For $t\in(-1,1)$, let
$$f(t):=\left(\frac{1+t}{1-t}\right)^{(1-t)/2}+\left(\frac{1-t}{1+t}\right)^{(1+t)/2}$$
and
$$g(t):=\frac1{f(t)}.$$
Note that the functions $f$ and $g$ are even.


**Question 1:** Is it true that all the even-order derivatives $f^{(2k)}$ of $f$ at $0$ are negative, except for $k=0$ and $k=2$?


**Question 2:** Is it true that all the even-order derivatives $g^{(2k)}$ of $g$ at $0$ are positive?


**Question 3:** Is there a simple, explicit, and accurate upper bound on the even-order derivatives $g^{(2k)}$?


A correct and complete answer to any one of these three questions will be considered as a correct and complete answer to this entire post.



Coefficients of certain Taylor series
Note that
$$\ln(2g(t))=\frac{1}{2} \,\ln \left(1-t^2\right)+ t \tanh ^{-1}(t)
=\sum\_{k=1}^\infty\frac{t^{2k}}{2k(2k-1)}.$$
This immediately yields the positive answer to Question 2.


