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This paper introduces the notion of probabilistic zero bounds for random polynomials. It presents new results regarding the probabilistic bounds of random polynomials whose coefficients are independently and identically distributed as standard normal variates. Additionally, the paper provides a clear exposition of the developed methodology. To establish our results, we develop a novel approach utilizing the classical Cauchy's bounds for the zeros of a deterministic polynomial with complex coefficients. We also corroborate our analytical results with extensive simulations. The methodology developed in the paper can potentially be applied to a broad class of problems regarding bounds and the distribution of zeros in the theory of random polynomials. [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]

In this paper, by using the residue method of complex analysis, we obtain a q-analogue for partial-fraction decomposition of the rational function x M (x + 1) n λ . As applications, we deduce the corresponding q-algebraic and q-combinatorial identities which are the q-extensions of Chu' results. [ABSTRACT FROM AUTHOR]

In a previous paper, we introduced the Collatz polynomials $ P_N\left (z \right) $ P N (z) , whose coefficients are the terms of the Collatz sequence of the positive integer N. Our work in this paper expands on our previous results, using the Eneström-Kakeya Theorem to tighten our old bounds of the roots of $ P_N\left (z \right) $ P N (z) and giving precise conditions under which these new bounds are sharp. In particular, we confirm an experimental result that zeros on the circle $ \ensuremath {\{z\in \ensuremath {\mathbb {C}}:\left | z \right | = 2\}} $ { z ∈ C : | z | = 2 } are rare: the set of N such that $ P_N\left (z \right) $ P N (z) has a root of modulus 2 is sparse in the natural numbers. We close with some questions for further study. [ABSTRACT FROM AUTHOR]

The zeros of the nth D'Arcais polynomial, also known in combinatorics as the Nekrasov–Okounkov polynomial, dictate the vanishing properties of the nth Fourier coefficients of all complex powers x of the Dedekind η -function. In this paper, we prove that these coefficients are non-vanishing for | x | > κ (n - 1) and κ ≈ 9.7225 . Numerical computations imply that 9.72245 is a lower bound for κ . This significantly improves previous results by Kostant, Han, and Heim–Neuhauser. The polynomials studied in this paper include Chebyshev polynomials of the second kind, 1-associated Laguerre polynomials, Hermite polynomials, and polynomials associated with overpartitions and plane partitions. [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]

In this paper, we establish some generalizations of the upper bound estimates for the modulus of the derivative of a polynomial on the unit disk while accounting for the positioning of the zeros and extremal coefficients of the underlying polynomial. We shall also extend the obtained results to the polar derivative of a polynomial. The estimates obtained sharpen as well as generalize some recently proved Erdös–Lax type inequalities. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate properties of certain class of logharmonic mappings. Initially, we establish the argument principle of sense-preserving mappings F = h g ¯ + H G ¯ , where h, g, H and G are analytic functions. As applications, we obtain a direct extension of Rouché’s theorem, open mapping theorem and minimal area image of sense-preserving logharmonic mappings. Furthermore, we present an estimate for the modulus of the partial derivative of K-quasiconformal logharmonic mappings, and by using it, we get a Schwarz type lemma of K-quasiconformal logharmonic mappings. Finally, we obtain estimates of h ′ ′ (0) 2 and g ′ ′ (0) 2 for univalent sense-preserving logharmonic mappings f = h g ¯ in the unit disk. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we use differential operator to present new identities and provide alternative proofs for certain established identities related to geometric polynomials. Additionally, by using the differential operator associated with geometric polynomials, Rolle’s theorem and a theorem of Wang and Yeh, we present some results on the real rootedness of polynomials. [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]

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]

In this paper, by using the residue method of complex analysis, we obtain an explicit partial fraction decomposition for the general rational function x M (x + 1) n λ (M is any nonnegative integer, λ and n are any positive integers). As applications, we deduce the corresponding algebraic identities and combinatorial identities which are the corresponding extensions of Chu' results. We also give some explicit formulas of Apostol-type polynomials and harmonic Stirling numbers of the second kind. [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]

This paper establishes conditions guaranteeing the reality of all the zeros of polynomials Pn(z) in the polynomial sequence { P n (z) } n = 1 ∞ satisfying a five-term recurrence relation P n (z) = z P n − 1 (z) + α P n − 2 (z) + β P n − 3 (z) + γ P n − 4 (z) with the standard initial conditions P 0 (z) = 1 , P − 1 (z) = P − 2 (z) = P − 3 (z) = 0 , where α, β, γ are real coefficients, γ ≠ 0 and z is a complex variable. We interpret this sequence of polynomials as principal minors of an appropriate banded Toeplitz matrix whose associated Laurent polynomial b(z) is holomorphic in ℂ {0}. We derive a criterion for the preimage b−1 (ℝ) to contain a Jordan curve, which, by a recent result of Shapiro and Stampach, is necessary and sufficient for every polynomial in the sequence { P n (z) } n = 1 ∞ to be hyperbolic. [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]

Brenti conjectured that, for any finite Coxeter group, the descent generating polynomial has only real zeros, and he left the type D case open. Dilks, Petersen, and Stembridge proposed a companion conjecture, which states that, for any irreducible finite Weyl group, the affine descent generating polynomial has only real zeros, and they left the type B and type D cases open. By developing the theory of s -Eulerian polynomials, Savage and Visontai confirmed the type D case of the former conjecture and the type B case of the latter conjecture. In this paper, based on the Hermite–Biehler theorem and the theory of linear transformations preserving Hurwitz stability, we obtain the Hurwitz stability of certain polynomials related to the descent generating polynomials of type D, and thus give an alternative proof of Savage and Visontai's results. This new approach also enables us to prove Hyatt's conjectures on the interlacing property of half Eulerian polynomials of type B and type D, and to prove that the h-polynomial of certain subcomplexes of Coxeter complexes of type D has only real zeros. As one of the most important application of this approach, we completely confirm Dilks, Petersen, and Stembridge's conjecture by proving the real-rootedness of the affine descent generating polynomials of type D. [ABSTRACT FROM AUTHOR]

In light of the Alexander transformation, the class of spirallike functions is significant. The characteristics of special functions also appear very frequently in Geometric function theory. In this paper, we find the radii of γ -spirallike and convex γ -spirallike of order α of certain special functions. [ABSTRACT FROM AUTHOR]

This paper treats the properties of polynomials generated by a four-term recurrence relation and linear coefficients. From the eigenvalue theory, we give information on the location of the zeros. Furthermore, we investigate the distribution and simplicity of the zeros in a particular case. [ABSTRACT FROM AUTHOR]

We study Bernstein polynomials for simple nonsmooth rational module functions. We show that these polynomials can be represented as special sums of regular structure. For historical reasons, the representations found are naturally called "generalized Popoviciu expansions." To write generalized expansions, we develop a special formalism based on combinatorial calculations. Based on the formulas obtained, we propose a complete description of the character of convergence of the Bernstein polynomials studied in the complex plane. We also discuss the relationship of generalized Popoviciu expansions with the distribution of zeros of Bernstein polynomials. In the final part of the paper, we present additional new relations for Bernstein polynomials of the rational module function. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove that Markov's theorem on variation of zeros of orthogonal polynomials on the real line (Markoff in Math Ann 27:177–182, 1886) remains essentially valid in the case of paraorthogonal polynomials on the unit circle. [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 discuss the necessary and sufficient conditions for a polynomial P(z) to have all its zeros inside the open unit disc. These results involve two associated polynomials namely, the derivative of the reciprocal polynomial of P(z) and the reciprocal of the derivative of P(z). We also derive some generalizations of the classical Theorem of Laguerre. [ABSTRACT FROM AUTHOR]

The main purpose of this study is to determine the radii of starlikeness and convexity of the generalized k-Bessel functions for three different kinds of normalization in such a way that the resulting functions are analytic in the unit disk of the complex plane. The characterization of entire functions from Laguerre–Pólya class plays a significant role in this paper. Moreover, the interlacing properties of the zeros of the k-Bessel function and its derivative is also useful in the proof of the main results. By making use of the Euler–Rayleigh inequalities for the real zeros of the generalized k-Bessel function, we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero. [ABSTRACT FROM AUTHOR]