• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.155-166
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• 2022

In this paper, the paired Hayman conjecture of different types are considered, namely, the zeros distribution of f(z)ⁿL(g) - a(z) and g(z)ⁿL(f) - a(z), where L(h) takes the derivatives h^(k)(z) or the shift h(z+c) or the difference h(z+c)-h(z) or the delay-differential h^(k)(z+c), where k is a positive integer, c is a non-zero constant and a(z) is a nonzero small function with respect to f(z) and g(z). The related uniqueness problems of complex delay-differential polynomials are also considered.

• The Transactions of the Korean Institute of Electrical Engineers D
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• v.50 no.2
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• pp.59-64
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• 2001

In this paper, we propose an alternate method for determining the uniqueness of the reconstruction of a complex sequence from its phase. Uniqueness constraints could be derived in terms of the zeros of a complex polynomial defined by the DFT of the sequence. However, rooting of complex polynomials of high order is a very difficult problem. Instead of finding zeros of a complex polynomial, the proposed uniqueness criteria show that non-singularity of a matrix can guarantee the uniqueness of the reconstruction of a complex sequence from its phase-only data. It has clear advantage over the rooting method in numerical stability and computational time.

• Korean Journal of Mathematics
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• v.31 no.2
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• pp.145-152
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• 2023

Let $P(z):=\sum\limits^{n}_{j=0}A_jz^j$, A_j ∈ ℂ^m×m, 0 ≤ j ≤ n be a matrix polynomial of degree n, such that A_n ≥ A_n-1 ≥ . . . ≥ A₀ ≥ 0, A_n > 0. Then the eigenvalues of P(z) lie in the closed unit disk. This theorem proved by Dirr and Wimmer [IEEE Trans. Automat. Control 52(2007), 2151-2153] is infact a matrix extension of a famous and elegant result on the distribution of zeros of polynomials known as Eneström-Kakeya theorem. In this paper, we prove a more general result which inter alia includes the above result as a special case. We also prove an improvement of a result due to Lê, Du, Nguyên [Oper. Matrices, 13(2019), 937-954] besides a matrix extention of a result proved by Mohammad [Amer. Math. Monthly, vol.74, No.3, March 1967].

• Korean Journal of Mathematics
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• v.30 no.3
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• pp.525-532
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• 2022

Let 𝓟_n denote the space of all complex polynomials $P(z)=\sum\limits_{j=0}^{n}{\alpha}_jz^j$ of degree n. Let P ∈ 𝓟_n, for any complex number α, D_αP(z) = nP(z) + (α - z)P'(z), denote the polar derivative of the polynomial P(z) with respect to α and B_n denote a family of operators that maps 𝓟_n into itself. In this paper, we combine the operators B and D_α and establish certain operator preserving inequalities concerning polynomials, from which a variety of interesting results can be obtained as special cases.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.135-143
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• 2011

As a zero set of some multivariate splines, the piecewise algebraic variety is a kind of generalization of the classical algebraic variety. This paper presents an algorithm for isolating real roots of the zero-dimensional piecewise algebraic variety which is based on interval evaluation and the interval zeros of univariate interval polynomials in Bernstein form. An example is provided to show the proposed algorithm is effective.

• Kyungpook Mathematical Journal
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• v.46 no.1
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• pp.111-118
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• 2006

The present paper investigates the weighted $L^1$-convergence of Gr$\ddot{u}$nwald interpolatory operators based on the zeros of the second Chebyshev polynomials $U_n(x)=\frac{sin(n+1)\theta}{sin\theta}$. The approximation rate is sharp.

• Journal of Integrative Natural Science
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• v.10 no.1
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• pp.40-42
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• 2017

In this note, we study a generalization of a certain polynomial $z^n-{\sum\limits_{k=0}^{n-1}}a_kz^k$, where $\sum\limits_{k=0}^{n-1}a_k=1$, $a_k{\geq}0$ for each k, whose all zeros except for z=1 lie on the circle of radius 1/2 with center at the origin.

• 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.57 no.4
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• pp.1061-1073
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• 2020

In this paper, the expressions of meromorphic solutions of the following nonlinear complex differential equation of the form $$f^n+Qd(z,f)=\sum\limits_{i=1}^{3}pi(z)e^{{\alpha}_i(z)}$$ are studied by using Nevanlinna theory, where n ≥ 5 is an integer, Q_d(z, f) is a differential polynomial in f of degree d ≤ n - 4 with rational functions as its coefficients, p₁(z), p₂(z), p₃(z) are non-vanishing rational functions, and α₁(z), α₂(z), α₃(z) are nonconstant polynomials such that α'₁(z), α'₂(z), α'₃(z) are distinct each other. Moreover, examples are given to illustrate the accuracy of the condition.

• Nonlinear Functional Analysis and Applications
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• v.27 no.3
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• pp.691-700
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• 2022

Let p(z) be a polynomial of degree n having no zero inside the unit circle. Then for 0 < r ≤ 1, the well-known inequality due to Rivlin [Amer. Math. Monthly., 67 (1960) 251-253] is $$\max\limits_{{\mid}z{\mid}=r}{\mid}p(z){\mid}{\geq}{\(\frac{r+1}{2}\)^n}\max\limits_{{\mid}z{\mid}=1}{\mid}p(z){\mid}$$. In this paper, we generalize as well as sharpen the above inequality. Also our results not only generalize, but also sharpen some known results proved recently.