$f(x)$ is a transcendental function over $\mathbb{Q}(x)$,and analytic in disk with natural boundary. If $a\gt 0$ and $a\in \mathbb{Q}$, then $f(a)$ is a transcendental number.
Has this assertion been proved? Any reference?
Update base on comments: $f(x)$ is a transcendental function over $\mathbb{Q}(x)$,and analytic in disk with natural boundary. If $a\gt 0$,and $a\neq 1$ and $a\in \mathbb{Q}$, then $f(a)$ is a transcendental number.
Has this assertion been proved? Any reference?
Let $g$ be any transcendental function over $\mathbb{Q}(x)$ with the unit circle as natural boundary (or whatever reasonable assumption you want to put). Since $g$ is not constant, $g$ assumes some value that is algebraic at some point in the disc, say $g(a) = x$ where $x$ is algebraic.
Define $$ f(z) = g(m(z)) $$ where $m$ is a biholomorphic automorphism of the disk taking some rational positive number $r$ to $a$. Then $f(r) = x$, so in general this fails for a very large class of transcendental functions.