The natural algorithmic approach of mixed trigonometric-polynomial problems. (English) Zbl 1373.42003
Summary: The aim of this paper is to present a new algorithm for proving mixed trigonometric-polynomial inequalities of the form
\[
\sum_{i=1}^{n}\alpha _{i}x^{p_{i}} \cos ^{q_{i}} x\sin ^{r_{i}} x>0
\]
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 \(\cos^{2}x\) 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.
MSC:
| 42A05 | Trigonometric polynomials, inequalities, extremal problems |
| 42A10 | Trigonometric approximation |
| 41A58 | Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) |
| 26D05 | Inequalities for trigonometric functions and polynomials |
| 12L05 | Decidability and field theory |
Keywords:
mixed trigonometric-polynomial functions; Taylor series; approximations; inequalities; algorithms; automated theorem provingReferences:
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