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

Abstract

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 $a\neq b$ .

Keywords

• modified Bessel function
• Toader-Qi mean
• logarithmic mean
• identric mean