Is there an analytic function on $\mathbb D$ with $\vert f\vert\leq1$ on $\mathbb D$ and $f(-1/2)=0$ and $f(1/2)=5/6$?

Question

Prove or disprove that there does not exists an analytic function f on $\mathbb D,$ the open unit disk in the complex plane, such that $\vert f\vert\leq1$ on $\mathbb D, f(-1/2) = 0$ and $f(1/2) = 5/6.$

I think this is an application of the Schwarz Lemma, but I fail to see how to bring it in since $f(0)\ne0$. Do I need to use conformal maps in this example?

I also tried using the Schwarz-Pick Theorem, but noticed that is not allowed as the function does not map $\mathbb D$ onto itself. If I could somehow make use of the Schwarz-Pick Theorem, them it would yield the contradiction that $5/6\leq1/2,$ and it would be solved. Is there a way to apply the Theorem here that I am not seeing?

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Any hints on how to proceed are much appreciated. Thank you for your time.