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New Sharp Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean

Author

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  • Zhen-Hang Yang

    (Engineering Research Center of Intelligent Computing for Complex Energy Systems of Ministry of Education, North China Electric Power University, Yonghua Street 619, Baoding 071003, China
    Zhejiang Society for Electric Power, Hangzhou 310014, China)

  • Jing-Feng Tian

    (Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, Baoding 071003, China)

  • Ya-Ru Zhu

    (Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, Baoding 071003, China)

Abstract

Let I v x be he modified Bessel function of the first kind of order v . We prove the double inequality sinh t t cosh 1 / q q t < I 0 t < sinh t t cosh 1 / p p t holds for t > 0 if and only if p ≥ 2 / 3 and q ≤ ln 2 / ln π . The corresponding inequalities for means improve already known results.

Suggested Citation

  • Zhen-Hang Yang & Jing-Feng Tian & Ya-Ru Zhu, 2020. "New Sharp Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean," Mathematics, MDPI, vol. 8(6), pages 1-13, June.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:6:p:901-:d:366530
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    References listed on IDEAS

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    1. Wang, Miao-Kun & Chu, Yu-Ming & Song, Ying-Qing, 2016. "Asymptotical formulas for Gaussian and generalized hypergeometric functions," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 44-60.
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    Cited by:

    1. Muhey U. Din & Mohsan Raza & Qin Xin & Sibel Yalçin & Sarfraz Nawaz Malik, 2022. "Close-to-Convexity of q -Bessel–Wright Functions," Mathematics, MDPI, vol. 10(18), pages 1-12, September.
    2. Andrés Martín & Ernesto Estrada, 2023. "Fractional-Modified Bessel Function of the First Kind of Integer Order," Mathematics, MDPI, vol. 11(7), pages 1-13, March.

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