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Inequalities for the derivative of polynomials with restricted zeros
N. A. Rather,Ishfaq Dar,A. Iqbal 강원경기수학회 2020 한국수학논문집 Vol.28 No.4
For a polynomial $\mathit{P(z)=\sum_{\nu =0}^{n}a_{\nu}z^{\nu}}$ of degree $\mathit{n}$ having all its zeros in $\mathit{|z|\leq k,k \geq 1}$, it was shown by Rather and Dar \cite{1} that $$\max_{|z|=1} |P^{\prime}(z)|\geq \frac{1}{1+k^n}\bigg(n+\frac{k^n|a_n|-|a_0|}{k^n|a_n|+|a_0|}\bigg)\max_{|z|=1}|P(z)|.$$ In this paper, we shall obtain some sharp estimates, which not only refine the above inequality but also generalize some well known Tur\'{a}n-type inequalities.
ON AN OPERATOR PRESERVING POLYNOMIAL INEQUALITIES
Rather, N.A.,Ali, Liyaqat,Dar, Ishfaq The Kangwon-Kyungki Mathematical Society 2021 한국수학논문집 Vol.29 No.2
In this paper, we consider an operator N : 𝓟<sub>n</sub> → 𝓟<sub>n</sub> on the space of polynomials 𝓟<sub>n</sub> of degree at most n and establish some compact generalizations of Bernstein-type polynomial inequalities, which include several well known polynomial inequalities as special cases.
INEQUALITIES CONCERNING THE POLAR DERIVATIVES OF POLYNOMIALS WITH RESTRICTED ZEROS
Rather N. A.,Iqbal A.,Dar Ishfaq 경남대학교 수학교육과 2019 Nonlinear Functional Analysis and Applications Vol.24 No.4
In this paper some inequalities for the maximum modulus of the polar derivative for polynomials with restricted zeros are obtained by using the boundary Schwarz lemma of Osserman. Our results generalize and re ne some well-known results concerning the polynomials due to Turan, Dubinin and others.