On zeros of solutions of higher order homogeneous linear differential equations

In this article, we investigate the exponent of convergence of zeros of solutions for some higher-order homogeneous linear differential equation, and prove that if A_k−1 is the dominant coefficient, then every transcendental solution f(z) of equation

f(k)+A_k-1 f(k-1)+?+A₀ f=0$f^{(k)}+A_{k-1}\hspace{0.17em}{f}^{(k-1)}+?+A_0\hspace{0.17em}f=0$

satisfies λ(f) = ∞, where λ(f) denotes the exponent of convergence of zeros of the meromorphic function f(z).     

Fatou sets of entire solutions of linear differential equations

This paper is devoted to studying the dynamical properties of solutions of f^″+A(z)f=0$f^{″}+A\left(z\right)f=0$, where A(z)$A\left(z\right)$ is an entire function of finite order. With some added conditions on A(z)$A\left(z\right)$, we prove that all the Fatou components of E(z)$E\left(z\right)$ are simply-connected provided that E=_f1f2$E=f_1{f}_2$ and _f1,_f2$f_1,f_2$ are two linearly independent solutions of such equations.     

A note on value distribution of difference polynomials of meromorphic functions

In this paper, we investigate the value distribution of the difference counterpart Δf(z)-afn(z)of f^′(z) -afn(z)$\Delta f(z)-af{(z)}^n\text{o}\text{f}\hspace{0.17em}{f}^{\prime }(z)\hspace{0.17em}-af{(z)}^n$ and obtain an almost direct difference analogue of result of Hayman.     

Muckenhoupt weights and Lindelöf theorem for harmonic mappings

We extend the result of Lavrentiev which asserts that the harmonic measure and the arc-length measure are _A∞$A_{\infty }$ equivalent in a chord-arc Jordan domain. By using this result we extend the classical result of Lindelöf to the class of quasiconformal (q.c.) harmonic mappings by proving the following assertion. Assume that f is a quasiconformal harmonic mapping of the unit disk U onto a Jordan domain. Then the function A(z)=arg?(_∂φ(f(z))/z)$A(z)=\arg ?({\partial }_{\phi }(f(z))/z)$ where z=re^iφ$z=r{e}^{i\phi }$, is well-defined and smooth in U^?={z:0<|z|<1}${\mathbf{U}}^{?}=\{z:0<|z|<1\}$ and has a continuous extension to the boundary of the unit disk if and only if the image domain has C¹$C^1$ boundary.     

On spaces of modular forms spanned by eta-quotients

An eta-quotient of level N   is a modular form of the shape f(z)=_∏δ|Nη(δz)^_rδ$f(z)={\prod }_{\delta |N}\eta {(\delta z)}^{r_{\delta }}$. We study the problem of determining levels N   for which the graded ring of holomorphic modular forms for _Γ0(N)${\Gamma }_0(N)$ is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level N  . In addition, we prove that if f(z)$f(z)$ is a holomorphic modular form that is non-vanishing on the upper half plane and has integer Fourier coefficients at infinity, then f(z)$f(z)$ is an integer multiple of an eta-quotient. Finally, we use our results to determine the structure of the cuspidal subgroup of _J0(2^k)(Q)$J_0(2^k)(\mathbb{Q})$.     

Parameter space for families of parabolic-like mappings

In this paper we study families of degree 2 parabolic-like mappings _(fλ)λ∈Λ${(f_{\lambda })}_{\lambda \in \Lambda }$ (as defined in [4]). We prove that the hybrid conjugacies between a nice analytic family of degree 2 parabolic-like mappings and members of the family _Per1(1)${\mathit{Per}}_1(1)$ induce a continuous map χ:Λ→C$\chi :\Lambda \to \mathbb{C}$, which under suitable conditions restricts to a ramified covering from the connectedness locus of _(fλ)λ∈Λ${(f_{\lambda })}_{\lambda \in \Lambda }$ to the connectedness locus _M1?{1}$M_1?\{1\}$ of _Per1(1)${\mathit{Per}}_1(1)$. As an application, we prove that the connectedness locus of the family _Ca(z)=z+az²+z³$C_a(z)=z+a{z}^2+z^3$, a∈C$a\in \mathbb{C}$ presents baby _M1$M_1$.     

The exact analytic solutions of a nonlinear differential iterative equation

This paper is concerned with a second-order nonlinear iterated differential equation of the form _c0x^″(z)+_c1x^′(z)+_c2x(z)=x(p(z)+bx(z))+h(z)$c_0{x}^{″}\left(z\right)+c_1{x}^{\prime }\left(z\right)+c_{2}x\left(z\right)=x(p\left(z\right)+bx\left(z\right))+h\left(z\right)$. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. We discuss not only the general case |β|≠1$\left|\beta \right|\ne 1$, but also the critical case |β|=1$\left|\beta \right|=1$, especially when β$\beta$ is a root of unity. Furthermore, the exact and explicit analytic solution of the original equation is investigated for the first time. Such equations are important in both applications and the theory of iterations.     

Simple geometry facilitates iterative solution of a nonlinear equation via a special transformation to accelerate convergence to third order

Direct substitution _xk+1=g(_xk)$x_{k+1}=g(x_k)$ generally represents iterative techniques for locating a root z   of a nonlinear equation f(x)$f(x)$. At the solution, f(z)=0$f(z)=0$ and g(z)=z$g(z)=z$. Efforts continue worldwide both to improve old iterators and create new ones. This is a study of convergence acceleration by generating secondary solvers through the transformation _gm(x)=(g(x)-m(x)x)/(1-m(x))$g_m(x)=(g(x)-m(x)x)/(1-m(x))$ or, equivalently, through partial substitution _gmps(x)=x+G(x)(g-x)$g_{m\mathrm{ps}}(x)=x+G(x)(g-x)$, G(x)=1/(1-m(x))$G(x)=1/(1-m(x))$. As a matter of fact, _gm(x)≡_gmps(x)$g_m(x)\equiv g_{m\mathrm{ps}}(x)$ is the point of intersection of a linearised g   with the g=x$g=x$ line. Aitken's and Wegstein's accelerators are special cases of _gm$g_m$. Simple geometry suggests that m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ is a good approximation for the ideal slope of the linearised g  . Indeed, this renders a third-order _gm$g_m$. The pertinent asymptotic error constant has been determined. The theoretical background covers a critical review of several partial substitution variants of the well-known Newton's method, including third-order Halley's and Chebyshev's solvers. The new technique is illustrated using first-, second-, and third-order primaries. A flexible algorithm is added to facilitate applications to any solver. The transformed Newton's method is identical to Halley's. The use of m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ thus obviates the requirement for the second derivative of f(x)$f(x)$. Comparison and combination with Halley's and Chebyshev's solvers are provided. Numerical results are from the square root and cube root examples.     

The fekete-szegö problem for close-to-convex functions with respect to the koebe function

An analytic function f   in the unit disk D :={z ∈ ? : |z| < 1}$\mathbb{D}\hspace{0.17em}:=\left\{z\hspace{0.17em}\in \hspace{0.17em}?\hspace{0.17em}:\hspace{0.17em}\left|z\right|\hspace{0.17em}<\hspace{0.17em}1\right\}$, standardly normalized, is called close-to-convex with respect to the Koebe function k(z) := z/(1−z)², z ∈ D$k\left(z\right)\hspace{0.17em}:=\hspace{0.17em}z/{\left(1-z\right)}^2,\hspace{0.17em}z\hspace{0.17em}\in \hspace{0.17em}\mathbb{D}$ if there exists δ∈(-π/2,π/2) such that Re{eiδ(1−z)²f^′(z)} > 0, ∈ D$\text{Re}\left\{{\text{e}}^{\text{i}\delta }{\left(1-z\right)}^2{f}^{\prime }\left(z\right)\right\}\hspace{0.17em}>\hspace{0.17em}0,\hspace{0.17em}\in \hspace{0.17em}\mathbb{D}$. For the class C(k) of all close-to-convex functions with respect to k, related to the class of functions convex in the positive direction of the imaginary axis, the Fekete-Szegö problem is studied.