The modified Bessel function of order n of the first kind is given by
$$I_n(x)=\sum_{m=0}^{\infty}\frac{(\frac{1}{2}x)^{2m+n}}{m!\Gamma(m+n+1)}$$
where $\Gamma$ is defined by an improper integral,
$$\Gamma(s)=\int_{0}^{+\infty}t^{s-1}e^{-t}dt$$
Now, how can I prove the following expression is true for all non-zero real value of $\alpha$?
$$\frac{\alpha I_n^\prime(\alpha)}{I_n(\alpha)}>0$$
where $n\geqslant0$ and $n$ is an integer, $I_n^\prime(\alpha)=\frac{dI_n(\alpha)}{d\alpha}$.
Over $\mathbb{R}^+$, $I_n(x)$ is a convergent series of non-negative, increasing functions. It follows that $I_n(x)$ is positive and increasing over $\mathbb{R}^+$, so $\log(I_n(x))$ is increasing, too, and: $$\forall x>0,\qquad \frac{d}{dx}\log I_n(x) = \frac{I_n'(x)}{I_n(x)}> 0.$$ Can you complete the proof by studying the behaviour of $I_n(x)$ over $\mathbb{R}^-$ in a similar way?