Una nueva prueba para una conjetura de Özban

En este artículo presentamos una prueba corta y elemental de la siguiente desigualdad algebraico-trigonométrica de tipo Laub-Ilani: cos(xy) + cos(yx) ≥ cos(xx) + cos(yy) para x, y ∈ [0, π/2] que fue conjeturada por Özban [‘New algebraic-trigonometric inequalities of Laub-Ilani type’, Bull. Aust. Math. Soc. 96 (2017), 87–97] y recientemente probada por Matejíčka [‘Proof of one open inequality of Laub-Ilani type’, Journal of Mathematical Inequalities, 14 (2020), 83–98]. La prueba se basa en las propiedades de las funciones potenciales-exponenciales y trigonométricas.

Simple efficient bounds for arcsine and arctangent functions

The aim of this paper is to present new, simple and sufficiently sharp bounds for arcsine and arctangent functions. Some of the bounds are computationally efficient while others are efficient to approximate the integrals Int_{a}^{b} (arcsin x)/x dx and Int_{a}^{b} (arctan x)/x dx. As a matter of interest, several other sharp and generalized inequalities for (arcsin x)/x and (arctan x)/x are also established which are efficient to give some known and other trigonometric inequalities

New Refinements and Improvements of Some Trigonometric Inequalities Based on Padé Approximant

A multiple-point Padé approximant method is presented for approximating and bounding some trigonometric functions in this paper. We give new refinements and improvements of some trigonometric inequalities including Jordan’s inequality, Kober’s inequality, and Becker-Stark’s inequality. The analysis results show that our conclusions are better than the previous conclusions.

ABOUT DEVELOPMENT OF STUDENTS ' THINKING WHEN SOLVING TRIGONOMETRIC EQUATIONS AND INEQUALITIES IN THE SCHOOL ALGEBRA COURSE

The article deals with solutions of trigonometric inequalities using the unit circle. Specific examples show its application for all trigonometric functions, namely sinus, cosine, tangent and cotangent. An explanation of how to correctly define the period for solving inequalities is also provided. Before analyzing the solution to trigonometric inequalities, the authors present the solution of trigonometric equations according to the formula, but his roots are depicted on the unit circle, where detailed explanation of the record of solutions of this equation. The pictures in the article demonstrate the images that should be presented by the teacher on the blackboard when solving trigonometric inequalities. The article is written in an accessible language, when reading which the unit circle method will be understandable not only to current teachers, but also to students of Junior courses of pedagogical universities.

On the Generalization for Some Power-Exponential-Trigonometric Inequalities

In this paper, we introduce and prove several generalized algebraic-trigonometric inequalities by considering negative exponents in the inequalities.

NEW ALGEBRAIC-TRIGONOMETRIC INEQUALITIES OF LAUB–ILANI TYPE

The Laub–Ilani inequality [‘A subtle inequality’, Amer. Math. Monthly97 (1990), 65–67] states that $x^{x}+y^{y}\geqslant x^{y}+y^{x}$ for nonnegative real numbers $x,y$. We introduce and prove new trigonometric and algebraic-trigonometric inequalities of Laub–Ilani type and propose some conjectural algebraic-trigonometric inequalities of similar forms.