In this paper, we generalise certain uniqueness results related to the partial sharing of small functions of a meromorphic function f, with a difference polynomial of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L(f)=b_k(z)f(z+kc)+\ldots +b_0(z)f(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>k</mi> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>b</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$b_i\in \mathscr {S}(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mi>i</mi> </msub> <mo>∈</mo> <mi mathvariant="script">S</mi> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$i=0,1,\ldots ,k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation> by considering the integrated counting functions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N_{p)}\displaystyle {\left( r,\frac{1}{f-a}\right) }$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <msub> <mi>N</mi> <mrow> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mfenced close=")" open="("> <mi>r</mi> <mo>,</mo> <mfrac> <mn>1</mn> <mrow> <mi>f</mi> <mo>-</mo> <mi>a</mi> </mrow> </mfrac> </mfenced> </mrow> </mstyle> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\overline{N}_{p)}\displaystyle {\left( r,\frac{1}{f-a}\right) }$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <msub> <mover> <mi>N</mi> <mo>¯</mo> </mover> <mrow> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mfenced close=")" open="("> <mi>r</mi> <mo>,</mo> <mfrac> <mn>1</mn> <mrow> <mi>f</mi> <mo>-</mo> <mi>a</mi> </mrow> </mfrac> </mfenced> </mrow> </mstyle> </math></EquationSource> </InlineEquation>, for any positive integer p. Thereafter, we have deduced the corresponding uniqueness results for the difference operator <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Delta _c^k f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="normal">Δ</mi> <mi>c</mi> <mi>k</mi> </msubsup> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> as corollaries.

Some generalisations on partial sharing of small functions

期刊论文
发表日期：

2025年9月2日

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Rupa Pal,
Audrija Choudhury

摘要

In this paper, we generalise certain uniqueness results related to the partial sharing of small functions of a meromorphic function f, with a difference polynomial of the form $L(f)=b_k(z)f(z+kc)+\ldots +b_0(z)f(z)$ $L (f) = b_{k} (z) f (z + k c) + \dots + b_{0} (z) f (z)$ , for $b_i\in \mathscr {S}(f)$ $b_{i} \in S (f)$ , $i=0,1,\ldots ,k$ $i = 0, 1, \dots, k$ by considering the integrated counting functions $N_{p)}\displaystyle {\left( r,\frac{1}{f-a}\right) }$ $N_{p)} (r, \frac{1}{f - a})$ , and $\overline{N}_{p)}\displaystyle {\left( r,\frac{1}{f-a}\right) }$ ${\bar{N}}_{p)} (r, \frac{1}{f - a})$ , for any positive integer p. Thereafter, we have deduced the corresponding uniqueness results for the difference operator $\Delta _c^k f$ $Δ_{c}^{k} f$ as corollaries.