Asymptotic series and inequalities associated to some expressions involving the volume of the unit ball

The aim of this work is to expose some asymptotic series associated to some expressions involving the volume of the n-dimensional unit ball. All proofs and the methods used for improving the classical inequalities announced in the final part of the first section are presented in an extended form in a paper submitted by the author to a journal for publication.Comment: 9 page

A subtly analysis of Wilker inequality

The aim of this work is to improve Wilker inequalities near the origin and {\pi}/2.Comment: 6 page

On an infinite series for (1+1/x)x(1+1/x)^x

The aim of this paper is to construct a new expansion of $(1+1/x)^x$ related to Carleman's inequality. Our results extend some results of Yang [Approximations for constant e and their applications J. Math. Anal. Appl. 262 (2001) 651-659]

Asymptotic formulas and inequalities for gamma function in terms of tri-gamma function

In the paper, the authors establish some asymptotic formulas and double inequalities for the factorial $n!$ and the gamma function $\Gamma$ in terms of the tri-gamma function $\psi'$.Comment: 6 page

On the coefficients of an expansion of (1+1/x)x(1+1/x)^x related to Carleman's inequality

In this note, we present new properties for a sequence arising in some refinements of Carleman's inequality. Our results extend some results of Yang [Approximations for constant e and their applications J. Math. Anal. Appl. 262 (2001) 651-659] and Alzer and Berg [some classes of completely monotonic functions Ann. Acad. Sci. Fennicae 27(2002) 445-460].Comment: 4 page

Some best approximation formulas and inequalities for Wallis ratio

In the paper, the authors establish some best approximation formulas and inequalities for Wallis ratio. These formulas and inequalities improve an approximation formula and a double inequality for Wallis ratio recently presented in ``S. Guo, J.-G. Xu, and F. Qi, \textit{Some exact constants for the approximation of the quantity in the Wallis' formula}, J. Inequal. Appl. 2013, \textbf{2013}:67, 7 pages; Available online at \url{http://dx.doi.org/10.1186/1029-242X-2013-67}''.Comment: 6 page

A survey on recent extensions of the Stirling formula

We present a survey on recent results about Stirling's formula. More exactly, we reffer to a method using a form of Cesaro-Stolz lemma firstly introduced in [C. Mortici Product approximations via asymptotic integration Amer. Math. Monthly 117 (5) (2010) 434-441]. As an example we improve a result obtained in [C. Mortici A substantial improvement of the Stirling formula Appl. Math. Lett. 24 (2011) no. 8 1351-1354]. Finally, some numerical computations are made.Comment: 7 page

Some inequalities for the trigamma function in terms of the digamma function

In the paper, the authors establish three kinds of double inequalities for the trigamma function in terms of the exponential function to powers of the digamma function. These newly established inequalities extend some known results. The method in the paper utilizes some facts from the asymptotic theory and is a natural way to solve problems for approximating some quantities for large values of the variable.Comment: 13 page

On some convergences to the constant e and improvements of Carlemans' inequality

We present inequalities and some applications to Kellers' limit and Carlemans' inequality.Comment: 7 page

Multiple-correction and summation of the rational series

The goal of this work is to formulate a systematical method for looking for the simple closed form or continued fraction representation of a class of rational series. As applications, we obtain the continued fraction representations for the alternating Mathieu series and some rational series. The main tools are multiple-correction and two of Ramanujan's continued fraction formulae involving the quotient of the gamma functions