Integral mean estimates for polynomials whose zeros are within a circle
Let p(z) be a polynomial of degree n having all its zeros in |z|≤k; k≤1, then for each r>0, p>1, q>1 with p−1+q−1=1, Aziz and Ahemad (1996) recently proved that n{∫02π|p(eiθ)|rdθ}1/r≤{∫02π|1+keiθ|prdθ}1/pr{∫02π|p′(eiθ)|qrdθ}1/qr. In this paper, we extend the above inequality to the class o...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S016117120100607X |
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| Summary: | Let p(z) be a polynomial of degree n having all its zeros in |z|≤k; k≤1, then for each r>0, p>1, q>1 with p−1+q−1=1, Aziz and Ahemad (1996) recently proved that n{∫02π|p(eiθ)|rdθ}1/r≤{∫02π|1+keiθ|prdθ}1/pr{∫02π|p′(eiθ)|qrdθ}1/qr. In this paper, we extend the above inequality to the class of polynomials p(z)=anzn+∑v=μnan−vzn−v;1≤μ≤n having all its zeros in |z|≤k; k≤1 and obtain a generalization as well as a refinement of the above
result. |
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| ISSN: | 0161-1712 1687-0425 |