• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.915-937
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• 2021

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, …, c_n ∈ ℂ\{0} and k, n ∈ ℕ. Suppose fⁿ(z), f(z+c₁)f(z+c₂) ⋯ f(z+c_n) share 0 CM and fⁿ(z)-Q₁(z), (f(z+c₁)f(z+c₂) ⋯ f(z+c_n))^(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 e^{λ(c₁+c₂+⋯+c_n)} = 1 and λ^k = 1. Also we exhibit some examples to show that the conditions of our result are the best possible.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.379-387
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• 2001

In this paper, we study the value distribution of meromorphic functions and prove the following theorem: Let f(z) be a transcendental meromorphic function. If f and f'have the same zeros, then f'(z) takes any non-zero value b infinitely many times.

• Communications of the Korean Mathematical Society
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• v.34 no.4
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• pp.1145-1155
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• 2019

Let f be a transcendental meromorphic function, defined in the complex plane $\mathbb{C}$. In this paper, we give a quantitative estimations of the characteristic function T(r, f) in terms of the counting function of a homogeneous differential polynomial generated by f. Our result improves and generalizes some recent results.

• Journal of applied mathematics & informatics
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• v.41 no.2
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• pp.373-386
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• 2023

In this paper we have obtained various forms of (p, q)-analogue of Aleph-Function satisfying Truesdell's ascending and descending F_p,q-equation. These structures have been employed to arrive at certain generating functions for (p, q)-analogue of Aleph-Function. Some specific instances of these outcomes as far as (p, q)-analogue of I-function, H-function and G-functions have likewise been obtained.

• Communications of the Korean Mathematical Society
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• v.32 no.2
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• pp.361-373
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• 2017

In this paper, we utilize the Nevanlinna theory and uniqueness theory of meromorphic function to investigate the differential-difference analogue of $Br{\ddot{u}}ck$ conjecture. In other words, we consider ${\Delta}_{\eta}f(z)=f(z+{\eta})-f(z)$ and f'(z) share one value or one small function, and then obtain the precise expression of transcendental entire function f(z) under certain conditions, where ${\eta}{\in}{\mathbb{C}}{\backslash}\{0\}$ is a constant such that $f(z+{\eta})-f(z){\not\equiv}0$.

• Bulletin of the Korean Mathematical Society
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• v.48 no.2
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• pp.247-260
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• 2011

Let h be a meromorphic function with few poles and zeros. By Nevanlinna's value distribution theory we prove some new properties on the polynomials in h with the coefficients being small functions of h. We prove that if f is a meromorphic function and if $f^m$ is identically a polynomial in h with the constant term not vanish identically, then f is a polynomial in h. As an application, we are able to find the entire solutions of the differential equation of the type $$f^n+P(f)=be^{sz}+Q(e^z)$$, where P(f) is a differential polynomial in f of degree at most n-1, and Q($e^z$) is a polynomial in $e^z$ of degree k $\leqslant$ max {n-1, s(n-1)/n} with small functions of $e^z$ as its coefficients.

• Bulletin of the Korean Mathematical Society
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• v.40 no.4
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• pp.565-576
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• 2003

In this paper, we determine the general solution of the quartic equation f(x+2y)+f(x-2y)+6f(x) = 4[f(x+y)+f(x-y)+6f(y)] for all x, $y\;\in\;\mathbb{R}$ without assuming any regularity conditions on the unknown function f. The method used for solving this quartic functional equation is elementary but exploits an important result due to M. Hosszu [3]. The solution of this functional equation is also determined in certain commutative groups using two important results due to L. Szekelyhidi [5].

• Communications of the Korean Mathematical Society
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• v.32 no.3
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• pp.689-707
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• 2017

In this paper, a boundary version of Schwarz lemma is investigated. We take into consideration a function f(z) holomorphic in the unit disc and f(0) = 0, f'(0) = 1 such that ${\Re}f^{\prime}(z)$ > ${\frac{1-{\alpha}}{2}}$, -1 < ${\alpha}$ < 1, we estimate a modulus of the second non-tangential derivative of f(z) function at the boundary point $z_0$ with ${\Re}f^{\prime}(z_0)={\frac{1-{\alpha}}{2}}$, by taking into account their first nonzero two Maclaurin coefficients. Also, we shall give an estimate below ${\mid}f^{{\prime}{\prime}}(z_0){\mid}$ according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and $z_1{\neq}0$. The sharpness of these inequalities is also proved.

• Korean Journal of Mathematics
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• v.28 no.4
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• pp.889-906
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• 2020

In this article, we consider the uniqueness problem of the shift polynomials $f^n(z)(f^m(z)-1){\prod\limits_{j=1}^{s}}f(z+c_j)^{{\mu}_j}$ and $f^n(z)(f(z)-1)^m{\prod\limits_{j=1}^{s}}f(z+c_j)^{{\mu}_j}$, where f(z) is a transcendental entire function of finite order, cj (j = 1, 2, …, s) are distinct finite complex numbers and n(≥ 1), m(≥ 1), s and µj (j = 1, 2, …, s) are integers. With the concept of weakly weighted sharing and relaxed weighted sharing we obtain some results which extend and generalize some results due to P. Sahoo [Commun. Math. Stat. 3 (2015), 227-238].

• Development and Reproduction
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• v.25 no.3
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• pp.193-197
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• 2021

In previous reports, bisphenol A (BPA) exposure affects reproductive function in Drosophila melanogaster females. To test the maternal effect of BPA exposure on fly reproductive function, F0 mothers were exposed to 0, 0.1, 1, and 10 mg/L of BPA and the fecundity in F1 and F2 generations were checked. In this experiment, 1 and 10 mg/L BPA significantly decreased the fecundity of F1 females. Moreover, 0.1 and 1 mg/L BPA substantially reduced egg production in the F2 generation. These results suggested that maternal exposure to BPA at enviromentally relavant concnetrations reduces reproductive function in Drosophila melanogaster females and that this effect is transgenerational.