In this paper, we investigate the uniqueness problem of P(f)L(g) and P(g)L(f) when they share <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha (z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> counting multiplicities, where L(h) represents any one of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$h^{(k)}(z),\; h(z+c),\; h(z+c)-h(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="0.277778em" /> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="0.277778em" /> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$h^{(k)}(z+c)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> where k is a positive integer, c is a nonzero constant and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha (z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a nonzero small function with respect to both f(z) and g(z). The results of the paper generalize the recent results of Gao and Liu (Bull Korean Math Soc 59:155–166, 2022).

Hayman conjecture and uniqueness of some delay-differential polynomials sharing a small function

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发表日期：

2026年3月4日

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Pulak Sahoo,
Suman Pal

摘要

In this paper, we investigate the uniqueness problem of P(f)L(g) and P(g)L(f) when they share $\alpha (z)$ $α (z)$ counting multiplicities, where L(h) represents any one of $h^{(k)}(z),\; h(z+c),\; h(z+c)-h(z)$ $h^{(k)} (z), h (z + c), h (z + c) - h (z)$ and $h^{(k)}(z+c)$ $h^{(k)} (z + c)$ where k is a positive integer, c is a nonzero constant and $\alpha (z)$ $α (z)$ is a nonzero small function with respect to both f(z) and g(z). The results of the paper generalize the recent results of Gao and Liu (Bull Korean Math Soc 59:155–166, 2022).