The function (b x - A x )/x: Ratio's properties

Abstract

In the present paper, after reviewing the history, background, origin, and applications of the functions and, we establish sufficient and necessary conditions such that the special function is monotonic, logarithmic convex, logarithmic concave, 3-log-convex, and 3-log-concave on, where α, β, λ, and μ are real numbers satisfying (α, β) ≠ (λ, μ), (α, β) ≠ (μ, λ), α ≠ β, and λ ≠ μ.