• Abdul Aziz

      Articles written in Proceedings – Mathematical Sciences

    • On the zeros of polynomials

      Abdul Aziz B A Zargar

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      In this paper we extend a classical result due to Cauchy and its improvement due to Datt and Govil to a class of lacunary type polynomials.

    • Inequalities for the derivative of a polynomial

      Abdul Aziz Nisar Ahmad

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      Let P(z) be a polynomial of degreen which does not vanish in ¦z¦ <k, wherek > 0. Fork ≤ 1, it is known that$$\mathop {\max }\limits_{|z| = 1} |P'(z)| \leqslant \frac{n}{{1 + k^n }}\mathop {\max }\limits_{|z| = 1} |P(z)|$$, provided ¦P’(z)¦ and ¦Q’(z)¦ become maximum at the same point on ¦z¦ = 1, where$$Q(z) = z^n \overline {P(1/\bar z)} $$. In this paper we obtain certain refinements of this result. We also present a refinement of a generalization of the theorem of Tuŕan.

    • On self-reciprocal polynomials

      Abdul Aziz B A Zargar

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      In this paper we establish a sharp result concerning integral mean estimates for self-reciprocal polynomials.

    • Some inequalities for the polar derivative of a polynomial

      Abdul Aziz Wali Mohammad Shah

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      LetP(z) be a polynomial of degreen which does not vanish in |z|<1. In this paper, we estimate the maximum and minimum moduli of thekth polar derivative ofP(z) on |z|=1 and thereby obtain compact generalizations of some known results, which among other results, yields interesting refinements of Erdos-Lax theorem and a theorem of Ankeny and Rivlin.

    • Lp inequalities for polynomials with restricted zeros

      Abdul Aziz W M Shah

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      LetP(z) be a polynomial of degreen which does not vanish in the disk |z|<k. It has been proved that for eachp>0 andk≥1,$$\begin{gathered} \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P^{(s)} (e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} \leqslant n(n - 1) \cdots (n - s + 1) B_p \hfill \\ \times \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P(e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} , \hfill \\ \end{gathered} $$ where$$B_p = \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {k^s + e^{i\alpha } } \right|^p d\alpha } } \right\}^{ - 1/p} $$ andP(s)(z) is thesth derivative ofP(z). This result generalizes well-known inequality due to De Bruijn. Asp→∞, it gives an inequality due to Govil and Rahman which as a special case gives a result conjectured by Erdös and first proved by Lax.

    • New integral mean estimates for polynomials

      Abdul Aziz Nisar Ahmad Rather

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      In this paper we prove someLP inequalities for polynomials, wherep is any positive number. They are related to earlier inequalities due to A Zygmund, N G De Bruijn, V V Arestov, etc. A generalization of a polynomial inequality concerning self-inversive polynomials, is also obtained.

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