Uniqueness of derivatives of a meromorphic function and its shift - Pramanik, D. C.; Dey, N. (Matematychni Studii, 2025)
Let \( f \) be a meromorphic function in the complex plane \( \mathbb{C} \). The uniqueness problems concerning \( f(z) \) and its shifted counterpart \( f(z+c) \) under various shared-value and growth conditions have been extensively studied over the past two decades. Many researchers have explored the uniqueness of the derivatives of these functions when they share values, considering both cases: ignoring multiplicities (IM) and counting multiplicities (CM). Recently, attention has been directed toward the uniqueness of \( f^{(j)}(z) \) and \( f^{(k)}(z+c) \) in the context of three shared values—specifically, one shared value in the IM sense and two shared values in the partial CM~sense.The objective of this study is to refine and extend the existing sharing conditions. In this article, we investigate the uniqueness between \( f^{(j)}(z) \) and \( f^{(k)}(z+c) \) when sharing two values in the CM sense. We provide examples to demonstrate that the proposed conditions are optimal. Additionally, we examine how deficiency conditions affect value sharing and their influence on the unicity results. Our final result extends our investigation by proving the uniqueness between \( f^{(j)}(z) \) and \( f^{(k)}(z+c) \) for one CM shared value, subject to suitable deficiency conditions. We also show that the same result holds when these derivatives share a value in the IM sense, along with a stronger deficiency condition.