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Absolutely and completely monotonic functions and sequences
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<p>In mathematics, the notions of an absolutely monotonic function and a completely monotonic function are two very closely related concepts. Both imply very strong monotonicity properties. Both types of functions have derivatives of all orders. In the case of an absolutely monotonic function, the function as well as its derivatives of all orders must be non-negative in its domain of definition which would imply that the function as well as its derivatives of all orders are monotonically increasing functions in the domain of definition. In the case of a completely monotonic function, the function and its derivatives must be alternately non-negative and non-positive in its domain of definition which would imply that function and its derivatives are alternately monotonically increasing and monotonically decreasing functions. </p><p>Such functions were first studied by S. Bernshtein in 1914 and the terminology is also due to him. There are several other related notions like the concepts of almost completely monotonic function, logarithmically completely monotonic function, strongly logarithmically completely monotonic function, strongly completely monotonic function and almost strongly completely monotonic function. Another related concept is that of a completely/absolutely monotonic sequence. This notion was introduced by Hausdorff in 1921. </p><p>The notions of completely and absolutely monotone function/sequence play an important role in several areas of mathematics. For example, in classical analysis they occur in the proof of the positivity of integrals involving Bessel functions or the positivity of Cesàro means of certain Jacobi series. Such functions occur in other areas of mathematics such as probability theory, numerical analysis, and elasticity. </p>