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http://dx.doi.org/10.4134/BKMS.b200670

A RESULT ON AN OPEN PROBLEM OF LÜ, LI AND YANG  

Majumder, Sujoy (Department of Mathematics Raiganj University)
Saha, Somnath (Mehendipara Jr. High School)
Publication Information
Bulletin of the Korean Mathematical Society / v.58, no.4, 2021 , pp. 915-937 More about this Journal
Abstract
In this paper we deal with the open problem posed by Lü, Li and Yang [10]. In fact, we prove the following result: Let f(z) be a transcendental meromorphic function of finite order having finitely many poles, c1, c2, …, cn ∈ ℂ\{0} and k, n ∈ ℕ. Suppose fn(z), f(z+c1)f(z+c2) ⋯ f(z+cn) share 0 CM and fn(z)-Q1(z), (f(z+c1)f(z+c2) ⋯ f(z+cn))(k) - Q2(z) share (0, 1), where Q1(z) and Q2(z) are non-zero polynomials. If n ≥ k+1, then $(f(z+c_1)f(z+c_2)\;{\cdots}\;f(z+c_n))^{(k)}\;{\equiv}\;{\frac{Q_2(z)}{Q_1(z)}}f^n(z)$. Furthermore, if Q1(z) ≡ Q2(z), then $f(z)=c\;e^{\frac{\lambda}{n}z}$, where c, λ ∈ ℂ \ {0} such that eλ(c1+c2+⋯+cn) = 1 and λk = 1. Also we exhibit some examples to show that the conditions of our result are the best possible.
Keywords
Meromorphic function; derivative; small function;
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Times Cited By KSCI : 3  (Citation Analysis)
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