In this paper, two double Jordan-type inequalities are introduced that generalize some previously established inequalities. As a result, some new upper and lower bounds and approximations of the sinc function are obtained. This extension of Jordan’s inequality is enabled by considering the corresponding inequalities through the concept of stratified families of functions. Based on this approach, some optimal approximations of the sinc function are derived by determining the corresponding minimax approximants.

pdf/jordan-type-inequalities-and-stratification-hggfajosvq.pdf

Jordan-Type Inequalities and Stratification

In this paper, two double Jordan-type inequalities are introduced based on the papers [1]-[5]. These inequalities generalize the inequalities obtained in [1]-[5]. As a result, some new upper and lower bounds of the sinc function are obtained. This extension of Jordan&amp;rsquo;s inequality is enabled by considering the corresponding inequalities through the concept of stratified families of functions elaborated in [6]. Based on this approach, some optimal approximations of the sinc function are derived by determining corresponding minimax approximants, also described in the paper [6]. 

In this paper, we generalize Cristinel Mortici's results on Wilker-Cusa-Huygens inequalities using stratified families of functions and SimTheP - a system for automated proving of MTP inequalities.

The best possible constants approach for Wilker-Cusa-Huygens inequalities via stratification

In this paper, a parametric method for proving inequalities is described. The method is based on associating a considered inequality with the corresponding stratified family of functions. Many inequalities from the theory of analytic inequalities can be interpreted using families of functions that are stratified with respect to some parameter. By discussing the sign of the functions from the family by the parameter according to which the family is stratified, inequalities are obtained that contain the best possible constants, if they exist. The application of this method is demonstrated for four inequalities: the Cusa–Huygens inequality, the Wilker-type inequality and the two Mitrinović–Adamović-type inequalities. Significantly simpler proofs and improvements of all these inequalities are provided. 

A Parametric Method for Proving Some Analytic Inequalities

/pdf/a-solution-to-an-open-problem-for-wilker-type-inequalities-hujgehc13a.pdf

A solution to an open problem for Wilker-type inequalities

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A solution to an open problem for Wilker-type inequalities