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Completely monotonic functions. (English) Zbl 1035.26012
The paper deals with completely monotonic functions $f\left(x\right)$ defined on $\left(0,+\infty \right)$ and possessing derivatives ${f}^{\left(n\right)}\left(x\right)$ for all $n=0,1,2,\cdots$ such that ${\left(-1\right)}^{n}{f}^{\left(n\right)}\left(x\right)\ge 0$ for all $x\ge 0$. 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.
MSC:
 26A48 Monotonic functions, generalizations (one real variable) 33E12 Mittag-Leffler functions and generalizations 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 33C15 Confluent hypergeometric functions, Whittaker functions, ${}_{1}{F}_{1}$ 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$ 44A10 Laplace transform 44A15 Special transforms (Legendre, Hilbert, etc.)