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...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 inequality concerning the polar derivative of a polynomial.展开更多
Let P（z） be a polynomial of degree n and for any complex number α, let D;P（z） = nP（z）+（α- z） P′（z） denote the polar derivative of the polynomial P（z） with respect to α. In this paper, we obtain inequal...Let P（z） be a polynomial of degree n and for any complex number α, let D;P（z） = nP（z）+（α- z） P′（z） denote the polar derivative of the polynomial P（z） with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all zeros inside a circle. Our results shall generalize and sharpen some well-known results of Turan, Govil, Dewan et al. and others.展开更多
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. Ou...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.展开更多
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...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.展开更多
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...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.展开更多
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...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.展开更多
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...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 a circle. Further the paper corrects the proofs of some already known results.展开更多
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）...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）.展开更多
In this paper we consider a class of polynomials P（z）=a0＋∑v^n=tavZv, t≥ 1, not vanishing in ｜z｜ 〈k, k≥1 and investigate the dependence of max｜z｜=1 ｜P（Rz） -P（rz）] on max｜z｜=1｜P（z）｜, where 1≤ r 〈...In this paper we consider a class of polynomials P（z）=a0＋∑v^n=tavZv, t≥ 1, not vanishing in ｜z｜ 〈k, k≥1 and investigate the dependence of max｜z｜=1 ｜P（Rz） -P（rz）] on max｜z｜=1｜P（z）｜, where 1≤ r 〈 R. Our result generalizes and refines some known polynomial inequalities.展开更多
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 pa...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.展开更多
基金supported by the University of Kashmir vide No: F (Seed Money Grant) RES/KU/13
文摘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 inequality concerning the polar derivative of a polynomial.
文摘Let P（z） be a polynomial of degree n and for any complex number α, let D;P（z） = nP（z）+（α- z） P′（z） denote the polar derivative of the polynomial P（z） with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all zeros inside a circle. Our results shall generalize and sharpen some well-known results of Turan, Govil, Dewan et al. and others.
文摘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.
文摘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.
文摘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.
基金supported by UGC under major research project scheme vide No. MRP-MAJOR-MATH-2013-29143
文摘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.
基金sponsored by UGC, Goverment of India under the Major Research Project Scheme vide No. MRP-MAJOR-MATH-2013-29143
文摘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 a circle. Further the paper corrects the proofs of some already known results.
文摘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）.
文摘In this paper we consider a class of polynomials P（z）=a0＋∑v^n=tavZv, t≥ 1, not vanishing in ｜z｜ 〈k, k≥1 and investigate the dependence of max｜z｜=1 ｜P（Rz） -P（rz）] on max｜z｜=1｜P（z）｜, where 1≤ r 〈 R. Our result generalizes and refines some known polynomial inequalities.
文摘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.
