Let $B\subseteq \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ centered at $(0,0)$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real analytic and have only one zero, namely $(0,0)$. Moreover, assume that $f$ is strictly increasing along lines intersecting the origin.
Claim. Let $$ \tilde{f}(r,\theta) = f(r\cos(\theta), r\sin(\theta)) $$ be the polar form of $f$. Moreover, define $n_{\theta}\in \Bbb{N}$ to be the smallest non-negative integer such that $\tilde{f}^{(n_{\theta})}(0,\theta)\ne 0$. Lastly, let $\sup_{\theta\in [0,2\pi)}n_{\theta}\leq N <\infty$. Then, there exists some $r' \leq \delta$ and $\theta'\in[0,2\pi)$ such that for all $r\leq r'$,
$$ \tilde{f}(r,\theta) \geq \tilde{f}(r,\theta'). $$
Ideas. I'm thinking that maybe there's some sort of "cardioid" function where it decreases as it approaches $\theta = 0$. However, I'm unsure if such a function is real-analytic. Any advice would be appreciated.
