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

Since 1970’s B. Fuglede and others have been studying finely holomorhic functions, i.e., ‘holomorphic’ functions defined on the so-called fine domains which are not necessarily open in the usual sense. This note is a survey of finely monogenic functions which were introduced in (Lávička, R., A generalisation of monogenic functions to fine domains, preprint.) like a higher dimensional analogue of finely holomorphic functions.

In this paper, we investigate the growth of solutions of a certain class of linear differential equation where the coefficients are analytic functions in the closed complex plane except at a finite singular point. For that, we will use the value distribution theory of meromorphic functions developed by Rolf Nevanlinna with adapted definitions.

In this paper, we estimate the Douglas-Dirichlet functionals of harmonic mappings, namely Euclidean harmonic mapping and flat harmonic mapping, by using the extremal dilatation of finite distortion functions with given boundary value on the unit circle. In addition, $\overline{\partial }$-Dirichlet functionals of harmonic mappings are also investigated.

This note deals with interpolation of values of analytic functions belonging to a given space, on finite sets of consecutive points of sequences in the disc, performed by rational functions and polynomials. Our goal is to identify sequences and spaces whose functions provide a bound of the error at the first uninterpolated point that is as small as desired. For certain sequences, we prove that this happens for bounded functions, Lipschitz functions and those that have derivatives in the disc algebra....

Firstly we study the growth of meromorphic solutions of linear difference equation of the form$$A_k\left(z\right)f(z+c_k)+\cdots +A_1\left(z\right)f(z+c_1)+A_0\left(z\right)f\left(z\right)=F\left(z\right),$$ where $A_k\left(z\right),...,A_0\left(z\right)$ and $F\left(z\right)$ are meromorphic functions of finite logarithmic order, $c_i$$(i=1,......$