The Function (b x − a x )∕x: Ratio’s Properties

Feng Qi (), Qiu-Ming Luo () and Bai-Ni Guo ()
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Feng Qi: Henan Polytechnic University, School of Mathematics and Informatics
Qiu-Ming Luo: Chongqing Normal University, Department of Mathematics
Bai-Ni Guo: Henan Polytechnic University, School of Mathematics and Informatics

A chapter in Analytic Number Theory, Approximation Theory, and Special Functions, 2014, pp 485-494 from Springer

Abstract: Abstract In the present paper, after reviewing the history, background, origin, and applications of the functions $$\frac{{b}^{t}-{a}^{t}} {t}$$ and $$\frac{{e}^{-\alpha t}-{e}^{-\beta t}} {1-{e}^{-t}}$$ , we establish sufficient and necessary conditions such that the special function $$\frac{{e}^{\alpha t}-{e}^{\beta t}} {{e}^{\lambda t}-{e}^{\mu t}}$$ is monotonic, logarithmic convex, logarithmic concave, 3-log-convex, and 3-log-concave on $$\mathbb{R}$$ , where α, β, λ, and μ are real numbers satisfying (α, β) ≠ (λ, μ), (α, β) ≠ (μ, λ), α ≠ β, and λ ≠ μ.

Keywords: Logarithmic Convexity; Logarithmic Concavity; Complete Monotonicity; Extended Mean Value; Generalized Bernoulli Numbers (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4939-0258-3_16

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DOI: 10.1007/978-1-4939-0258-3_16

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