On approximating the modified Bessel function of the first kind and Toader-Qi mean

• Main Authors: Zhen-Hang Yang, Yu-Ming Chu
• Format: Article
• Language: English
• Published: SpringerOpen 2016-02-01
• Series: Journal of Inequalities and Applications
• Subjects: modified Bessel function; Toader-Qi mean; logarithmic mean; identric mean
• Online Access: http://link.springer.com/article/10.1186/s13660-016-0988-1

Abstract In the article, we present several sharp bounds for the modified Bessel function of the first kind I 0 ( t ) = ∑ n = 0 ∞ t 2 n 2 2 n ( n ! ) 2 $I_{0}(t)=\sum_{n=0}^{\infty}\frac{t^{2n}}{2^{2n}(n!)^{2}}$ and the Toader-Qi mean T Q ( a , b ) = 2 π ∫ 0 π / 2 a cos 2 θ b sin 2 θ d θ $TQ(a,b)=\frac{2}{\pi}\int_{0}^{\pi/2}a^{\cos^{2}\theta }b^{\sin^{2}\theta}\,d\theta$ for all t > 0 $t>0$ and a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ .