Sharp bounds for gamma function in terms of
Section snippets
Introduction and motivation
The problem of approximating the factorial function and its extension gamma function to positive real numbershas been intensively studied in the recent past. There were discovered many new, increasingly accurate approximation formulas, but a sacrifice of simplicity. The latter trend is to present estimates having a simple form.
In this paper we refer to the estimates of the formand others related.
Anderson and Qiu [4, Theorem 1.5] proved the following inequality
An asymptotic series
We solve in this section the open problem posed in [6] on the function defined by (1.6). This conjecture asks for finding the general term in the asymptotic expansion (1.5). The first coefficients were computed using Maple software and the results are the following (see [6, Rel. 5]):
In order to solve this problem, we use the standard asymptotic expansion of the gamma function as :
Monotonicity and convexity arguments and applications
Our new results announced in the first section are mainly established using a result of Alzer [2, Theorem 8], who proved that for every integers m, , the functionsandare completely monotonic on .
From and , we get
From and , we deducewhile and
Acknowledgments
This work was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS – UEFISCDI, Project number PN-II-ID-PCE-2011-3-0087. Computations made in this paper were performed using Maple software.
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