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Abstract
The aim of this paper is to present a new algorithm for proving mixed trigonometric-polynomial inequalities of the form \t\t\t∑i=1nαixpicosqixsinrix>0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\sum_{i=1}^{n}\\alpha _{i}x^{p_{i}} \\cos ^{q_{i}} x\\sin ^{r_{i}} x>0 $$\\end{document} 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 cos2x 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.
Highlights
Introduction and motivationIn this paper, we propose a general computational method for reducing some inequalities involving trigonometric functions to the corresponding polynomial inequalities
The aim of this paper is to develop a new algorithm, based on the natural approach method, for proving MTP inequalities by reducing to polynomial inequalities
3 Some applications of the algorithm We present an application of the Natural Approach algorithm in the proof (Application Theorem ) of certain new rational (Padé) approximations of the function cos x, as well as in the improvement of a class of inequalities ( ) by Yang (Application, Theorem )
Summary
Introduction and motivationIn this paper, we propose a general computational method for reducing some inequalities involving trigonometric functions to the corresponding polynomial inequalities. We mention the work of Mortici [ ] who extended Wilker-Cusa-Huygens inequalities using the method he called the natural approach method. This method consists in comparing and replacing sin x and cos x by their corresponding Taylor polynomials as follows: s+ (– )ix i+ s (– )ix i+ < sin x < ( i + )! Lutovac et al Journal of Inequalities and Applications (2017) 2017:116 In this way, complicated trigonometric expressions can be reduced to polynomial or rational expressions, which can be, at least theoretically, easier studied (this can be done using some software for symbolic computation, such as Maple)
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