Research Problems in Function Theory — Problem 1.43

Definitions and mathematical statement

For a nonconstant meromorphic function $f$, let $n(t,a)$ count solutions of $f(z)=a$ in $|z|\le t$, with multiplicity; $a=\infty$ means poles. Put \[ N(r,a)=\int_0^r\frac{n(t,a)-n(0,a)}t\,dt+n(0,a)\log r, \quad T(r,f)=N(r,\infty)+\frac1{2\pi}\int_0^{2\pi}\log^+|f(re^{i\theta})|\,d\theta, \] where $\log^+x=\max(0,\log x)$. At a boundary radius $R\in\{1,\infty\}$ with $T(r,f)\to\infty$, define the deficiency $\delta(a,f)=1-\limsup_{r\to R}N(r,a)/T(r,f)$. A deficient value has positive deficiency. Let $f$ be meromorphic in the plane, with lower order $\lambda=\liminf_{r\to\infty}\log T(r,f)/\log r$. Define $m_0(r,f)=\inf_{|z|=r}|f(z)|$ and $M(r,f)=\sup_{|z|=r}|f(z)|$. Use circles without poles when a quotient involving $M$ is evaluated, and set $\log m_0=-\infty$ on a circle containing a zero. Assume $\log r=o(\log M(r,f))$ as $r\to\infty$ along the eligible circles. The source establishes the following inequality for $0<\lambda<1/2$. Does it continue to hold for $1/2\le\lambda<1$? \[ \limsup_{r\to\infty}\frac{\log m_0(r,f)}{\log M(r,f)} +\pi\lambda\sin(\pi\lambda) \limsup_{r\to\infty}\frac{N(r,\infty)}{\log M(r,f)} \ge\cos(\pi\lambda). \] The two upper limits are taken separately. At $\lambda=1/2$ the right side is zero. The pole term is essential because the proposed extension is for meromorphic functions, not only entire ones.

Short English statement

Does the stated minimum-modulus inequality, with a correction for poles, extend from lower order below one-half to lower order below one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2301043 (AMR-022-1043), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 1.43. Original source indexed by AMR-022-1043. https://arxiv.org/abs/1809.07200.