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SOME INTEGRAL INEQUALITIES FOR THE POLAR DERIVATIVE OF A POLYNOMIAL

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Analysis in Theory and Applications 2011年第4期27卷 340-350页

作者： abdullah mir Sajad Amin Baba Kashmir University India Govt.Hr.Sec.Institute India

If P（z） is a polynomial of degree n which does not vanish in ｜z｜〈 1, then it is recently proved by Rather [Jour. Ineq. Pure andAppl. Math., 9 （2008）, Issue 4, Art. 103] that for every γ 〉 0 and every real or complex number a with ｜ α ｜ ≥ 1,{∫ 2π 0｜DαP（e^iθ）｜γdθ｜}^1/γ≤n（｜α｜＋1）Cγ{∫2π0｜P（e^iθ）｜γ^dθ}^1/γ,Cγ={1/2π∫2π 0｜1＋e^iβ｜^γdβ}^-1/γ,where DaP（z） denotes the polar derivative of P（z） with respect to α. In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J. Approx. Theory, 54 （1988）, 306-313] as a special case.

关键词： polar derivative polynomial Zygmund inequality zeros

On Growth of Polynomials with Restricted Zeros

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Analysis in Theory and Applications 2016年第2期32卷 181-188页

作者： abdullah mir G.N.Parrey Department of Mathematics University of Kashmir Department of Mathematics Islamia College of Science and Commerce

Let P（z） be a polynomial of degree n which does not vanish in ｜z｜〈k, k≥ 1. It is known that for each 0 〈 s 〈 n and 1 ≤ R ≤ k,M（P（s）,R）≤（1/（R^s＋k^s）[{d（s）/dx（s）（1＋x^n）}x=1]（（R＋k）/（1＋k）^nM（P,1）. In this paper, we obtain certain extensions and refinements of this inequality by in- volving binomial coefficients and some of the coefficients of the polynomial P（z）.

关键词： Polynomial maximum modulus princple zeros.

Some Results on the Growth of Polynomials

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Analysis in Theory and Applications 2017年第4期33卷 316-322页

作者： abdullah mir Ajaz Wani Imtiaz Hussain Department of Mathematics University of Kashmir

In this paper, we have studied the Lacunary type of polynomials and proved a result which generalizes as well as refines some well-known polynomial inequalities regarding the growth of polynomials not vanishing inside... 详细信息

关键词： Lacunary polynomial Growth Bernstien-inequality.

Some Lγ Inequalities for the Polar Derivative of a Polynomial

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Analysis in Theory and Applications 2017年第1期33卷 1-10页

作者： abdullah mir M. Bidkham Bilal Dar Department of Mathematics University of KashmirSrinagar190006India Department of Mathematics Semnan UniversitySemnanIran Govt.Degree College Pulwama

In this paper, we consider an operator Da which maps a polynomial P（z） in to DaP（z）：= np（z）＋（a-z）P＇（z）, where and obtain some Lγ inequalities for lucanary polynomials having zeros in /z/ 〈 k 〈 1. Our results yields several generaliza- tions and refinements of many known results and also provide an alternative proof of a result due to Dewan et al. [7], which is independent of Laguerre＇s theorem.

关键词： Polar derivative polynomials Lγ-inequalities in the complex domain Laguerre'stheorem.

A Note on a Theorem of ***

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Analysis in Theory and Applications 2018年第4期34卷 293-296页

作者： abdullah mir Ajaz Wani Imtiaz Hussain Department of Mathematics University of KashmirSrinagar190006India

In this paper, we obtain a result that improves the results of Govil and Nwaeze, Qazi and the classical result of Rivlin.

关键词： Polynomial maximum modulus zeros

Some Integral Mean Estimates for Polynomials with Restricted Zeros

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Analysis in Theory and Applications 2015年第1期31卷 81-91页

作者： abdullah mir Q.M.Dawood Bilal Dar Department of Mathematics University of Kashmir

Let P（z） be a polynomial of degree n having all its zeros in ｜z｜≤ k. For k = 1, it is known that for each r 〉 0 and ｜a｜≥1,n（｜a｜-1）{∫0^2π｜P（e^iθ）｜^rdθ}^1/r≤{∫0^2π｜1＋e^iθ｜^rdθ}^1/rmax｜z｜=1｜DzP（z）｜.In this paper, we shall first consider the case when k≥1 and present certain generaliza- tions of this inequality. Also for k≤ 1, we shall prove an interesting result for Lacunary type of polynomials from which many results can be easily deduced.

关键词： Polynomial zeros polar derivative.

Some Results on the Polar Derivative of a Polynomial

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Analysis in Theory and Applications 2014年第3期30卷 306-317页

作者： abdullah mir Bilal Dar Department of Mathematics University of Kashmir

Let P（z） be a polynomial of degree n and for any complex number α, let DαP（z） = nP（z）＋（α-z）P＇（z） denote the polar derivative of P（z） with respect to α. In this paper, we obtain certain inequalities for the polar derivative of a polynomial with restricted zeros. Our results generalize and sharpen some well-known polynomial inequalities.

关键词： Polynomial zeros polar derivative Bemstein inequality.

Some Inequalities Concerning the Polar Derivative of a Polynomial-II

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Analysis in Theory and Applications 2013年第4期29卷 384-389页

作者： abdullah mir Bilal Dar Department of Mathematics University of KashmirSrinagar190006 India

In this paper, we consider the class of polynomials P（z）= anz^n＋ ∑vn=μan-vz^n-v,1≤μ≤n , having all zeros in ｜z｜≤k, k ≤1 and thereby present an alternative proof, independent of Laguerre＇s theorem, of an... 详细信息

关键词： Polar derivative of a polynomial.

New Inequalities on L_γ-Spaces

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Analysis in Theory and Applications 2013年第4期29卷 390-400页

作者： abdullah mir Bilal Dar Sajad Amin Baba Department of Mathematics University of KashmirSrinagar190006India Govt.Hr.Sec.Institute KurhamaGandarbal-191201India

We consider for a fixed μ, the class of polynomialsPn,μ,s：＋{P（z）=z^2（anz^n-s＋∑j=μ^n-s an-jz^n-j-s）;1≤μ≤n-s}of degree n, having all zeros in ｜ z｜ ≤ k, k≤ 1, with s-fold zeros at the origin. In this paper, we have obtained inequalities in the reverse direction for the above class of polynomials. Besides, extensions of some Turan-type inequalities for the polar derivative of polynomials have been considered.

关键词： Polynomial Zygmund inequality polar derivative.