Optimal inequalities for bounding Toader mean by arithmetic and quadratic means

Area of Science:

• Mathematical Analysis
• Inequalities
• Special Functions

Background:

• The study of inequalities and means is fundamental in mathematical analysis.
• Complete elliptic integrals of the second kind have significant applications in various scientific fields.
• Previous research has focused on bounding these integrals using different mean types.

Purpose of the Study:

• To determine the best possible parameters for a specific double inequality.
• To establish new, tighter bounds for the complete elliptic integral of the second kind.
• To explore the relationship between different means (Toader, arithmetic, quadratic) and elliptic integrals.

Main Methods:

• Derivation of optimal parameters through analytical methods.
• Application of inequality theory to establish bounds.
• Utilizing definitions of Toader, arithmetic, and quadratic means.

Main Results:

• Identification of the sharpest constants [Formula: see text] and [Formula: see text] for the inequality.
• Presentation of novel inequalities for the complete elliptic integral of the second kind.
• Demonstration that the established bounds are the best possible.

Conclusions:

• The determined parameters provide the most precise conditions for the inequality.
• The new bounds offer improved accuracy for the complete elliptic integral of the second kind.
• The findings contribute to the understanding of the interplay between means and special functions.