Let $-\ln(2)\leq x<0$ and $n\geq 1$ a natural number then define :
$$f(x)=\left(0.5+\sum_{k=1}^{2n}e^{k^2x}\right)\left(0.5+\sum_{k=1}^{2n}(-1)^ke^{k^2x}\right)$$
Then it seems we have the following claim :
$$f''(x)>0$$
Proof for the case $n=1$ :
this case is very simple if we differentiate one time and substitute $y=e^x$ we get a fourth degree polynomial in subsituting again and to show that the derivative increase we differentatiate the polynomial to get a third degree polynomial .The rest is smooth using Cardano'formula.
Question :
How to (dis)prove the claim ?
Thanks !