Abstract. The aim of this paper is to provide a natural approach of Wilker-Cusa-Huygens inequalities. This new approach permits us to give new proofs then to refine much these inequalities and we are convinced that it is suitable to establish many other similar inequalities. To attain these purposes, computer softwares such as Maple are of great importance throughout this work.Mathematics subject classification (2010): 26D05; 26D15; 33B10.

a b s t r a c tThe goal of this paper is to prove the following asymptotic formulawhere is the Euler Gamma function and ψ is the digamma function, namely, the logarithmic derivative of . Moreover, optimal values of parameters b, c are calculated in such a way that this asymptotic convergence is the best possible.

In this paper we propose a new method for sharpening and refinements of some trigonometric inequalities. We apply these ideas to some inequalities of Wilker–Cusa–Huygens type.

A new class of sequences convergent to Euler's constant is investigated. Special choices of parameters show that the class includes the original sequence defined by Euler, as well as more recently defined sequences due to DeTemple [1] and Vernescu [9]. It is shown how the rate of convergence of the sequences can be improved by computing optimal values of the parameters.

The aim of this paper is to refine Jordan-Stečkin and Becker-Stark inequalities.Mathematics Subject Classification. 26D15, 41A10, 42A16.

The aim of this paper is to present a new algorithm for proving mixed trigonometric-polynomial inequalities of the form by reducing them to polynomial inequalities. Finally, we show the great applicability of this algorithm and, as an example, we use it to analyze some new rational (Padé) approximations of the function cos2 x and to improve a class of inequalities by Yang. The results of our analysis could be implemented by means of an automated proof assistant, so our work is a contribution to the library of automatic support tools for proving various analytic inequalities.

We prove in this paper that for every x ≥ 0,

No abstract