Some integral inequalities for the polar derivative of a polynomial |
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| Authors: | Abdullah Mir Sajad Amin Baba |
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| Affiliation: | 1.Kashmir University,Hazratbal, Srinagar,India;2.Govt. Hr. Sec. Institute,Kurhama Ganderbal,India;3.P. G. Department of Mathematics,Kashmir University,Hazratbal, Srinagar,India;4.Department of Mathematics,Govt. Hr. Sec. Institute,Kurhama Ganderbal,India |
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| Abstract: | 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 and Appl. Math., 9 (2008), Issue 4, Art. 103] that for every γ> 0 and every real or complex number α with |α| ≥ 1, | $begin{gathered}
left{ {int_0^{2pi } {left| {D_alpha P(e^{itheta } )} right|^gamma dtheta } } right}^{{1 mathord{left/
{vphantom {1 gamma }} right.
kern-nulldelimiterspace} gamma }} leqslant n(|alpha | + 1)C_gamma left{ {int_0^{2pi } {left| {P(e^{itheta } )} right|^gamma dtheta } } right}^{{1 mathord{left/
{vphantom {1 gamma }} right.
kern-nulldelimiterspace} gamma }} , hfill
C_gamma left{ {frac{1}
{{2pi }}int_0^{2pi } {left| {1 + e^{ibeta } } right|^gamma dbeta } } right}^{ - {1 mathord{left/
{vphantom {1 gamma }} right.
kern-nulldelimiterspace} gamma }} hfill
end{gathered}
$begin{gathered}
left{ {int_0^{2pi } {left| {D_alpha P(e^{itheta } )} right|^gamma dtheta } } right}^{{1 mathord{left/
{vphantom {1 gamma }} right.
kern-nulldelimiterspace} gamma }} leqslant n(|alpha | + 1)C_gamma left{ {int_0^{2pi } {left| {P(e^{itheta } )} right|^gamma dtheta } } right}^{{1 mathord{left/
{vphantom {1 gamma }} right.
kern-nulldelimiterspace} gamma }} , hfill
C_gamma left{ {frac{1}
{{2pi }}int_0^{2pi } {left| {1 + e^{ibeta } } right|^gamma dbeta } } right}^{ - {1 mathord{left/
{vphantom {1 gamma }} right.
kern-nulldelimiterspace} gamma }} hfill
end{gathered}
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