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Let p (z) be a polynomial of degree n having no zero in | z | < k , k ≤ 1 , then Govil [Proc. Nat. Acad. Sci., 50(1980), 50-52] proved max | z | = 1 | p ′ (z) | ≤ n 1 + k n max | z | = 1 | p (z) | , provided | p ′ (z) | and | q ′ (z) | attain their maxima at the same point on the circle | z | = 1 , where q (z) = z n p (1 z ‾) ‾. In this paper, we present integral mean inequalities of Turán- and Erdös-Lax-type for the polar derivative of a polynomial by involving some coefficients of the polynomial, which refine some previously proved results and one of our results improves the above Govil inequality as a special case. These results incorporate the placement of the zeros and some coefficients of the underlying polynomial. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to some previously established results. [ABSTRACT FROM AUTHOR]

By using the principle of mathematical induction, it was shown by Singh and Chanam (J. Math. Inequal 15:1663–1675, 2021) that if p(z) is a polynomial of degree n having all its zeros in | z | ≤ 1 , then for all z on | z | = 1 for which p (z) ≠ 0 , ℜ z p ′ (z) p (z) ≥ n + 1 2 - 1 2 | a 0 | | a n | . In this paper, by using simple techniques we generalize the above inequality, thereby give a simple proof of the above inequality. As an application of our result, we obtain improvements of the well-known result due to Malik (J. Lond. Math. Soc 1(2):57–60, 1969). Further, we obtain some sharp refinements of a result due to Aziz and Rather (J. Math. Inequal. Appl 1:231–238, 1998). These results take into account the placement of the coefficients of the underlying polynomial. Moreover, a concrete numerical example is presented in order to graphically illustrate and compare the obtained inequalities with a classical result, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known. [ABSTRACT FROM AUTHOR]

Let P (z) = ∑ j = 0 n a j z j be a polynomial of degree n such that an ≥ an−1 ≥... ≥ a1 ≥ a0 ≥ 0. Then according to Eneström-Kakeya theorem all the zeros of P (z) lie in |z| ≤ 1. This result has been generalized in various ways (see [1, 3, 4, 6, 7]). In this paper we shall prove some generalizations of the results due to Aziz and Zargar [1, 2] and Nwaeze [7]. [ABSTRACT FROM AUTHOR]

When exploring the literature, it can be observed that the operator obtained when applying Newton-like root finding algorithms to the quadratic polynomials z2 − c has the same form regardless of which algorithm has been used. In this paper, we justify why this expression is obtained. This is done by studying the symmetries of the operators obtained after applying Newton-like algorithms to a family of degree d polynomials p(z) = zd − c. Moreover, we provide an iterative procedure to obtain the expression of new Newton-like algorithms. We also carry out a dynamical study of the given generic operator and provide general conclusions of this type of methods. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new inequalities in the plane that are inspired by some classical Turán-type inequalities that relate the norm of a univariate complex coefficient polynomial and its derivative on the unit disk. The obtained results produce various inequalities in the integral-norm of a polynomial that are sharper than the previous ones while taking into account the placement of the zeros and some of the extremal coefficients of the underlying polynomial. [ABSTRACT FROM AUTHOR]

If P (z) : = ∑ v = 0 n a v z v is a polynomial of degree n having all its zeros in | z | ≤ k , k ≥ 1 , then Govil (Proc Am Math Soc 41:543–546, 1973) proved max | z | = 1 | P ′ (z) | ≥ n 1 + k n max | z | = 1 | P (z) |. Recently, Govil and Kumar (Appl Anal Discrete Math 13:711–720, 2019) gave a polar derivative generalization of this inequality and proved under the same hypotheses for a polynomial P (z) = z s (a 0 + a 1 z + ⋯ + a n - s z n - s) , 0 ≤ s ≤ n , that max | z | = 1 | D α P (z) | ≥ (| α | - k) { n + s 1 + k n + (k n - s | a n - s | - | a 0 |) (1 + k n) (k n - s | a n - s | + | a 0 |) } max | z | = 1 | P (z) | , where | α | ≥ k . In this paper, we generalize as well as improve upon the above inequalities and related results. [ABSTRACT FROM AUTHOR]

For a rational function r(z) having no zeros outside a unit disk, Li et al. (J Lond Math Soc 51:523–531, 1995) proved: | r ′ (z) | ≥ | B ′ (z) | | r (z) | , for | z | = 1 , where B (z) is the Blashke product. This result was improved and generalized in various ways and a recent generalization was obtained by Mir (Indian J Pure Appl Math 51:749–760, 2020). In this paper, we prove some results which include as special cases many Turán-type polynomial inequalities proved earlier. [ABSTRACT FROM AUTHOR]

Sendov's Conjecture asserts that if all the zeros of a polynomial p lie inside the closed unit disk then a closed unit disk centered at each of its zeros contains a critical point of p. In this paper, we show that the Sendov's Conjecture holds for a polynomial if all its zeros lie on a line or a circle inside the unit disk. [ABSTRACT FROM AUTHOR]

Let g t be a loop in the space of monic complex polynomials in one variable of fixed degree n. If the roots of g t are distinct for all t, they form a braid B 1 on n strands. Likewise, if the critical points of g t are distinct for all t, they form a braid B 2 on n - 1 strands. In this paper we study the relationship between B 1 and B 2 . Composing the polynomials g t with the argument map defines a pseudo-fibration map on the complement of the closure of B 1 in C × S 1 , whose critical points lie on B 2 . We prove that for B 1 a T-homogeneous braid and B 2 the trivial braid this map can be taken to be a fibration map. In the case of homogeneous braids we present a visualization of this fact. Our work implies that for every pair of links L 1 and L 2 there is a mixed polynomial f : C 2 → C in complex variables u, v and the complex conjugate v ¯ such that both f and the derivative f u have a weakly isolated singularity at the origin with L 1 as the link of the singularity of f and L 2 as a sublink of the link of the singularity of f u . [ABSTRACT FROM AUTHOR]

If $ P(z)=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a n ∏ v = 1 n (z − z v) is a polynomial of degree n having all its zeros in $ |z|\le k, k\ge 1 $ | z | ≤ k , k ≥ 1 then Aziz [Inequalities for the derivative of a polynomial. Proc Am Math Soc. 1983;89(2):259–266] proved that \[ \max_{|z|=1}|P'(z)|\ge \frac{2}{1+k^n}\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \] max | z | = 1 | P ′ (z) | ≥ 2 1 + k n ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. Recently, Kumar [On the inequalities concerning polynomials. Complex Anal Oper Theory. 2020;14(6):1–11 (Article ID 65)] established a generalization of this inequality and proved under the same hypothesis for a polynomial $ P(z)=a_0+a_1z+a_2z^2+\cdots +a_nz^n=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a 0 + a 1 z + a 2 z 2 + ⋯ + a n z n = a n ∏ v = 1 n (z − z v) , that $$\begin{align*} & \max_{|z|=1}|P'(z)| \\ & \ge \left(\frac{2}{1+k^n}+\frac{(|a_n|k^n-|a_0|)(k-1)}{(1+k^n)(|a_n|k^n+k|a_0|)}\right)\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \end{align*} $$ max | z | = 1 | P ′ (z) | ≥ (2 1 + k n + (| a n | k n − | a 0 |) (k − 1) (1 + k n) (| a n | k n + k | a 0 |)) ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. In this paper, we sharpen the above inequalities and further extend the obtained results to the polar derivative of a polynomial. As a consequence, our results also sharpens considerably some results of Dewan and Upadhye [Inequalities for the polar derivative of a polynomial. J Ineq Pure Appl Math. 2008;9:1–9 (Article ID 119)]. [ABSTRACT FROM AUTHOR]

In this paper, we extend some results on Eneström–Kakeya theorem for quaternionic polynomials by relaxing the hypothesis which leads to their applicability to a larger class of polynomials that are beneficial in determining the regions containing all the zeros of a polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of derivatives of algebraic polynomials in bounded and unbounded regions of the complex plane. At the same time, both interior and exterior cusp points are allowed on the boundary of such regions. Bernstein-Walsh-type estimates are obtained for derivatives of algebraic polynomials in the specified region with corners for exterior points, as well as Markoff-type estimates for closure of the region. As a result, estimates are found for the derivatives of algebraic polynomials in the whole complex plane. It is also shown that the inequality for the closed region is exact in order for the given region. [ABSTRACT FROM AUTHOR]

The famous T. Suffridge polynomials have many extremal properties: the maximality of coefficients when the leading coefficient is maximal; the zeros of the derivative are located on the unit circle; the maximum radius of stretching the unit disk with the schlicht normalization F (0) = 0 , F ′ (0) = 1 ; the maximum size of the unit disk contraction in the direction of the real axis for univalent polynomials with the normalization F (0) = 0 , F (1) = 1. However, under the standard symmetrization method F (z T) T , these polynomials become functions which are not polynomials. How can we construct the polynomials with fold symmetry that have properties similar to those of the Suffridge polynomial? What values will the corresponding extremal quantities take in the above-mentioned extremal problems? The paper is devoted to solving these questions for the case of the trinomials F (z) = z + a z 1 + T + b z 1 + 2 T . Also, there are suggested hypotheses for the general case in the work. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n with zero of multiplicity s at the origin and the remaining zeros in | z | ≥ k or in | z | ≤ k , k > 0 . In this paper, first we obtain inequalities about the dependence of |p(Rz)| on |p(rz)|, where | z | = 1 , for r 2 ≤ R r ≤ k 2 or k 2 ≤ R r ≤ R 2 . Further, another similar inequality for the class of polynomials having all zeros in | z | ≥ k , k > 0 is also proved for 0 < r ≤ R ≤ k . Our results improve as well as generalize certain well-known polynomial inequalities. [ABSTRACT FROM AUTHOR]

The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev p-norm ( 1 < p < ∞ ) for the case p = 1 . Some relevant examples are indicated. The second part deals with the location of zeros of polynomials of least deviation in discrete Sobolev p-norm. The asymptotic distribution of zeros is established on general conditions. Under some order restriction in the discrete part, we prove that the n-th polynomial of least deviation has at least n - d ∗ zeros on the convex hull of the support of the measure, where d ∗ denotes the number of terms in the discrete part. [ABSTRACT FROM AUTHOR]

In this paper, we prove some inequalities that relate the sup-norm of a univariate complex coefficient polynomial and its derivative, when there is a restriction on its zeros. We further obtain the polar derivative generalizations of the obtained results. The obtained results produce various inequalities that are sharper than the previous ones while taking into account the placement (absolute value) of the zeros and the extremal coefficients of the polynomial. Moreover, some concrete numerical examples are presented, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known. [ABSTRACT FROM AUTHOR]

Most of the problems in mathematical and engineering sciences can be studied in the context of nonlinear equations. In this paper, we develop a new family of iterative methods for the approximation of the zeros of mathematical models whose governing equations are nonlinear in nature. The proposed methods are based on decomposition technique due to Daftardar-Gejji and Jaffri [1]. The new family gives several iterative schemes as special cases. The convergence analysis of proposed methods is also presented. In order to determine the performance of newly developed methods, numerical as well as graphical analysis of four complex mathematical models from diverse fields of science and engineering are considered. We also consider some complex polynomials to visualize the roots through polynomiography in the context of proposed methods. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n. The polar derivative of p(z) with respect to a complex number α is defined by D α p (z) = n p (z) + (α - z) p ′ (z). If p (z) = z s ∑ j = 0 n - s c j z j , 0 ≤ s ≤ n , has all its zeros in | z | ≤ k , k ≥ 1 , then for | α | ≥ k , Kumar and Dhankhar [Bull, Math. Soc. Sci. Math., 63(4), 359-367 (2020)] proved max | z | = 1 | D α p (z) | ≥ n (| α | - k) 1 + k n - s 1 + (| c n - s | k n - | c 0 | k s) (k - 1) 2 (| c n - s | k n + | c 0 | k s + 1) max | z | = 1 | p (z) |. In this paper, we first improve the above inequality. Besides, we are able to prove an improvement of a result due to Govil and Mctume [Acta. Math. Hungar., 104, 115-126 (2004)] and also prove an inequality for a subclass of polynomials having no zero in | z | < k , k ≤ 1 . [ABSTRACT FROM AUTHOR]

According to Gauss-Lucas theorem, every convex set containing all the zeros of a polynomial also contains all its critical points. This result is of central importance in the geometry of critical points in the analytic theory of polynomials. In this paper, an extension of Gauss-Lucas theorem is obtained and as an application some generalizations of Bernstein-type polynomial inequalities are also established. [ABSTRACT FROM AUTHOR]