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Record Nr. |
UNINA9910583594703321 |
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Autore |
Marichal Jean-Luc |
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Titolo |
A generalization of Bohr-Mollerup's theorem for higher order convex functions / / Jean-Luc Marichal, Naïm Zenaïdi |
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Pubbl/distr/stampa |
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Cham, : Springer Nature, 2022 |
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Cham : , : Springer International Publishing AG, , 2022 |
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©2022 |
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ISBN |
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Descrizione fisica |
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1 online resource (xviii, 323 pages) |
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Collana |
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Developments in mathematics ; v.70 |
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Altri autori (Persone) |
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Soggetti |
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Convex functions |
Gamma functions |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Nota di contenuto |
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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. Defining 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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Sommario/riassunto |
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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 |
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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. |
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