Uniqueness of entire function concerning derivatives and shifts
Sujoy Majumder, Nabadwip Sarkar
TL;DR
The paper addresses the uniqueness problem for entire functions in relation to their derivatives and shifts. It develops two main results: a t1-type theorem showing that if $f(z+c)$ and $f^{(k)}(z)$ share two distinct small functions $a_1$ and $a_2$ of $f$ IM (with $a_1a_2\notin\mathbb{C}$), then $f(z+c)\equiv f^{(k)}(z)$; and a t2-type strengthening under the condition $\rho_2(f)<1$ allowing similar sharing conclusions even when $a_1$ and $a_2$ have more general small-function behavior. The proofs rely on Nevanlinna theory, especially the second main theorem for small functions, and involve careful case analyses (polynomial vs. transcendental) and auxiliary constructions to force equality. The results extend and sharpen prior work by Majumder et al. and Huang–Fang, include sharpness aspects via counterexamples, and contribute to the broader understanding of unicity phenomena for entire functions under shifts and differential operators.
Abstract
In the paper, we investigate the uniqueness problem of entire function concerning its derivative and shift and obtain two results. On of our result solves the open problem posed by Majumder et al. (On a conjecture of Li and Yang, Hiroshima Math. J., 53 (2023), 199-223) and the other result improves and generalizes the recent result due to Huang and Fang (Unicity of entire functions concerning their shifts and derivatives, Comput. Methods Funct. Theory, 21 (2021), 523-532) in a large extend.