• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.807-816
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• 2015

Let $P(z)={\limits\sum_{j=0}^{n}}a_jz^j$ be a polynomial of degree n and Re $a_j={\alpha}_j$, Im $a_j=B_j$. In this paper, we have obtained a zero-free region for polynomials in terms of ${\alpha}_j$ and ${\beta}_j$ and also obtain the bound for number of zeros that can lie in a prescribed region.

• The Pure and Applied Mathematics
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• v.31 no.1
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• pp.49-56
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• 2024

For a Polynomial P(z) = Σⁿ_j=0 a_jz^j with a_j ≥ a_j-1, a₀ > 0 (j = 1, 2, ..., n), a classical result of Enestrom-Kakeya says that all the zeros of P(z) lie in |z| ≤ 1. This result was generalized by A. Joyal et al. [3] where they relaxed the non-negative condition on the coefficents. This result was further generalized by Dewan and Bidkham [9] by relaxing the monotonicity of the coefficients. In this paper, we use some techniques to obtain some more generalizations of the results [3], [8], [9].

• 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.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.

• Nonlinear Functional Analysis and Applications
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• v.28 no.4
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• pp.943-956
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• 2023

Let p(z) be a polynomial of degree n and for any complex number 𝛽, let D_𝛽p(z) = np(z) + (𝛽 - z)p'(z) denote the polar derivative of the polynomial with respect to 𝛽. In this paper, we consider the class of polynomial $$p(z)=(z-z_0)^s \left(a_0+\sum\limits_{{\nu}=0}^{n-s}a_{\nu}z^{\nu}\right)$$ of degree n having a zero of order s at z₀, |z₀| < 1 and the remaining n - s zeros are outside |z| < k, k ≥ 1 and establish upper bound estimates for the maximum of |D_𝛽p(z)| as well as |p(Rz) - p(rz)|, R ≥ r ≥ 1 on the unit disk.

• Korean Journal of Mathematics
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• v.30 no.2
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• pp.199-203
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• 2022

Let f(z) be an entire function of exponential type τ such that ║f║ = 1. Also suppose, in addition, that f(z) ≠ 0 for ℑz > 0 and that $h_f(\frac{\pi}{2})=0$. Then, it was proved by Gardner and Govil [Proc. Amer. Math. Soc., 123(1995), 2757-2761] that for y = ℑz ≤ 0 $${\parallel}D_{\zeta}[f]{\parallel}{\leq}\frac{\tau}{2}({\mid}{\zeta}{\mid}+1)$$, where D_ζ[f] is referred to as polar derivative of entire function f(z) with respect to ζ. In this paper, we prove an inequality in the opposite direction and thereby obtain some known inequalities concerning polynomials and entire functions of exponential type.

• The Korean Journal of Applied Statistics
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• v.24 no.3
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• pp.485-494
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• 2011

When the observations can take only the non-negative integer values, it is called the count data such as the numbers of car accidents, earthquakes, or insurance coverage. In general, the Poisson regression model has been used to model these count data; however, this model has a weakness in that it is restricted by the equality of the mean and the variance. On the other hand, the count data often tend to be too dispersed to allow the use of the Poisson model in practice because the variance of data is significantly larger than its mean due to heterogeneity within groups. When overdispersion is not taken into account, it is expected that the resulting parameter estimates or standard errors will be inefficient. Since coverage is the main issue for insurance, some accidents may not be covered by insurance, and the number covered by insurance may be zero. This paper considers the zero-inflated model for the count data including many zeros. The performance of this model has been investigated by using of real data with overdispersion and many zeros. The results indicate that the Zero-Inflated Negative Binomial Regression Model performs the best for model evaluation.