Related papers: On Solution of Second Order Complex Differential E…
We classify transcendental entire functions that are compositions of a polynomial and the exponential for which all singular values escape on disjoint rays. The construction involves an iteration procedure on an infinite-dimensional…
In this paper, utilizing Nevanlinna theory, we study existence and forms of the entire solutions $ f $ of the quadratic trinomial-type partial differential-difference equations in $ \mathbb{C}^n $ \begin{align*} a\left(\alpha\dfrac{\partial…
Meromorphic solutions of non-linear differential equations of the form $f^n+P(z,f)=h$ are investigated, where $n\geq 2$ is an integer, $h$ is a meromorphic function, and $P(z,f)$ is differential polynomial in $f$ and its derivatives with…
In this paper by exploiting critical point theory, the existence of two distinct nontrivial solutions for a nonlinear algebraic system with a parameter is established. Our goal is achieved by requiring an appropriate behavior of the…
Complex linear differential equations with entire coefficients are studied in the situation where one of the coefficients is an exponential polynomial and dominates the growth of all the other coefficients. If such an equation has an…
Normality arguments are applied to study the oscillation of solutions of $f''+Af=0$, where the coefficient $A$ is analytic in the unit disc $\mathbb{D}$ and $\sup_{z\in\mathbb{D}} (1-|z|^2)^2|A(z)| < \infty$. It is shown that such…
In this paper, we study the following complex Schr\"{o}dinger equation with a $q$-difference term: \begin{align}\tag{{\dag}}\label{dagger} f'(z) = a(z)f(qz) + R(z, f(z)), \quad R(z, f(z)) = \frac{P(z, f(z))}{Q(z, f(z))}, \end{align} where…
In this note a critical point result for differentiable functionals is exploited in order to prove that a suitable class of one-dimensional fractional problems admits at least one non-trivial solution under an asymptotical behaviour of the…
Recently Brownawell and the second author proved a "non-degenerate" case of the (unproved) "Zilber Nullstellensatz" in connexion with "Strong Exponential Closure". Here we treat some significant new cases. In particular these settle…
We prove necessary and sufficient conditions for a system $\dot z_i=z_ip_i(z)$ ($p_i$ a polynomial) to have only entire analytic functions as solutions.
This article studies the singular values of entire functions of the form $E^k (z)+P(z)$ where $E^k$ denotes the $k-$times composition of $e^z$ with itself and $P$ is any non-constant polynomial. It is proved that the full preimage of each…
Assuming that $A_0,\ldots,A_{n-1}$ are entire functions and that $p\in \{0,\ldots,n-1\}$ is the smallest index such that $A_p$ is transcendental, then, by a classical theorem of Frei, each solution base of the differential equation…
This paper presents several results concerning second and third-order differential subordination for the class $\mathcal{S}^{*}_{e}:=\{f\in \mathcal{A}:zf'(z)/f(z)\prec e^z\}$, which represents the class of starlike functions associated…
We prove the real non-attractive fixed point conjecture for complex polynomial and rational harmonic functions. A harmonic function $f=h+\overline{g}$ is polynomial (rational) if both $h$ and $g$ are polynomials (rational functions) of…
One of the fourth-order analog to the first Painlev\'{e} equation is studied. All power expansions for solutions of this equation near points $z=0$ and $z=\infty$ are found by means of the power geometry method. The exponential additions to…
In this paper, we show that any polynomial of zeta or $L$-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the…
We treat the following "polynomial moment problem": for a complex polynomial P(z) and distinct complex numbers a,b such that P(a)=P(b) to describe polynomials q(z)=Q'(z) orthogonal to all degrees of P(z) on the segment [a,b]. We show that…
In this work we prove the existence of infinitely many nonradial solutions that change signal to the problem $-\Delta u=f(u)$ in $B$ with $u=0$ on $\partial B$, where $B$ is the unit ball in $\mathbb{R}^2$ and $f$ is a continuous and odd…
Several years ago, Aziz and Zargar, while considering some questions related to Sendov's conjecture, solved a problem posed by Brown (see [1,2]), showing that any complex polynomial of degree $n$ with a single zero at $z=0$ does not have…
Let $p(z)$ be a nonconstant polynomial and $\beta(z)$ be a small entire function of $e^{p(z)}$ in the sense of Nevanlinna. We first describe the growth behavior of the entire function $H(z):=e^{p(z)}\int_0^{z}\beta(t)e^{-p(t)}dt$ on the…