Real-rootedness, log-concavity, and unimodality are intertwined properties. It's in this light that I was prompted to ask the question below.

Suppose the roots of a polynomial $p(x)$ are all real and $p(0)>0$. Fix an integer $k\geq0$ and consider the function $f=\frac1{p^{2k+1}}$. I like to consider the partial sums (polynomial) $$\sum_{j=0}^{2k}\frac{f^{(j)}(0)}{j!}x^j; \tag1$$ where $f^{(j)}$ means the $j$-th derivative.

QUESTION.Is it true that the polynomial in (1) has no real roots?