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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.

26A48Monotonic functions, generalizations (one real variable)
33E12Mittag-Leffler functions and generalizations
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
33C15Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$
33C05Classical hypergeometric functions, ${}_2F_1$
44A10Laplace transform
44A15Special transforms (Legendre, Hilbert, etc.)
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