We consider uniqueness results for meromorphic functions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f :\mathbb {C}\rightarrow \widehat{\mathbb {C}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">→</mo> <mover accent="true"> <mi mathvariant="double-struck">C</mi> <mo stretchy="true">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> such that for certain values <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a \in \mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> the implication <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f(z)=a \Rightarrow f'(z)=a$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mo stretchy="false">⇒</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> </mrow> </math></EquationSource> </InlineEquation> holds, i.e. that f and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>f</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> share values partially. In particular, we give a result for four partially shared values.

Meromorphic Functions that Partially Share Values with Their First Derivative

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2026年3月12日

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Andreas Sauer,
Andreas Schweizer

摘要

We consider uniqueness results for meromorphic functions $f :\mathbb {C}\rightarrow \widehat{\mathbb {C}}$ $f : C \to \hat{C}$ such that for certain values $a \in \mathbb {C}$ $a \in C$ the implication $f(z)=a \Rightarrow f'(z)=a$ $f (z) = a \Rightarrow f^{'} (z) = a$ holds, i.e. that f and $f'$ $f^{'}$ share values partially. In particular, we give a result for four partially shared values.