Approximation Methods for Solving the Cauchy Problem

In this paper we give some new results concerning solvability of the 1-dimensional differential equation y′ = f(x, y) with initial conditions. We study the basic theorem due to Picard. First we prove that the existence and uniqueness result remains true if f is a Lipschitz function with respect to the first argument. In the second part we give a contractive method for the proof of Picard theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.     

On Some Mean Value Points Defined by Divided Differences and Their Hyers–Ulam Stability

The aim of this paper is to establish some stability results involving generalized divided differences.     

Asymptotic Formulas and Inequalities for the Gamma Function in Terms of the Tri-Gamma Function

Results in Mathematics - In the paper, the authors establish some asymptotic formulas and double inequalities for the factorial n! and the gamma function Γ in terms of the tri-gamma function...     

Ramanujan’s estimate for the gamma function via monotonicity arguments

The aim of this paper is to extend and refine an approximation formula of the gamma function by Ramanujan.     

Estimates for Wallis’ ratio and related functions

We present improvements of approximation formula for Wallis ratio related to a class of inequalities stated in [D.-J. Zhao, On a two-sided inequality involving Wallis’s formula, Math. Practice Theory, 34 (2004), 166-168], [Y. Zhao and Q. Wu, Wallis inequality with a parameter, J. Inequal. Pure Appl. Math., 7(2) (2006), Art. 56] and [C. Mortici, Completely monotone functions and the Wallis ratio, Applied Mathematics Letters, 25 (2012), 717-722]. Some sharp inequalities are obtained as a result of monotonicity of some functions involving gamma function.     

Improved convergence towards generalized Euler-Mascheroni constant

We propose new simple sequences approximating the Euler-Mascheroni constant and its generalization, which converge faster towards their limits than those considered by DeTemple [D.W. DeTemple, A quicker convergence to Euler’s constant, Am. Math. Monthly 100 (5) (1998) 468-470], Sînt?m?rian [A. Sînt?m?rian, A generalization of Euler’s constant, Numer. Algorithms 46 (2) (2007) 141-151] and Vernescu [A. Vernescu, A new accelerate convergence to the constant of Euler, Gazeta Matem., Ser. A, Bucharest XVII(XCVI) (4) (1999) 273-278].     

Monotonicity properties of the volume of the unit ball in $${\mathbb{R}^{n}}$$

The aim of this paper is to establish new bounds for ratios involving the volume of the unit ball in \mathbbRⁿ{\mathbb{R}^{n}}.     

Accurate approximations of the Mathieu series

The aim of this paper is to establish new bounds for the Mathieu series.     

A new Stirling series as continued fraction

We introduce the following new Stirling series

$ n!\sim \sqrt{2\pi n}\left( \frac{n}{e}\right) ^{n}\exp \frac{1}{12n+\frac{ \frac{2}{5}}{n+\frac{\frac{53}{210}}{n+\frac{\frac{195}{371}}{n+\frac{\frac{ 22,\!999}{22,\!737}}{n+\ddots}}}}}, $

as a continued fraction, which is faster than the classical Stirling series.