In the beginning of the twentieth century, Plemelj introduced the notion of factorization of matrix functions. The matrix factorization finds applications in many fields such as in the diffraction theory, in the theory of differential equations and in the theory of singular integral operators. However, the explicit formulas for the factors of the factorization are known only in a few classes of matrices. In the present paper we consider a new approach to obtain the factorization of a rational matrix...

We show that for every open Riemann surface $X$ with non-abelian fundamental group there is a multiple-valued function $f$ on $X$ such that the fiberwise convex hull of the graph of $f$ fails to contain the graph of a single-valued holomorphic function on $X$.

Ortega-Cerdà-Seip demonstrated that there are bounded multiplicative Hankel forms which do not arise from bounded symbols. On the other hand, when such a form is in the Hilbert-Schmidt class ₂, Helson showed that it has a bounded symbol. The present work investigates forms belonging to the Schatten classes between these two cases. It is shown that for every $p>{(1-log\pi /log4)}^{-1}$ there exist multiplicative Hankel forms in the Schatten class ${}_p$ which lack bounded symbols. The lower bound on p is in a certain sense optimal...

We show that the Fatou components of a certain transcendental entire function have a common curve in their boundaries.

We give a complete characterization of the positive trigonometric polynomials $Q(\theta ,\varphi )$ on the bi-circle, which can be factored as $Q(\theta ,\varphi )=|p(e^{i\theta },e^{i\varphi }){|}^2$ where $p(z,w)$ is a polynomial nonzero for $\left|z\right|=1$ and $\left|w\right|\le 1$. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight $\frac{1}{4{\pi }^{2}Q(\theta ,\varphi )}$ on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities relating...

MSC 2010: 30C45The universally prestarlike functions of order α ≤ 1 in the slit domain Λ = C [1;∞) have been recently introduced by S. Ruscheweyh. This notion generalizes the corresponding one for functions in the unit disk Δ (and other circular domains in C). In this paper, we obtain the coefficient inequalities and the Fekete-Szegö inequality for such functions.

In this paper, we obtain Fekete–Szegö inequalities for a generalized class of analytic functions $f\left(z\right)\in 𝒜$ for which $1+\frac{1}{b}\left(\frac{z{\left(D_{\alpha ,\beta ,\lambda ,\delta }^{n}f\left(z\right)\right)}^{\text{'}}}{D_{\alpha ,\beta ,\lambda ,\delta }^{n}f\left(z\right)}-1\right)$ ($\alpha ,\beta ,\lambda ,\delta \ge 0$; $\beta >\alpha$; $\lambda >\delta$; $b\in {ℂ}^{*}$; $n\in {\mathbb{N}}_0$; $z\in U$) lies in a region starlike with respect to $1$ and is symmetric with respect to the real axis.

The authors obtain the Fekete-Szegő inequality (according to parameters $s$ and $t$ in the region $s^2+st+t^2<3$, $s\ne t$ and $s+t\ne 2$, or in the region $s^2+st+t^2>3,$$s\ne t$ and $s+t\ne 2$) for certain normalized analytic functions $f\left(z\right)$ belonging to $k{\text{-UST}}_{\lambda ,\mu }^n(s,t,\gamma )$ which satisfy the condition $$\Re \left\{\frac{(s-t)z{\left(D_{\lambda ,\mu }^{n}f\left(z\right)\right)}^{\text{'}}}{D_{\lambda ,\mu }^{n}f\left(sz\right)-D_{\lambda ,\mu }^{n}f\left(tz\right)}\right\}>k\left|\frac{(s-t)z{\left(D_{\lambda ,\mu }^{n}f\left(z\right)\right)}^{\text{'}}}{D_{\lambda ,\mu }^{n}f\left(sz\right)-D_{\lambda ,\mu }^{n}f\left(tz\right)}-1\right|+\gamma ,z\in 𝒰.$$ Also certain...

MSC 2010: 30C45