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In this article, we extend inequalities concerning the polar derivative of a polynomial to integral mean for the class of polynomials with s-fold zero at the origin and the remaining zeros inside some closed disk of radius kk for k≥1k\ge 1 and k≤1k\le 1, and we thereby obtain bounds that depend on some coefficients of the underlying polynomial. Our results not only extend some known polynomial inequalities but also reduce some interesting results in particular cases. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to previously established results.

In this paper, we establish some upper bound estimates for the maximum modulus of the higher order derivative of a polynomial with restricted zeros on a circle of |z| = R, R ≥ 1, in terms of the maximum modulus of the same polynomial on the unit circle |z| = 1. These results generalize and sharpen some known results in this direction. [ABSTRACT FROM AUTHOR]

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]

In this paper, we extend and generalize an improved bound on the unit circle jzj = 1 concerning polar derivative of a polynomial of degree n having no zero in jzj < k, 0 < k ≤ 1 to an inequality involving the maximum modulus of the polar derivative of the polynomial on circles of radii r;R with 1 ≤ r ≤ R < 1. As consequences of our result, we are able to extend and improve some known polynomial inequalities as well. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n with zero of multiplicity s at the origin and the remaining zeros be 0. In this paper, we investigate the relative growth of a polynomial p(z) with respect to two circles z=r and z = R and obtain inequalities about the dependence of p(rz) on p(Rz), where z = 1, for 0 while taking into account the placement of the zeros of the underlying polynomial. Our results improve as well as generalize certain well-known polynomial inequalities. Some numerical examples are also given in order to illustrate and compare graphically the obtained inequalities with some recent results. [ABSTRACT FROM AUTHOR]

If p(z) is a polynomial of degree n having no zero in |z| < k, k ≥ 1, and ||p||r = ( 1/2π ∫ 2π0|p(eiθ) | r dθ)1/r, 0 < r < ∞ then Govil and Rahman in [16] proved ||p||r ≤ n/|| k+ z||r ||p||r, r ≥ 1. In this paper, by involving certain co-efficients of the polynomial, we not only prove an integral inequality which improves as well as generalizes the above inequality, but also gives a number of interesting consequences as special cases [ABSTRACT FROM AUTHOR]

If is a polynomial of degree n having all its zeros in |z| 6 k, k > 1, V.K. Jain [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 59 (2016), 339-347] proved We first obtain a generalization as well as improvement of the above inequality. Further, we extend our first result to a more generalized result which yields improved results of some known inequalities as particular case. [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]

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]

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]