A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions

Harald Bohr and Johannes Mollerup characterized the Euler gamma function in 1922 using its log-convexity property. Emil Artin derived the basic properties of the gamma function using elementary methods of calculus. Nicolas Bourbaki adopted Bohr-Mollerup's theorem as the starting point for his exposition of the gamma function. This open access book develops a generalization of Bohr-Mollerup's theorem to higher order convex functions, showing that a rich spectrum of functions satisfy analogues of classical properties of the gamma function, including Bohr-Mollerup's theorem, Euler's reflection formula, Gauss's multiplication theorem, Stirling's formula, and Weierstrass's canonical factorization. The theory is illustrated through various examples, ranging from the gamma function and its variants to important special functions. This volume also honors the 100th anniversary of Bohr-Mollerup's theorem and sparks the interest of researchers in this beautiful theory.

Format: Hardback
Length: 323 pages
Publication date: 07 July 2022
Publisher: Springer Nature Switzerland AG

Harald Bohr and Johannes Mollerup made a significant breakthrough in 1922 by characterizing the Euler gamma function through its log-convexity property. A decade later, Emil Artin delved into this result and utilized it to derive fundamental properties of the gamma function using elementary calculus techniques. Nicolas Bourbaki then embraced Bohr-Mollerup's theorem as the foundation for his comprehensive exposition of the gamma function.

This open-access book extends Bohr-Mollerup's theorem to higher-order convex functions, building upon the pioneering work of Wolfgang Krull, Roger Webster, and others. It demonstrates, through elementary methods, that a wide range of functions satisfy analogues of several classical properties of the gamma function, including Bohr-Mollerup's theorem itself, Euler's reflection formula, Gauss's multiplication theorem, Stirling's formula, and Weierstrass's canonical factorization. The theory developed in this work is showcased through various examples, encompassing the gamma function and its variants, as well as important special functions such as the Hurwitz zeta function and the generalized Stieltjes constants.

Furthermore, this volume serves as a tribute to the 100th anniversary of Bohr-Mollerup's theorem, aimed at captivating a significant number of researchers in this captivating theory.

Weight: 682g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030950873
Edition number: 1st ed. 2022