \phi(x),\ \ x>0,

for

k>0

allow to get a characterization of the Beta function. This fact and the notion of the beta-type function lead to a new characterization of the Gamma function.

Author information

Contact details are reproduced from the original publication and may be historical.

Martin Himmel

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4A, 65-516 Zielona Góra, Poland

himmel@mathematik.uni-mainz.de

Janusz Matkowski

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4A, 65-516 Zielona Góra, Poland

j.matkowski@wmie.uz.zgora.pl

Keywords

Gamma function
Beta function
beta-type function
logarithmical convexity
geometrical convexity
directional convexity
functional equation

Mathematics Subject Classification

33B15, 26B25, 39B22

References

J. Aczél: Lectures on Functional Equations and their Applications, Academic Press, New York (1966).Google Scholar
H. Alzer, J. Matkowski: Convexity property and a new characterization of Euler’s Gamma function, Arch. Math. (Basel) 100(2) (2013) 131–137.https://doi.org/10.1007/s00013-013-0487-2 Google Scholar
H. Bohr, J. Mollerup: Lærebog i Matematisk Analyse III, Grænseprocessor, Jul. Gjellerup, København (1922).Google Scholar
D. S. Dragomir, R. P. Agarwal, N. S. Barnett: Inequalities for Beta and Gamma functions via some classical and new integral inequalities, J. Inequalities Appl. 5(2) (2000) 103–165.https://doi.org/10.1155/S1025583400000084 Google Scholar
W. Krull: Bemerkungen zur Differenzengleichung g(x + 1) − g(x) = F (x) I, II, Math. Nachr. 1 (1948) 365–376; 2 (1949) 251–262.https://doi.org/10.1002/mana.19490020503 Google Scholar
D. Gronau, J. Matkowski: Geometrical convexity and generalization of the Bohr-Mollerup theorem on the Gamma function, Math. Pannonica 4/2 (1993) 153–160.Google Scholar
D. Gronau, J. Matkowski: Geometrically convex solutions of certain difference equation and generalized Bohr-Mollerup type theorems, Results in Math. 26 (1994) 290–297.https://doi.org/10.1007/BF03323051 Google Scholar
W. Hengartner, G. Schober: A remark on level curves for domains convex in one direction, Applicable Anal. 3 (1973) 101–106.https://doi.org/10.1080/00036817308839059 Google Scholar
G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities, Second Edition, Cambridge University Press, Cambridge (1952).Google Scholar
M. Kuczma: Functional Equations in a Single Variable, Monografie Matematyczne 46, PWN, Warszawa (1968).Google Scholar
M. Kuczma: An Introduction to the Theory of Functional Equations and Inequalities, Uniwersytet Śląski, PWN, Warszawa (1985).Google Scholar
J. Matoušek: On directional convexity, Discrete Comput. Geom. 25(3) (2001) 389–403.https://doi.org/10.1007/s004540010069 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2018. Rights now held by Banach Press.

[1] J. Aczél: Lectures on Functional Equations and their Applications, Academic Press, New York (1966).Google Scholar

[2] H. Alzer, J. Matkowski: Convexity property and a new characterization of Euler’s Gamma function, Arch. Math. (Basel) 100(2) (2013) 131–137.https://doi.org/10.1007/s00013-013-0487-2 Google Scholar

[3] H. Bohr, J. Mollerup: Lærebog i Matematisk Analyse III, Grænseprocessor, Jul. Gjellerup, København (1922).Google Scholar

[4] D. S. Dragomir, R. P. Agarwal, N. S. Barnett: Inequalities for Beta and Gamma functions via some classical and new integral inequalities, J. Inequalities Appl. 5(2) (2000) 103–165.https://doi.org/10.1155/S1025583400000084 Google Scholar

[5] W. Krull: Bemerkungen zur Differenzengleichung g(x + 1) − g(x) = F (x) I, II, Math. Nachr. 1 (1948) 365–376; 2 (1949) 251–262.https://doi.org/10.1002/mana.19490020503 Google Scholar

[6] D. Gronau, J. Matkowski: Geometrical convexity and generalization of the Bohr-Mollerup theorem on the Gamma function, Math. Pannonica 4/2 (1993) 153–160.Google Scholar

[7] D. Gronau, J. Matkowski: Geometrically convex solutions of certain difference equation and generalized Bohr-Mollerup type theorems, Results in Math. 26 (1994) 290–297.https://doi.org/10.1007/BF03323051 Google Scholar

[8] W. Hengartner, G. Schober: A remark on level curves for domains convex in one direction, Applicable Anal. 3 (1973) 101–106.https://doi.org/10.1080/00036817308839059 Google Scholar

[9] G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities, Second Edition, Cambridge University Press, Cambridge (1952).Google Scholar

[10] M. Kuczma: Functional Equations in a Single Variable, Monografie Matematyczne 46, PWN, Warszawa (1968).Google Scholar

[11] M. Kuczma: An Introduction to the Theory of Functional Equations and Inequalities, Uniwersytet Śląski, PWN, Warszawa (1985).Google Scholar

[12] J. Matoušek: On directional convexity, Discrete Comput. Geom. 25(3) (2001) 389–403.https://doi.org/10.1007/s004540010069 Google Scholar