Related papers: Perturbing rational harmonic functions by poles
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.
A rational function is the ratio of two complex polynomials in one variable without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into…
Rouch\'e's Theorem is among the most useful results in complex analysis for counting zeros of analytic functions. Rouch\'e's Theorem also admits a harmonic analogue for counting zeros of complex harmonic functions. Previously, this analogue…
This study deals with certain harmonic zeta functions, one of them occurs in the study of the multiplication property of the harmonic Hurwitz zeta function. The values at the negative even integers are found and Laurent expansions at poles…
We study Rado functionals and the maximal condition (first introduced by J. M. Barret et al.) in terms of the partition regularity of mixed systems of linear equations and inequalities. By strengthening the maximal Rado condition, we…
In this paper, we study a Dirichlet series generated by powers of harmonic numbers. As an application of these functions, we derive certain series involving harmonic numbers. We also study the analytic properties of these Dirichlet series…
For rational functions, we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erd\"os-Lax and Tur\'an to rational functions R. In return these…
Given $n\geq1$ and $r\in[0, 1),$ we consider the set $\mathcal{R}_{n, r}$ of rational functions having at most $n$ poles all outside of $\frac{1}{r}\mathbb{D},$ were $\mathbb{D}$ is the unit disc of the complex plane. We give an…
A family of Zeta functions built as Dirichlet series over the Riemann zeros are shown to have meromorphic extensions in the whole complex plane, for which numerous analytical features (the polar structure, plus countably many special…
This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a binomial-type denominator. We show that every member of a two-parameter family consisting of such generating functions…
When one integrates the q-exponential function of Tsallis' so as to get the partition function $Z$, a gamma function inevitably emerges. Consequently, poles arise. We investigate here here the thermodynamic significance of these poles in…
This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a denominator of the form $G(z,t)=P(t)+zt^{r}$, where the zeros of $P$ are positive and real. We show that every member of…
In this paper we sharpen significantly several known estimates on the maximal number of zeros of complex harmonic polynomials. We also study the relation between the curvature of critical lemniscates and its impact on geometry of caustics…
In this note, we use Rouch\'e's theorem and the pleasant properties of the arithmetic of the logarithmic derivative to establish several new results regarding the geometry of the zeros, poles, and critical points of a rational function.…
Let f be a function transcendental and meromorphic in the plane, and define g(z) by g(z) = f(z+1) - f(z). A number of results are proved concerning the existence of zeros of g(z) or g(z)/f(z), in terms of the growth and the poles of f.
We give an example of a sequence of positive harmonic functions on $\mathbb{Z}^d$, $d\geq 2$, that converges pointwise to a non-harmonic function.
We establish a one-to-one correspondence between conjugacy classes of any Hecke group and irreducible systems of poles of rational period functions for automorphic integrals on the same group. We use this correspondence to construct…
In this work we extend the theory of the classical Hardy space $H^1$ to the rational Dunkl setting. Specifically, let $\Delta$ be the Dunkl Laplacian on a Euclidean space $\mathbb{R}^N$. On the half-space $\mathbb{R}_+\times\mathbb{R}^N$,…
A natural kind of compactification of the virtual moduli spaces of rational functions of one complex variable is given. To describe the boundary points geometrically, the authors introduce the concept of rational functions with nodes,…
We consider the set \mathcal{R}_{n} of rational functions of degree at most n\geq1 with no poles on the unit circle \mathbb{T} and its subclass \mathcal{R}_{n,\, r} consisting of rational functions without poles in the annulus \left\{\xi:\;…