Show that $f(z) := e^{z+\frac{z^3}{3}}$ is Hayman admissable.
Remark: You can find the relevant definition from p.565 of the Floyd & Sedgewick book below.
From the lecture I know the following theorem:
Let $P(z)$ be a polynomial with real coefficients. If the Taylor coefficients of $e^{P(z)}$ are positive for large ennough $n$, then $e^{P(z)}$ is Hayman admissable.
I would say that this already solves the exercise, since
$$\exp(z) := \sum_{n \ge 0} \frac{z^n}{n!}$$
is always positive and thus also $e^{z+\frac{z^3}{3}}$. Is it really that easy or am I missing something here?