If an entire function is bounded for all $z \in \mathbb{C}$, than it's a constant by Liouville's theorem. Of course an entire function can be bounded on lines through the origin $z=r \exp(i \phi), \phi= const., r \in \mathbb{R}$ without being constant (e.g. $\cos(z^n)$ is bounded on $n$ lines).

What is the maximum cardinality of the set of "directions" $\phi$ for which an entire function can be bounded without being constant?

From intuition I would expect only finitely many directions. Is this correct?

(Picard's second theorem says that in any open set containing $\infty$ every value with possibly a single exception is taken infinitely often by an entire non-constant function. Here I'm asking a somehow "orthogonal" question, looking for lines through $\infty$ where an entire non-constant function is bounded.)

If an entire function is bounded for all $z \in \mathbb{C}$, than it's a constant by Liouville's theorem. Of course an entire function can be bounded on lines through the origin $z=r \exp(i \phi), \phi= \text{const.}, r \in \mathbb{R}$ without being constant (e.g. $\cos(z^n)$ is bounded on $n$ lines).

What is the maximum cardinality of the set of "directions" $\phi$ for which an entire function can be bounded without being constant?

From intuition I would expect only finitely many directions. Is this correct?

(Picard's second theorem says that in any open set containing $\infty$ every value with possibly a single exception is taken infinitely often by an entire non-constant function. Here I'm asking a somehow "orthogonal" question, looking for lines through $\infty$ where an entire non-constant function is bounded.)

If an entire function is bounded for all $z \in \mathbb{C}$, than it's a constant by Liouville's theorem. Of course an entire function can be bounded on lines through the origin $z=r \exp(i \phi), \phi= const., r \in \mathbb{R}$ without being constant (e.g. $\cos(z^n)$ is bounded on $n$ lines).

What is the maximum cardinality of the set of "directions" $\phi$ for which an entire function can be bounded without being constant?

From intuition I would expect only finitely many directions. Is this correct?

(Picard's second theorem says that in any open set conztaining $\infty$ every value with possible a single exception is taken infinitely often by an entire non-constant function. Here I'm asking a somehow "orthogonal" question, looking for lines through $\infty$ where an entire non-constant function is bounded.)

If an entire function is bounded for all $z \in \mathbb{C}$, than it's a constant by Liouville's theorem. Of course an entire function can be bounded on lines through the origin $z=r \exp(i \phi), \phi= const., r \in \mathbb{R}$ without being constant (e.g. $\cos(z^n)$ is bounded on $n$ lines).

What is the maximum cardinality of the set of "directions" $\phi$ for which an entire function can be bounded without being constant?

From intuition I would expect only finitely many directions. Is this correct?

(Picard's second theorem says that in any open set containing $\infty$ every value with possibly a single exception is taken infinitely often by an entire non-constant function. Here I'm asking a somehow "orthogonal" question, looking for lines through $\infty$ where an entire non-constant function is bounded.)