Suppose I have a real analytic function $f(z) = \sum_{j=3}^{\infty} a_j z^j$ defined for $|z|< \epsilon$ where $\epsilon > 0$. And $a_3 \not = 0$. I was wondering is it possible to prove that $f$ is strictly convex (https://en.wikipedia.org/wiki/Convex_function) on $|z|< \epsilon$, provided $\epsilon > 0$ is sufficiently small? THank you!
Edit: I would like to fix $f(z) = \sum_{j=2}^{\infty} a_j z^j$ and $a_2 > 0$.
Since $a_2 > 0$, you have $f''(0) > 0$, and since $f''$ is continuous, there is some $\epsilon > 0$ such that $f''(z) > 0$ for all $z$ with $|z| < \epsilon$. From this you can show that $f$ is strictly convex on $|z| < \epsilon$ (for instance, by using the mean value theorem 3 times).