Elsevier

Applied Mathematics and Computation

Volume 249, 15 December 2014, Pages 278-285
Applied Mathematics and Computation

Sharp bounds for gamma function in terms of xx-1

https://doi.org/10.1016/j.amc.2014.10.027Get rights and content

Abstract

The aim of this paper is to present new approximations for the gamma function in terms of xx-1. The monotonicity and convexity of some involved functions are studied. As applications, sharp bounds are presented. It is also solved an open problem posed by Mortici (2013).

Section snippets

Introduction and motivation

The problem of approximating the factorial function and its extension gamma function to positive real numbersΓ(x)=0tx-1e-tdthas 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 formΓ(x)xx-1,asxand 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 θ defined by (1.6). This conjecture asks for finding the general term ak in the asymptotic expansion (1.5). The first coefficients were computed using Maple software and the results are the following (see [6, Rel. 5]):a1=12ln2π-1a2=a3=12ln2π-1112a4=a5=12ln2π-331360a6=a7=12ln2π-463504a8=a9=12ln2π-46335040,etc.

In order to solve this problem, we use the standard asymptotic expansion of the gamma function as x:lnΓ(x)x-12lnx-x+

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, n1, the functionsFm(x)=lnΓ(x)-x-12lnx+x-12ln2π-i=12mB2i2i(2i-1)x2i-1andGn(x)=-lnΓ(x)+x-12lnx-x+12ln2π+i=12n-1B2i2i(2i-1)x2i-1are completely monotonic on (0,).

From Fm>0 and Gn>0, we geti=12mB2i2i2i-1x2i-1<lnΓ(x)-x-12lnx+x-12ln2π<i=12n-1B2i2i2i-1x2i-1.

From Fm<0 and Gn<0, we deducei=12m-B2i2ix2i<ψ(x)-lnx+12x<i=12n-1-B2i2ix2i,while Fm>0 and Gn

Acknowledgments

This work was supported by a Grant of the Romanian National Authority for Scientific Research, CNCSUEFISCDI, Project number PN-II-ID-PCE-2011-3-0087. Computations made in this paper were performed using Maple software.

References (6)

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    On some inequalities for the gamma and psi functions

    Math. Comput.

    (1997)
  • H. Alzer

    Inequalities for the gamma function

    Proc. Am. Math. Soc.

    (1999)
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