• Abdul Aziz

Articles written in Proceedings – Mathematical Sciences

• On the zeros of polynomials

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

Let P(z) be a polynomial of degreen which does not vanish in ¦z¦ &lt;k, wherek &gt; 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

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

LetP(z) be a polynomial of degreen which does not vanish in |z|&lt;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

LetP(z) be a polynomial of degreen which does not vanish in the disk |z|&lt;k. It has been proved that for eachp&gt;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

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.

• Proceedings – Mathematical Sciences

Volume 133, 2023
All articles
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• Editorial Note on Continuous Article Publication

Posted on July 25, 2019