Holomorphic curves with bounded spherical derivative

Definitions and mathematical statement

Let $n\geq2$ and let $f=[F_0:\cdots:F_n]$ be a holomorphic map from $\mathbb C$ to $\mathbb P^n$, with entire coordinates having no common zero. Define \[ \|f'\|^2= \frac{\sum_{i<j}|F_i'F_j-F_iF_j'|^2} {(\sum_i|F_i|^2)^2}. \] Suppose $\|f'\|(z)\leq C|z|^\sigma$ for $|z|\geq1$, with some $\sigma>-1$ and finite $C$. Let $H_1,\ldots,H_q$ be distinct hyperplanes in general position: every at most $n+1$ of their defining linear forms are independent. Require that $f(\mathbb C)$ is contained in none of these hyperplanes.

Write $g_j=\ell_j(F)$ for a defining linear form and let $n_j(t)$ count its zeros in $|z|\leq t$ with multiplicity. Put \[ N_j(r)=n_j(0)\log r+ \int_0^r\frac{n_j(t)-n_j(0)}t\,dt,\qquad T(r)=\frac1{2\pi}\int_0^{2\pi}\log\|F(re^{it})\|\,dt -\log\|F(0)\|. \]

Must there be constants $C',R>0$, depending on the curve and hyperplanes, such that \[ \sum_{j=1}^q N_j(r) \geq(q+1-n)T(r)-C'r^{\sigma+1} \quad(r\geq R)? \] No exceptional set of radii is allowed. All intersection multiplicities are counted; the functions are not truncated counting functions. Linear nondegeneracy relative to every possible hyperplane is not assumed, only avoidance of containment in the specified targets. The growth condition controls the spherical derivative throughout the plane at infinity.

Short English statement

Does a polynomial bound on the spherical derivative force the strengthened hyperplane-counting inequality with the stated error term?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700041 (AMR-036-0041), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Duval–da Costa conjecture. Original source indexed by AMR-036-0041. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Holomorphic curves with bounds on the spherical derivative. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/dacosta.pdf.