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

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Analytic Number Theory, Approximation Theory, and Special Functions

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.

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Acknowledgements

The first author was partially supported by the China Scholarship Council and the Science Foundation of Tianjin Polytechnic University. The second author was supported in part by the Natural Science Foundation Project of Chongqing under Grant CSTC2011JJA00024, the Research Project of Science and Technology of Chongqing Education Commission under Grant KJ120625, and the Fund of Chongqing Normal University under Grant 10XLR017 and 2011XLZ07, China.

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Authors and Affiliations

  1. School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

    Feng Qi & Bai-Ni Guo

  2. Department of Mathematics, Chongqing Normal University, Chongqing City, 401331, China

    Qiu-Ming Luo

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  1. Feng Qi
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  3. Bai-Ni Guo
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Correspondence to Feng Qi .

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Editors and Affiliations

  1. Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia

    Gradimir V. Milovanović

  2. Department of Mathematics, ETH Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Dedicated to Professor Hari M. Srivastava

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Qi, F., Luo, QM., Guo, BN. (2014). The Function (b x − a x)∕x: Ratio’s Properties. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_16

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