The natural algorithmic approach of mixed trigonometric-polynomial problems

Area of Science:

• Mathematical Analysis
• Computational Mathematics

Background:

• Proving mixed trigonometric-polynomial inequalities can be complex.
• Existing methods may lack efficiency or broad applicability.

Purpose of the Study:

• To present a new algorithm for proving mixed trigonometric-polynomial inequalities.
• To demonstrate the algorithm's applicability in analyzing rational approximations and improving existing inequalities.
• To contribute to automated tools for proving analytic inequalities.

Main Methods:

• Reducing mixed trigonometric-polynomial inequalities to polynomial inequalities.
• Applying the algorithm to analyze rational (Padé) approximations of cos^2(x).
• Utilizing the algorithm to improve inequalities by Yang.

Main Results:

• A novel algorithm for proving mixed trigonometric-polynomial inequalities has been developed.
• The algorithm effectively simplifies these inequalities by transforming them into polynomial inequalities.
• New rational approximations of cos^2(x) were analyzed, and existing inequalities were improved.

Conclusions:

• The presented algorithm is highly applicable and versatile for analytic inequalities.
• The findings can be integrated into automated proof assistants.
• This work advances the development of automatic support tools for mathematical proofs.