Completely monotonic functions. (English) Zbl 1035.26012
The paper deals with completely monotonic functions \(f(x)\) defined on \((0,+\infty)\) and possessing derivatives \(f^{(n)}(x)\) for all \(n=0,1,2,\dots \) such that \((-1)^{n}f^{(n)}(x)\geq 0\) for all \(x\geq 0\). Conditions are given when arithmetic operations, compositions and power series of functions and integral transforms with general kernel yield the completely monotonic functions. The results obtained are applied to establish the complete monotonicity for the confluent and Gauss hypergeometric functions, for functions of Bessel and Mittag-Leffler type, and for the one-dimensional Laplace, Stieltjes, Lambert and Meijer integral transforms.
Reviewer: Anatoliy Aleksandrovich Kilbas (Minsk)
MSC:
| 26A48 | Monotonic functions, generalizations |
| 33E12 | Mittag-Leffler functions and generalizations |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
| 33C15 | Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 44A10 | Laplace transform |
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
Keywords:
completely monotonic functions; hypergeometric functions; Bessel and Mittag-Leffler type functions; Laplace integral transforms; Stieltjes integral transforms; Lambert integral transforms; Meijer integral transformsReferences:
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