Erdős problem 514

For a transcendental entire function, how fast can $|f(z)|$ be forced to grow along a path to infinity, and how short can such a path be in terms of the maximum modulus $M(r, f)$ ?

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For a transcendental entire function, how fast can $|f(z)|$ be forced to grow along a path to infinity, and how short can such a path be in terms of the maximum modulus $M(r, f)$?

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