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Let ... be a polynomial of degree n having all its zeros in ..., then Aziz [Proc. Am. Math. Soc., 89, (1983) 259--266] proved ... . In this paper, we prove a polar derivative extension which sharpens the above inequality. As a consequence, we also derive a result on Bernstein type inequality for the class of polynomials having all its zeros in ... . [ABSTRACT FROM AUTHOR]

Let P(z) = a0 + Σn ν=µ aνzν, 1 ≤ µ ≤ n, be a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1. We obtain an improvement and a generalization of an inequality in polar derivative proved by Somsuwan and Nakprasit [1]. Further, we also extend a result proved by Chanam and Dewan [2] to its polar version. Besides, our results are also found to generalize and improve some known inequalities [ABSTRACT FROM AUTHOR]

If p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1; then for any real r ≥ 1, Aziz [J. Approx. Theory, 55(1988), 232-239] proved the integral mean inequality ... In this paper, we obtain an improved extension of the above inequality by using the polar derivative instead of the ordinary derivative and involving the leading coefficient and the constant term of the polynomial. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n. The polar derivative of p(z) with respect to a complex number a is defined by Da p(z) = np(z)+(a -z)p'(z). If p(z) has all its zeros in |z| ≥ k, k ≤ 1, then for |a| ≥ k, Aziz and Rather [Math. Inequal. Appl., 1, (1998), 231-238] proved. In this paper, we first improve as well as generalize the above inequality. Besides, we are able to prove an improvement of a result due to Govil and Mctume [Acta Math. Hungar., 104, (2004), 115-126] and also prove an inequality for a subclass of polynomials having all its zeros in |z| ≥ k, k ≤ 1. [ABSTRACT FROM AUTHOR]

In this paper we present two new examples of shift spaces of infinite type over two symbols. One of the spaces is shown to be densely Li-Yorke chaotic and Robinson chaotic, but not chaotic in the sense of Devaney, while the other one is found out to be Devaney chaotic. [ABSTRACT FROM AUTHOR]

We point out a aw in a result proved by Singh and Shah [Kyungpook Math. J., 57(2017), 537-543] which was recently published in Kyungpook Mathematical Journal. Further, we point out an error in another result of the same paper which we correct and obtain integral extension of the corrected form. [ABSTRACT FROM AUTHOR]