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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.
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참고문헌 (17)
참고문헌 신청
Abdul Aziz / 1983 / Inequalities for the derivative of a polynomial / Proc. Amer. Math. Soc 89(2):259~266
A. Aziz / 1998 / Inequalities for a polynomial and its derivatives / J. Approx. Theory 54:306~313
A. Aziz / 2004 / Some Zygmund type L q inequalities for polynomials / J. Math. Anal. Appl 289:14~29
A. Aziz / 2003 / Inequalities for the polar derivative of a polynomial with restricted zeros / Math. Bulk 17:15~28
A. Aziz / 1998 / A refinement of a theorem of Paul Tur´an concerning polynomials / Math. Ineq. Appl 1:231~238
A. Aziz / 1998 / Inequalities for a polynomial and its derivatives / J. Approx. Theory 54:306~313
A. Aziz / 2004 / Some Zygmund type L q inequalities for polynomials / J. Math. Anal. Appl 289:14~29
A. Aziz / 2003 / Inequalities for the polar derivative of a polynomial with restricted zeros / Math. Bulk 17:15~28
A. Aziz / 1998 / A refinement of a theorem of Paul Tur´an concerning polynomials / Math. Ineq. Appl 1:231~238
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