Theorem 2 (p. 498): a path to infinity of length at most u(z)^{K+o(1)}

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Source. Theorem 2, p. 498, with the definitions on pp. 497--498, Remarks 3 and 4 of section II on pp. 499--500 and the proof in section IV, pp. 501--504, of Jang-Mei Wu, Length of paths for subharmonic functions, J. London Math. Soc. (2) 32 (1985), 497--505, doi:10.1112/jlms/s2-32.3.497, the edition named on the source card.

Read depth. Claims checked: the statement, the definitions it uses and Remarks 3 and 4 were read clause by clause on the page images of the print; the proof was not checked. Nothing here is independently reviewed.

Statement

For $u$ subharmonic in $\mathbb C$ let $M(r)=\sup\{u(z):\lvert z\rvert=r\}$ and let

$$\lambda=\liminf_{r\to\infty}\frac{\log M(r)}{\log r}$$

be its lower order (p. 497). For a path $\Gamma$ starting at a point $P$ and a point $z$ on $\Gamma$, $\Gamma(z)$ is the part of $\Gamma$ from $P$ to $z$, and $L$ denotes length (p. 498).

Let $u$ be subharmonic in $\mathbb C$ of lower order $\lambda$ with $0<\lambda\le+\infty$, and let $K=\max\{1/\lambda,2\}$ and $k=\min\{\lambda,\tfrac12\}$. Then there is a path $\Gamma$ from a finite point to $\infty$ on which

$$L(\Gamma(z))\le u(z)^{K+o(1)}\quad\text{as }z\to\infty, \qquad (1.5)$$ $$u(z)>\lvert z\rvert^{k-o(1)}\quad\text{as }z\to\infty, \qquad (1.6)$$ $$\int_\Gamma u^{-(K+\alpha)}\,\lvert dz\rvert<+\infty\quad \text{for any }\alpha>0. \qquad (1.7)$$

The path does not depend on $\alpha$ (p. 501).

The paper says (p. 498) that the theorem improves Theorem B when $u$ has positive lower order. It also generalizes Theorem C (p. 498), which the paper attributes to Rossi and Weitsman: for nonconstant harmonic $u$ in $\mathbb C$ there are paths $\Gamma_1$ and $\Gamma_2$ to $\infty$ on which $u(z)\ge\lvert z\rvert^{1/2-o(1)}$ (the growth that Barth, Brannan and Hayman obtained along a path), with $\int_{\Gamma_1}u^{-(2+\alpha)}\lvert dz\rvert<\infty$ for any $\alpha>0$ and $L(\Gamma_2(z))\le u(z)^{2+B+o(1)}$, $B$ a positive absolute constant. The paper says Theorem 2 generalizes Theorem C because every nonconstant harmonic function in $\mathbb C$ has lower order at least $1$, and that it shows the constant $B$ can be eliminated.

Sharpness and conjecture (pp. 499--500)

• Remark 3: the theorem is sharp for $0<\lambda\le\tfrac12$. A modification of an example of Barth, Brannan and Hayman gives a harmonic $u$ of lower order $\infty$ with $\int_\Gamma\lvert u\rvert^{-2}\lvert dz\rvert=\infty$ on every path $\Gamma$ to $\infty$ on which $u>0$. For any $\lambda$ and $\rho$ with $1\le\lambda\le\rho<\infty$, Eremenko constructed an entire $f$ of lower order $\lambda$ and order $\rho$ with $\int_\Gamma(\log\lvert f(z)\rvert)^{-(2-1/\rho)}\lvert dz\rvert=\infty$ on every path $\Gamma$ to $\infty$ on which $\lvert f\rvert>1$. From these the paper concludes that $K=2$ cannot be reduced when $1\le\lambda\le\infty$. For $\tfrac12<\lambda<1$ the paper has no example and thinks $K=2$ is not best possible.
• Remark 4: the paper conjectures that (1.7) holds with $K=1/\lambda$ for $0<\lambda<1$ and $K=2-1/\rho$ for $1\le\lambda<\infty$, where $\rho$ is the order of $u$.

Proof, as a pointer

Section IV, pp. 501--504. Only (1.5) needs proof; (1.6) and (1.7) follow from it (p. 501). The path is a union of curves $\Gamma_n$ from $P_n$ to $P_{n+1}$, each a chain of paths given by Theorem 1 in nested components of sublevel sets of $u$, with $\sum_1^nL(\Gamma_k)\le u(P_{n+1})^{K+1/(n+1)}$; the cases $\lambda>\tfrac12$ and $\lambda\le\tfrac12$ are estimated separately. The proof was not checked here.

Dependencies

Theorem 1 (p. 497), through its constant $c_2$ in (1.2).

Bears on

Problem 514: the paper does not mention Erdős or the problem. Its theorem is stated for subharmonic $u$ of positive lower order, with $M(r)$ the maximum of $u$ on $\lvert z\rvert=r$, and bounds the length of the path up to $z$ by a power of $u(z)$; the paper draws no consequence for entire functions.