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<p>This paper investigates the relationship between the shift of an entire function with hyper-order less than 1 and its <i>k</i>th order derivative, focusing on cases where both functions share a doubleton set of small functions. To the best of the author’s knowledge, the most significant existing result in this context is due to Huang and Fang [Comput Methods Funct Theory, 21:523–532, 2021), who studied two-value (IM) sharing between the shift of an entire function <i>f</i> and its first derivative <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f^{\prime }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>f</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>. However, when it comes to the sharing properties between the shift of an entire function and its <i>k</i>th order derivative, particularly in both the value sharing and set sharing contexts with a minimal cardinality of two, no significant work has been done. Therefore, the main result of this paper provides a significant contribution to this area of research. Additionally, this finding refines a result from Banerjee and Roy (J Anal 32:1265–1280, 2024) by reducing the cardinality of the shared set.</p>

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The kth order derivative of an entire function and its shift operator sharing a doubleton of small functions CM

  • 期刊论文

  • 发表日期：

    2026年4月16日

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  • Arpita Roy

摘要

This paper investigates the relationship between the shift of an entire function with hyper-order less than 1 and its kth order derivative, focusing on cases where both functions share a doubleton set of small functions. To the best of the author’s knowledge, the most significant existing result in this context is due to Huang and Fang [Comput Methods Funct Theory, 21:523–532, 2021), who studied two-value (IM) sharing between the shift of an entire function f and its first derivative \(f^{\prime }\) f ′ . However, when it comes to the sharing properties between the shift of an entire function and its kth order derivative, particularly in both the value sharing and set sharing contexts with a minimal cardinality of two, no significant work has been done. Therefore, the main result of this paper provides a significant contribution to this area of research. Additionally, this finding refines a result from Banerjee and Roy (J Anal 32:1265–1280, 2024) by reducing the cardinality of the shared set.