Some Useful Properties of Modified Bessel Functions of the First Kind

I have to deal with Modified Bessel functions of the first kind frequently in my research. Here I list some useful properties of them for future reference. $$I_0'(x) = I_1(x)$$I_1'(x) = \frac{I_0(x)+I_2(x)}{2}$$\frac{I_1(x)}{I_0(x)}\uparrow \Leftrightarrow I_0(x)(I_0(x)+I_2(x))-2I_1^2(x)>0,\forall x>0$$\frac{I_1(x)-I_2(x)}{I_0(x)-I_1(x)}\uparrow$$where$I_n(x)$is the modified Bessel function of order$n$.

For more properties about modified Bessel functions of the first kind, please refer to Wolfram MathWorld.