Convexity of a ${\rm tanh}^{-1}\ (x) / x$?

Question

I want to prove that the function:

$f(x) = \displaystyle \lim_{y\rightarrow x} \dfrac{\text{tanh}^{-1}(y)}{y}$

is convex in $(-1,1)$, where $\text{tanh}^{-1}()$ is the inverse hyperbolic tangent function. Any help would be appreciated.

Thanks!

Answer 1

$\operatorname{arctanh} x= \sum_{k=1,3,5,...} {1 \over k} x^k$, so $f(x) = \sum_{k=1,3,5,...} {1 \over k} x^{k-1}$.

Hence $f''(x) = \sum_{k=3,5,7,...} {1 \over k} (k-1)(k-2) x^{k-3}$ and it is straightforward to observe that $f''(x) \ge 0$.