Question regarding an inequality of a function which is holomorphuc in first variable and anti-holomorphic in second variable

Question

Let $D$ be a domain in $\mathbb{C}^{n}$, where $n>1$. Let $F:D×D\to\mathbb{C}$ be a function defined as: $F(z,w)$ is holomorphic function in first variable $z$ and $F(z,w)$ is anti-holomorphic in second variable, $w$. My question is if $F(z,z)\leq0$ for all $z\in \{z:\operatorname{dist}(z,\partial D)<\delta\}$, where dist($z,\partial D$) denote the Euclidean distance of $z$ to the $\partial D$ for sufficiently small positive real number, $\delta>0$. Then can we imply that $F(z,z)\leq 0$ for all $z\in D$? My attempt was if there exist a point $p\in D$ with $F(p,p)>0$, then there exist a ball with centre $p$ and radius $r$, $\mathbb{B}^{n}(p,r)$ such that $F(w,w)>0$ for all $w\in\mathbb{B}^{n}(p,r)$ by continuity of $F$. Then I am not able to move forward. It would be a great help to me if someone can give a hint or something.