Complex analysis — a functional equation with one holomorphic solution

Let $\mathbb{D}$ denote the open unit disc $\{z\in\mathbb{C}:|z|<1\}$ and

$$X=\left\{f:\mathbb{D}\to\mathbb{C} \middle| f\text{ is holomorphic and satisfies } f(2z)=\frac{f(z)}{1-(f(z))^2}\text{ for all }|z|<\tfrac12\right\}.$$

Which of the following statements is true?