arXiv subjects
Publications and source records attributed to Stamatis Koumandos.
We prove that certain functions involving ratios of Gamma functions and the Psi-function belong to generalized Bernstein classes and new properties of generalized Bernstein functions are given.
We establish sharp inequalities involving the incomplete Beta and Gamma functions. These inequalities arise in the approximation of generalized Bernstein functions by higher order Thorin-Bernstein functions. Furthermore, new properties of a related function, namely $x^λΓ(x)/Γ(x+λ)$ are derived.
We investigate subclasses of generalized Bernstein functions related to complete Bernstein and Thorin-Bernstein functions. Representations in terms of incomplete beta and gamma as well as hypergeometric functions are presented. Several special cases and examples are discussed.
We show that a large collection of special functions, in particular Nielsen's beta function, are generalized Stieltjes functions of order 2, and therefore logarithmically completely monotonic. This includes the Laplace transform of functions of the form $xf(x)$, where $f$ is itself the Laplace transform of a sum of dilations and translations of periodic functions. Our methods are also applied to ratios of Gamma functions, and to the remainders in asymptotic expansions of the double Gamma function of Barnes.
It is shown that a function $f$ is a generalized Stieltjes function of order $λ>0$ if and only if $x^{1-λ}(x^{λ-1+k}f(x))^{(k)}$ is completely monotonic for all $k\geq 0$, thereby complementing a result due to Sokal. Furthermore, a characterization of those completely monotonic functions $f$ for which $x^{1-λ}(x^{λ-1+k}f(x))^{(k)}$ is completely monotonic for all $k\leq n$ is obtained in terms of properties of the representing measure of $f$.
We obtain necessary and sufficient conditions on a function in order that it be the Laplace transform of an absolutely monotonic function. Several closely related results are also given.
In this paper certain Turán type inequalities for some Lommel functions of the first kind are deduced. The key tools in our proofs are the infinite product representation for these Lommel functions of the first kind, a classical result of G. Pólya on the zeros of some particular entire functions, and the connection of these Lommel functions with the so-called Laguerre-Pólya class of entire functions. Moreover, it is shown that in some cases J. Steinig's results on the sign of Lommel functions of the first kind combined with the so-called monotone form of l'Hospital's rule can be used in the proof of the corresponding Turán type inequalities.
We introduce completely monotonic functions of order $r>0$ and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function give rise to completely monotonic functions of any positive integer order.