Let $f(z)$ be an entire function (on $\mathbb{C}$). Assume it has a power series of the form $$\displaystyle \sum_{n=0}^\infty (-1)^nc_{2n}z^{2n},$$ where $c_{2n}\geq 0$ for all $n$.
Is there a sufficient condition on the coefficients $c_{2n}$ under which the sum is bounded for all $z=x\in \mathbb{R}?$