If $f$ and $g$ admit power series in the ball $B(0,1)$ and $f(x)=g(x) \ \forall x\in (-1,1)$, does it follow that $f(iy)=g(iy) \ \forall y\in (-1,1)$?
My answer is true. As $f$ and $g$ both admit power series in the ball $B(0,1)$, then $f\in H(B(0,1))$ and $g\in H(B(0,1))$.
Now if $f(x)=g(x) \ \forall x\in (-1,1)$, then $f(z)=g(z) \ \forall z\in B(0,1)$. So it follows that $ f(iy)=g(iy) \ \forall y\in (-1,1)$.
Is this correct?