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.
Abstract. 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.
Abstract. Let L(x) := x − Γ(x+t) Γ(x+s)x s−t+1 , where Γ(x) is Euler's gamma function. We determine conditions for the numbers s, t so that the functionis strongly completely monotonic on (0, ∞). Through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation. We also give several other examples of strongly completely monotonic functions defined in terms of Γ and ψ := Γ /Γ functions. Some limiting and particular cases are also considered.
Applying the Euler-Maclaurin summation formula, we obtain upper and lower polynomial bounds for the function x e x −1 , x > 0, with coefficients the Bernoulli numbers B k . This enables us to give simpler proofs of some results of H. Alzer and F. Qi et al., concerning complete monotonicity of certain functions involving the gamma function (x), the psi function ψ(x) and the polygamma functions ψ (n) (x), n = 1, 2, . . . .
Let m (x) = −x m (m) (x), where denotes the logarithmic derivative of Euler's gamma function. Clark and Ismail prove in a recently published article that if m ∈ {1, 2, . . . , 16}, then (m) m is completely monotonic on (0, ∞), and they conjecture that this is true for all natural numbers m. We disprove this conjecture by showing that there exists an integer m 0 such that for all m m 0 the function (m) m is not completely monotonic on (0, ∞).
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