Ratio ’ S Properties

In the paper, after reviewing the history, background, origin, and applications of the functions b t −a t t and e −αt −e −βt 1−e−t , we establish sufficient and necessary conditions such that the special function e αt −e βt eλt−eμt are monotonic, logarithmic convex, logarithmic concave, 3-log-convex and 3-log-concave on R, where α, β, λ and μ are real numbers satisfying (α, β) 6= (λ, μ), (α, β) 6= (μ, λ), α 6= β and λ 6= μ.