A generalization of Bohr-Mollerup's theorem for higher order convex functions / Jean-Luc Marichal, Naïm Zenaïdi.
Marichal, Jean-Luc, author.; Zenaïdi, Naïm, author.
2022
QA331.5
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Title
A generalization of Bohr-Mollerup's theorem for higher order convex functions / Jean-Luc Marichal, Naïm Zenaïdi.
Author
Marichal, Jean-Luc, author.
ISBN
9783030950880 (electronic bk.)
3030950883 (electronic bk.)
9783030950873
DOI
https://doi.org/10.1007/978-3-030-95088-0
Published
Cham : Springer, [2022]
Copyright
©2022
Language
English
Description
1 online resource (xviii, 323 pages).
Item Number
10.1007/978-3-030-95088-0 doi
Call Number
QA331.5
Dewey Decimal Classification
515/.8
Summary
In 1922, Harald Bohr and Johannes Mollerup established a remarkable characterization of the Euler gamma function using its log-convexity property. A decade later, Emil Artin investigated this result and used it to derive the basic properties of the gamma function using elementary methods of the calculus. Bohr-Mollerup's theorem was then adopted by Nicolas Bourbaki as the starting point for his exposition of the gamma function. This open access book develops a far-reaching generalization of Bohr-Mollerup's theorem to higher order convex functions, along lines initiated by Wolfgang Krull, Roger Webster, and some others but going considerably further than past work. In particular, this generalization shows using elementary techniques that a very rich spectrum of functions satisfy analogues of several classical properties of the gamma function, including Bohr-Mollerup's theorem itself, Euler's reflection formula, Gauss' multiplication theorem, Stirling's formula, and Weierstrass' canonical factorization. The scope of the theory developed in this work is illustrated through various examples, ranging from the gamma function itself and its variants and generalizations (q-gamma, polygamma, multiple gamma functions) to important special functions such as the Hurwitz zeta function and the generalized Stieltjes constants. This volume is also an opportunity to honor the 100th anniversary of Bohr-Mollerup's theorem and to spark the interest of a large number of researchers in this beautiful theory.
Bibliography, etc. Note
Includes bibliographical references and index.
Access Note
Open access
Source of Description
Online resource; title from PDF title page (SpringerLink, viewed July 12, 2022).
Added Author
Zenaïdi, Naïm, author.
Series
Developments in mathematics ; v. 70. 2197-795X
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Preface
List of main symbols
Table of contents
Chapter 1. Introduction
Chapter 2. Preliminaries
Chapter 3. Uniqueness and existence results
Chapter 4. Interpretations of the asymptotic conditions
Chapter 5. Multiple log-gamma type functions
Chapter 6. Asymptotic analysis
Chapter 7. Derivatives of multiple log-gamma type functions
Chapter 8. Further results
Chapter 9. Summary of the main results
Chapter 10. Applications to some standard special functions
Chapter 11. Definining new log-gamma type functions
Chapter 12. Further examples
Chapter 13. Conclusion
A. Higher order convexity properties
B. On Krull-Webster's asymptotic condition
C. On a question raised by Webster
D. Asymptotic behaviors and bracketing
E. Generalized Webster's inequality
F. On the differentiability of \sigma_g
Bibliography
Analogues of properties of the gamma function
Index.

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