New Sharp Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean
Zhen-Hang Yang,
Jing-Feng Tian,
Ya-Ru Zhu
Affiliations
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
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
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.
