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A generalization of Bohr-Mollerup's theorem for higher order convex functions

Record Type:	Electronic resources : Monograph/item
Title/Author:	A generalization of Bohr-Mollerup's theorem for higher order convex functionsby Jean-Luc Marichal, Naim Zenaidi.
Author:	Marichal, Jean-Luc.
other author:	Zenaidi, Naim.
Published:	Cham :Springer International Publishing :2022.
Description:	xviii, 323 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Convex functions.
Online resource:	https://doi.org/10.1007/978-3-030-95088-0
ISBN:	9783030950880$q(electronic bk.)

A generalization of Bohr-Mollerup's theorem for higher order convex functions
Marichal, Jean-Luc.

A generalization of Bohr-Mollerup's theorem for higher order convex functions[electronic resource] /by Jean-Luc Marichal, Naim Zenaidi. - Cham :Springer International Publishing :2022. - xviii, 323 p. :ill., digital ;24 cm. - Developments in mathematics,v. 702197-795X ;. - Developments in mathematics ;v.25..

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.

Open access.

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.

ISBN: 9783030950880$q(electronic bk.)

Standard No.: 10.1007/978-3-030-95088-0doiSubjects--Topical Terms:

191128
Convex functions.

LC Class. No.: QA331.5 / .M37 2022

Dewey Class. No.: 515.8

A generalization of Bohr-Mollerup's theorem for higher order convex functions
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505 0 $a 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.
506 $a Open access.
520 $a 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.
650 0 $a Convex functions. $3 191128
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