Inequalities for the Polar Derivative of a Polynomial

Ahmad Zireh

Abstract and Applied Analysis, 2012, vol. 2012, 1-13

Abstract:

For a polynomial ð ‘ ( 𠑧 ) of degree ð ‘› , we consider an operator ð · ð ›¼ which map a polynomial ð ‘ ( 𠑧 ) into ð · ð ›¼ ð ‘ ( 𠑧 ) ∶ = ( ð ›¼ − 𠑧 ) ð ‘ â€² ( 𠑧 ) + ð ‘› ð ‘ ( 𠑧 ) with respect to ð ›¼ . It was proved by Liman et al. (2010) that if ð ‘ ( 𠑧 ) has no zeros in | 𠑧 | < 1 , then for all ð ›¼ , ð ›½ ∈ â„‚ with | ð ›¼ | ≥ 1 , | ð ›½ | ≤ 1 and | 𠑧 | = 1 , | 𠑧 ð · ð ›¼ ð ‘ ( 𠑧 ) + ð ‘› ð ›½ ( ( | ð ›¼ | − 1 ) / 2 ) ð ‘ ( 𠑧 ) | ≤ ( ð ‘› / 2 ) { [ | ð ›¼ + ð ›½ ( ( | ð ›¼ | − 1 ) / 2 ) | + | 𠑧 + ð ›½ ( ( | ð ›¼ | − 1 ) / 2 ) | ] m a x | 𠑧 | = 1 | ð ‘ ( 𠑧 ) | − [ | ð ›¼ + ð ›½ ( ( | ð ›¼ | − 1 ) / 2 ) | − | 𠑧 + ð ›½ ( ( | ð ›¼ | − 1 ) / 2 ) | ] m i n | 𠑧 | = 1 | ð ‘ ( 𠑧 ) | } . In this paper we extend the above inequality for the polynomials having no zeros in | 𠑧 | < 𠑘 , where 𠑘 ≤ 1 . Our result generalizes certain well-known polynomial inequalities.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:181934

DOI: 10.1155/2012/181934

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