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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 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.
Convex functions. --- Gamma functions. --- Functions, Convex --- Functions of real variables --- Functions, Gamma --- Transcendental functions --- Difference Equation --- Higher Order Convexity --- Bohr-Mollerup's Theorem --- Principal Indefinite Sums --- Gauss' Limit --- Euler Product Form --- Raabe's Formula --- Binet's Function --- Stirling's Formula --- Euler's Infinite Product --- Euler's Reflection Formula --- Weierstrass' Infinite Product --- Gauss Multiplication Formula --- Euler's Constant --- Gamma Function --- Polygamma Functions --- Hurwitz Zeta Function --- Generalized Stieltjes Constants
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Inequalities appear in various fields of natural science and engineering. Classical inequalities are still being improved and/or generalized by many researchers. That is, inequalities have been actively studied by mathematicians. In this book, we selected the papers that were published as the Special Issue ‘’Inequalities’’ in the journal Mathematics (MDPI publisher). They were ordered by similar topics for readers’ convenience and to give new and interesting results in mathematical inequalities, such as the improvements in famous inequalities, the results of Frame theory, the coefficient inequalities of functions, and the kind of convex functions used for Hermite–Hadamard inequalities. The editor believes that the contents of this book will be useful to study the latest results for researchers of this field.
quantum estimates --- Montgomery identity --- power inequalities --- positive linear map --- Hilbert C*-module --- Hermite–Hadamard type inequality --- Steffensen’s inequality --- Hilbert space --- Hadamard fractional integrals --- K-dual --- adjointable operator --- analytic functions --- special means --- geometrically convex function --- h2)-convex --- proportional fractional derivative --- commutator --- quasi-convex --- Katugampola fractional integrals --- Euler-Maclaurin summation formula --- starlike functions --- strongly ?-convex functions --- g-frame --- interval-valued functions --- twice differentiable convex functions --- Taylor theorem --- exponential inequalities --- g-Bessel sequence --- Riemann–Liouville and Caputo proportional fractional initial value problem --- frame --- Fejér’s inequality --- weight function --- Hermite-Hadamard type inequalities --- Gronwall–Bellman inequality --- ?-variation --- Hölder’s inequality --- majorization inequality --- alternate dual frame --- half-discrete Hardy-Hilbert’s inequality --- parameter --- Power mean inequality --- Riemann–Liouville fractional integrals --- reverse inequality --- weaving frame operator --- Fink’s identity --- pseudo-inverse --- operator inequality --- Hermite-Hadamard inequality --- one-sided weighted Morrey space --- Green functions --- weaving K-frame --- operator Kantorovich inequality --- higher order convexity --- weaving frame --- (h1 --- one-sided weighted Campanato space --- Fekete-Szegö inequality --- convex functions --- refined inequality --- trigonometric inequalities --- one-sided singular integral
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