• Nonlinear Functional Analysis and Applications
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• v.28 no.3
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• pp.813-830
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• 2023

Let p(z) be a polynomial of degree n having no zero in |z| < 1. In this paper, by involving some coefficients of the polynomial, we prove an inequality that not only improves as well as generalizes the well-known result proved by Rivlin but also has some interesting consequences.

• Nonlinear Functional Analysis and Applications
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• v.26 no.2
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• pp.373-380
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• 2021

If $p(z)={\sum\limits_{{\nu}=0}^{n}}a_{\nu}z^{\nu}$ is a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then Govil [3] proved that $${\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p^{\prime}(z){\mid}\;{\leq}\;{\frac{n}{k^n+k^{n-1}}}\;{\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p(z){\mid}$$. In this paper, by involving certain coefficients of p(z), we not only improve the above inequality but also improve a result proved by Dewan and Mir [2].

• Nonlinear Functional Analysis and Applications
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• v.28 no.4
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• pp.919-928
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• 2023

Let P(z) be a polynomial of degree n and P(z) ≠ 0 in |z| < 1. Then for every real α and R > 1, Aziz [1] proved that $$\max\limits_{{\mid}z{\mid}=1}{\mid}P(Rz)-P(z){\mid}{\leq}{\frac{R^n-1}{2}}(M^2_{\alpha}+M^2_{{\alpha}+{\pi}})^{\frac{1}{2}}{\mid},$$ where $$M{\alpha}={\max\limits_{1{\leq}k{\leq}n}}{\mid}P(e^{i({\alpha}+2k{\pi})n}){\mid}.$$ In this paper, we establish some improvements and generalizations of the above inequality concerning the polynomials and their ordinary derivatives.

• 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.

• Nonlinear Functional Analysis and Applications
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• v.27 no.2
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• pp.323-329
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• 2022

Let $p(z)=\sum\limits_{\nu=0}^{n}a_{\nu}z^{\nu}$ be a polynomial of degree n and $p^{\prime}(z)$ its derivative. If $\max\limits_{{\mid}z{\mid}=r}{\mid}p(z){\mid}$ is denoted by M(p, r). If p(z) has all its zeros on |z| = k, k ≤ 1, then it was shown by Govil [3] that $$M(p^{\prime},\;1){\leq}\frac{n}{k^n+k^{n-1}}M(p,\;1)$$. In this paper, we first prove a result concerning the sth derivative where 1 ≤ s < n of the polynomial involving some of the co-efficients of the polynomial. Our result not only improves and generalizes the above inequality, but also gives a generalization to higher derivative of a result due to Dewan and Mir [2] in this direction. Further, a direct generalization of the above inequality for the sth derivative where 1 ≤ s < n is also proved.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.455-465
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• 2022

We point out a flaw in a result proved by Singh and Shah [Kyungpook Math. J., 57(2017), 537-543] which was recently published in Kyungpook Mathematical Journal. Further, we point out an error in another result of the same paper which we correct and obtain integral extension of the corrected form.

• Nonlinear Functional Analysis and Applications
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• v.27 no.4
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• pp.797-805
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• 2022

Let P(z) be a polynomial of degree n. A well-known inequality due to S. Bernstein states that if P ∈ P_n, then $$\max_{{\mid}z{\mid}=1}\,{\mid}P^{\prime}(z){\mid}\,{\leq}n\,\max_{{\mid}z{\mid}=1}\,{\mid}P(z){\mid}$$. In this paper, we establish some extensions and refinements of the above inequality to polar derivative and some other well-known inequalities concerning the polynomials and their ordinary derivatives.

• Nonlinear Functional Analysis and Applications
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• v.28 no.2
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• pp.421-437
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• 2023

Let p(z) be a polynomial of degree n having no zero in |z| < k, k ≥ 1. Then Malik [12] obtained the following inequality: $${_{max \atop {\mid}z{\mid}=1}{\mid}p{\prime}(z){\mid}{\leq}{\frac{n}{1+k}}{_{max \atop {\mid}z{\mid}=1}{\mid}p(z){\mid}.$$ In this paper, we shall first improve as well as generalize the above inequality. Further, we also improve the bounds of two known inequalities obtained by Govil et al. [8].

• 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.

• Bulletin of the Korean Chemical Society
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• v.28 no.1
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• pp.115-117
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• 2007