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RISS 인기검색어
- 검색키워드
- 저자 : Majumder Sujoy (검색결과 4 건)
- 무료
- 기관 내 무료
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SOME RESULTS ON THE QUESTIONS OF KIT-WING YU
Majumder, Sujoy Korean Mathematical Society 2016 대한수학회논문집 Vol.31 No.2
The paper deals with the problem of meromorphic functions sharing a small function with its differential polynomials and improves the results of Liu and Gu [9], Lahiri and Sarkar [8], Zhang [13] and Zhang and Yang [14] and also answer some open questions posed by Kit-Wing Yu [16]. In this paper we provide some examples to show that the conditions in our results are the best possible.
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A RESULT ON A CONJECTURE OF W. LÜ, Q. LI AND C. YANG
Majumder, Sujoy Korean Mathematical Society 2016 대한수학회보 Vol.53 No.2
In this paper, we investigate the problem of transcendental entire functions that share two values with one of their derivative. Let f be a transcendental entire function, n and k be two positive integers. If $f^n-Q_1$ and $(f^n)^{(k)}-Q_2$ share 0 CM, and $n{\geq}k+1$, then $(f^n)^{(k)}{\equiv}{\frac{Q_2}{Q_1}}f^n$. Furthermore, if $Q_1=Q_2$, then $f=ce^{\frac{\lambda}{n}z}$, where $Q_1$, $Q_2$ are polynomials with $Q_1Q_2{\not\equiv}0$, and c, ${\lambda}$ are non-zero constants such that ${\lambda}^k=1$. This result shows that the Conjecture given by W. $L{\ddot{u}}$, Q. Li and C. Yang [On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc. 51 (2014), no. 5, 1281-1289.] is true. Also we exhibit some examples to show that the conditions of our result are the best possible.
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A result on a conjecture of W. L\"{u}, Q. Li and C. Yang
Sujoy Majumder 대한수학회 2016 대한수학회보 Vol.53 No.2
In this paper, we investigate the problem of transcendental entire functions that share two values with one of their derivative. Let $f$ be a transcendental entire function, $n$ and $k$ be two positive integers. If $f^{n}-Q_{1}$ and $(f^{n})^{(k)}-Q_{2}$ share $0$ CM, and $n\geq k+1$, then $(f^{n})^{(k)}\equiv \frac{Q_{2}}{Q_{1}}f^{n}$. Furthermore, if $Q_{1}=Q_{2}$, then $f=ce^{\frac{\lambda}{n}z}$, where $Q_{1}$, $Q_{2}$ are polynomials with $Q_{1}Q_{2}\not\equiv 0$, and $c$, $\lambda$ are non-zero constants such that $\lambda^{k}=1$.This result shows that the Conjecture given by W. L\"u, Q. Li and C. Yang [{{On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc. 51 (2014), no. 5, 1281--1289.}}] is true. Also we exhibit some examples to show that the conditions of our result are the best possible.
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A RESULT ON AN OPEN PROBLEM OF LÜ, LI AND YANG
Majumder, Sujoy,Saha, Somnath Korean Mathematical Society 2021 대한수학회보 Vol.58 No.4
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, c<sub>1</sub>, c<sub>2</sub>, …, c<sub>n</sub> ∈ ℂ\{0} and k, n ∈ ℕ. Suppose f<sup>n</sup>(z), f(z+c<sub>1</sub>)f(z+c<sub>2</sub>) ⋯ f(z+c<sub>n</sub>) share 0 CM and f<sup>n</sup>(z)-Q<sub>1</sub>(z), (f(z+c<sub>1</sub>)f(z+c<sub>2</sub>) ⋯ f(z+c<sub>n</sub>))<sup>(k)</sup> - Q<sub>2</sub>(z) share (0, 1), where Q<sub>1</sub>(z) and Q<sub>2</sub>(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 Q<sub>1</sub>(z) ≡ Q<sub>2</sub>(z), then $f(z)=c\;e^{\frac{\lambda}{n}z}$, where c, λ ∈ ℂ \ {0} such that e<sup>λ(c<sub>1</sub>+c<sub>2</sub>+⋯+c<sub>n</sub>)</sup> = 1 and λ<sup>k</sup> = 1. Also we exhibit some examples to show that the conditions of our result are the best possible.
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