Let D be a disk of center $c$ and radius $R>0$ located in the complex plane, $\partial D$ be the boundary of the disk, $H(D)$ denote the collection of analytic functions defined on $D$, $$H^p(D)=\left\{f\in H(D):\exists C>0,\sup_{r\in[0,R)}\int_{0}^{2\pi}|f(c+re^{i\theta})|^pd\theta\leq C<\infty\right\},\quad 0<p<\infty.$$ Now, I want to prove that a function with the following properties belongs to $H^p(D)$. Let $F\in H(D)$ that can be analytically extended on $\partial D$ except for a finite number of points on $\partial D=\{c+Re^{i\theta}:\theta\in[0,2\pi)\}$ (and thus on neighborhoods of those points that carry $F$ analytically in the outer of $D$) where we don't know how $F$ behaves. But we do know that the following integral is bounded $$\int_{0}^{2\pi}|f(c+Re^{i\theta}|^pd\theta\leq C,$$ for some $0<p<1$. My question is if these conditions are enough to ensure that $F\in H^p(D)$ for the same $0<p<1$.
Of course, the problem is with these points on $\partial D$, which I cannot extend $F$, otherwise, it would be in $H^p(D)$, trivially.