The value of the extremal second-quotient constant for remainder-real entire functions

Let $\mathcal R$ be the set of all entire functions

$$f(z)=\sum_{k=0}^\infty a_k z^k$$

such that, for every $l\in\mathbb N\cup\{0\}$, the remainder

$$R_l[f](z)=\sum_{k=l}^\infty a_k z^k$$

has only real nonpositive zeros. Define

$$c=\inf\left\{q_n(f)\mid f\in\mathcal R,\ n=2,3,\ldots\right\}.$$

Extremal-constant problem. In the previous problem, $c=q_\infty$.

The preceding theorem gives the lower bound $q_n(f)>3$ for every function in $\mathcal R$ and every $n\geq2$. This statement identifies the infimum from the preceding problem with the limiting second quotient $q_\infty$, but the supplied text does not define $q_\infty$ further or establish the value of the constant.