A new proof for a Özban conjecture

In this paper, we present an elementary short proof of the following algebraic-trigonometric inequality of Laub-Ilani type: cos(xy)+cos(yx) ≥ cos(xx) + cos(yy) for x, y ∈ [0, π/2] which was conjectured by Özban [‘New algebraic-trigonometric inequalities of Laub-Ilani type’, Bull. Aust. Math. Soc. 96 (2017), 87–97] and recently proved by Matejíčka [‘Proof of one open inequality of Laub-Ilani type’, Journal of Mathematical Inequalities, 14 (2020), 83–98]. The proof is based on the properties of the power-exponential and trigonometric functions.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