• Bulletin of the Korean Mathematical Society
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• v.46 no.6
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• pp.1153-1158
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• 2009

Given two integral self-reciprocal polynomials having the same modulus at a point $z_0$ on the unit circle, we show that the minimal polynomial of $z_0$ is also self-reciprocal and it divides an explicit integral self-reciprocal polynomial. Moreover, for any two integral self-reciprocal polynomials, we give a sufficient condition for the existence of a point $z_0$ on the unit circle such that the two polynomials have the same modulus at $z_0$.

• Kyungpook Mathematical Journal
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• v.53 no.4
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• pp.541-552
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• 2013

This research is a continuation of a recent paper due to the first author in [9]. Different from previous results, we investigate the value distribution of difference polynomials of moromorphic functions in this paper. In particular, we are interested in the existence of zeros of $f(z)^n({\lambda}f(z+c)^m+{\mu}f(z)^m)-a$, where f is a moromorphic function, n, m are two non-negative integers, and ${\lambda}$, ${\mu}$ are non-zero complex numbers. However, the proof here is obviously different to the one in [9]. We also study difference polynomials of entire functions sharing a common value, which improves the result in [10, 13].

• Kyungpook Mathematical Journal
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• v.61 no.4
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• pp.845-858
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• 2021

We present new bounds for the numerical radius of bounded linear operators and 2 × 2 operator matrices. We apply upper bounds for the numerical radius to the Frobenius companion matrix of a complex monic polynomial to obtain new estimations for the zeros of that polynomial. We also show with numerical examples that our new estimations improve on the existing estimations.

• Kyungpook Mathematical Journal
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• v.55 no.3
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• pp.689-694
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• 2015

In this paper we consider the class of polynomials of the type $p(z)=z^s(a_0+{\Sigma}_{j={\mu}}^{n-s}ajz^j)$, $1{\leq}{\mu}{\leq}n-s$, $0{\leq}s{\leq}n-1$ having some zeros at origin and rest of zeros on or outside the boundary of a prescribed disk, and obtain the generalization of well known results.

• Korean Journal of Mathematics
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• v.30 no.2
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• pp.305-313
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• 2022

In this paper we consider a modified version of Smirnov operator and obtain some Bernstein-type inequalities preserved by this operator. In particular, we prove some results which in turn provide the compact generalizations of some well-known inequalities for polynomials.

• Korean Journal of Mathematics
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• v.29 no.4
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• pp.825-831
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• 2021

The derivative of a polynomial p(z) of degree n, with respect to point α is defined by D_αp(z) = np(z) + (α - z)p'(z). Let p(z) be a polynomial having all its zeros in the unit disk |z| ≤ 1. The Sendov conjecture asserts that if all the zeros of a polynomial p(z) lie in the closed unit disk, then there must be a zero of p'(z) within unit distance of each zero. In this paper, we obtain certain results concerning the location of the zeros of D_αp(z) with respect to a specific zero of p(z) and a stronger result than Sendov conjecture is obtained. Further, a result is obtained for zeros of higher derivatives of polynomials having multiple roots.

• Journal of applied mathematics & informatics
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• v.35 no.3_4
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• pp.303-311
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• 2017

In this paper we define (p, q)-analogue of Euler zeta function. In order to define (p, q)-analogue of Euler zeta function, we introduce the (p, q)-analogue of Euler numbers and polynomials by generalizing the Euler numbers and polynomials, Carlitz's type q-Euler numbers and polynomials. We also give some interesting properties, explicit formulas, a connection with (p, q)-analogue of Euler numbers and polynomials. Finally, we investigate the zeros of the (p, q)-analogue of Euler polynomials by using computer.

• Korean Journal of Mathematics
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• v.28 no.4
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• pp.931-942
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• 2020

For a polynomial $P(z)={\sum_{{\nu}=0}^{n}}\;a_{\nu}z^{\nu}$ of degree n having all its zeros in |z| ≤ k, k ≥ 1, it was shown by Rather and Dar [13] that ${\max_{{\mid}z{\mid}=1}}{\mid}P^{\prime}(z){\mid}{\geq}{\frac{1}{1+k^n}}\(n+{\frac{k^n{\mid}a_n{\mid}-{\mid}a_0{\mid}}{k^n{\mid}a_n{\mid}+{\mid}a_0{\mid}}}\){\max_{{\mid}z{\mid}=1}}{\mid}P(z){\mid}$. In this paper, we shall obtain some sharp estimates, which not only refine the above inequality but also generalize some well known Turán-type inequalities.

• International Journal of Internet, Broadcasting and Communication
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• v.6 no.1
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• pp.9-12
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• 2014

We present a mathematical method for calculation of transmission zero locations, determining a filtering characteristics of two-port systems. By adjusting element values based on the zero locations, the frequency-selectivity is characterized. The characteristic polynomial of ladder networks in externally-loaded feed-forward systems is considered by adopting chain matrices for subsystems. This method can be extended to other types of lumped systems with cross-coupled sections. We find out the zeros by solving characteristics polynomials of closed-form expressions in terms of Laplace impedances of elements. The pairs of complex zeros are shown to be solely from the cross-coupled portion of the system.

• Communications of the Korean Mathematical Society
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• v.28 no.3
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• pp.457-462
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• 2013

In the present paper, we analyse analytic continuation of weighted $q$-Genocchi numbers and polynomials. A novel formula for weighted $q$-Genocchi-zeta function $\tilde{\zeta}_{G,q}(s{\mid}{\alpha})$ in terms of nested series of $\tilde{\zeta}_{G,q}(n{\mid}{\alpha})$ is derived. Moreover, we introduce a novel concept of dynamics of the zeros of analytically continued weighted $q$-Genocchi polynomials.