Related papers: Inequalities for the derivative of Polynomials wit…

We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{\'e}lyi, and Littmann~\cite{BE1}…

We present a new Schwarz Lemma for bounded domains with Bergman metrics. The key ingredient of our proof is the Cauchy-Schwarz inequality from probability theory.

Sequences of discrete random variables are studied whose probability generating functions are zero-free in a sector of the complex plane around the positive real axis. Sharp bounds on the cumulants of all orders are stated, leading to…

We examine the maximal number of zeros a polynomial of degree at most n with constrained coefficients may have at 1. Our results are essentially sharp and extend earlier results of this variety. An interesting connection to certain…

We provide an index bound for character sums of polynomials over finite fields. This improves the Weil bound for high degree polynomials with small indices, as well as polynomials with large indices that are generated by cyclotomic mappings…

We use Turan type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the…

We obtain bounds for the numerical radius of $2 \times 2$ operator matrices which improve on the existing bounds. We also show that the inequalities obtained here generalize the existing ones. As an application of the results obtained here…

We establish some Schwarz type Lemmas for mappings defined on the unit disk with bounded Laplacian. Then we apply these results to obtain boundary versions of the Schwarz lemma.

In this paper, we establish some inequalities for rational functions with prescribed poles having s-fold zeros at origin and also show that it implies some inequalities for polynomials and their polar derivatives.

We show that with high probability the number of real zeroes of a random polynomial is bounded by the number of vertices on its Newton-Hadamard polygon times the cube of the logarithm of the polynomial degree. A similar estimate holds for…

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…

We sharpen the bound $n^{2k}$ on the maximum modulus of the $k^{{\rm th}}$ radial derivative of the Zernike circle polynomials (disk polynomials) of degree $n$ to $n^2(n^2-1^2)\cdot ... \cdot(n^2-(k-1)^2)/2^k(1/2)_k$. This bound is obtained…

We discuss existence of explicit search bounds for zeros of polynomials with coefficients in a number field. Our main result is a theorem about the existence of polynomial zeros of small height over the field of algebraic numbers outside of…

We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to approximation by…

In this paper, we derive new bounds for the zeros of quaternionic polynomials by applying localization theorems, which includes Gershgorin-type theorems for the left eigenvalues of matrices of left monic quaternionic polynomials. These…

In this article an upper bound for the first consecutive zeros of the Fermat quotient is given in terms of the zeros of a Mirimanoff polynomial. This bound is obtained by investigating a relation between these polynomials and the factor…

We study the problem of placing effective upper bounds for the number of zeros of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of…

Lower bounds are given for the number of non-real zeros of a second order linear differential polynomial with constant coefficients in a real entire function with finitely many non-real zeros.

Let S be a finite subset of a field. For multivariate polynomials the generalized Schwartz-Zippel bound [2], [4] estimates the number of zeros over Sx...xS counted with multiplicity. It does this in terms of the total degree, the number of…

We approximate the Riemann Zeta-Function by polynomials and Dirichlet polynomials with restricted zeros.