Sharpness of the Phragmen-Lindelof growth exponent

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Problem 3.329 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 5 "The Phragmén-Lindelöf Method"

Let $\omega (x)$ be a positive function of the positive variable $x$ that increases with $x$ and tends to $+\infty$ as $x$ increases to $+\infty$ . A function $f(z)$ , regular in the half-plane $\mathrm{Re}\left(z\right)\geq 0$ , that satisfies the inequality

\left|f(z)\right| \geq e^{\omega (\left|z\right|)\left|z\right|} \quad \text{for} \quad \mathrm{Re}\left(z\right)\geq 0

does not exist.

Problem 3.332 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 5 "The Phragmén-Lindelöf Method"

The function $g(z)$ is assumed to be an entire function, $M(r)$ be the maximum of $\left|g(z)\right|$ on the circle $\left|z\right|=r$ . If

\lim _{r\to \infty } \frac{\log M(r)}{\sqrt{r}} = 0

then $g(z)$ cannot be bounded along any ray. [E.g. $g(z)$ is not bounded along the negative real axis.]

References

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.