IM-Sharing Variants of Theorem H for Entire Functions

Definitions and mathematical statement

Let $f:{\mathbb{C}}\to{\mathbb{C}}$ be nonconstant and entire, let $k\geq2$ be an integer, and let $a,b\in{\mathbb{C}}$ satisfy $b\ne0$ and $b\ne a$. Sharing $a$ ignoring multiplicity (IM) means equality of the level sets: \[ \forall z\in{\mathbb{C}},\qquad f(z)=a\iff f'(z)=a, \] without requiring equal zero multiplicities of $f-a$ and $f'-a$. Sharing counting multiplicity (CM) additionally requires those multiplicities to agree. The separate hypothesis on $b$ is only the implication \[ \forall z\in{\mathbb{C}},\qquad f'(z)=b\Longrightarrow f^{(k)}(z)=b. \] The requested conclusion is retained exactly: \[ \exists c,d\in{\mathbb{C}}^\times,\quad c^{k-1}=1,\qquad f(z)=d e^{cz}+c^{-1}a\quad\text{for every }z\in{\mathbb{C}}. \] The original record attributes this conclusion under CM sharing to Theorem H and asks whether it remains true under IM sharing.

There is a material consistency problem in the supplied formula: with $k=2$, $a=1$, $b=2$ and $f(z)=e^z$, both displayed hypotheses hold, even with CM sharing, because $f=f'=f''$. But the proposed conclusion forces $c=1$ and $e^z=d e^z+1$, which cannot hold identically. Thus the printed CM theorem and proposed IM conclusion cannot have exactly these hypotheses and this formula simultaneously. The source must be checked for a missing condition or a different formula; no correction, such as deleting or changing the additive constant, is guessed here.

Short English statement

The source asks whether its CM-sharing uniqueness conclusion survives IM sharing, while retaining a separate one-way condition at $b$. The exact formula is preserved, but it already conflicts with $f=e^z$, $k=2$, $a=1$, $b=2$, so the intended theorem requires source correction.

Sources

• Supplied report: Normal Families and Complex Dynamics (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/09, pp. 487--548; proposed year 2007. https://doi.org/10.4171/owr/2007/09.
• Supplied reference labelled "IM-sharing special cases": https://citeseerx.ist.psu.edu/document?doi=1ad2e1a1746f584176b55465340661e21d712c62&repid=rep1&type=pdf.
• Supplied reference labelled "CM-sharing uniqueness results": https://www.sciencedirect.com/science/article/pii/S0022247X00970070.
• Local UnsolvedMath registry, permanent ID 30000700, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note calls the extracted hypotheses and conclusion complete. The original names Theorem H. Its supplied 21 August 2026 literature triage labels the IM replacement open and reports nearby implications and special cases without establishing that exact replacement with the separate $b$-condition. These dated labels are retained, but do not resolve the elementary inconsistency identified above. No PDF access or independent literature assessment is asserted here.