We obtain short and unified new proofs of two recent characterizations of hyperellipticity given by Maskit (2000) and Schaller (2000), as well as a way of establishing a relation between them.

2000 Mathematics Subject Classification: Primary 26A33, 30C45; Secondary 33A35Recently, many papers in the theory of univalent functions have been devoted to mapping and characterization properties of various linear integral or integro-differential operators in the class S (of normalized analytic and univalent functions in the open unit disk U), and in its subclasses (as the classes S∗ of the starlike functions and K of the convex functions in U). Among these operators, two operators introduced...

Let $\mathrm{T}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta :=\{z\in ℂ:|z|<1\}$, normalized by $f\left(0\right)=f^{\text{'}}\left(0\right)-1=0$ and such that $\mathrm{Im}z\mathrm{Im}f\left(z\right)\ge 0$ for $z\in \Delta$. In this paper we discuss the class ${\mathrm{T}}_g$ defined as $${\mathrm{T}}_g:=\{\sqrt{f\left(z\right)g\left(z\right)}:f\in \mathrm{T}\},g\in \mathrm{T}.$$ We determine the sets ${\bigcup }_{g\in \mathrm{T}}{\mathrm{T}}_g$ and ${\bigcap }_{g\in \mathrm{T}}{\mathrm{T}}_g$. Moreover, for a fixed $g$, we determine the superdomain of local univalence of ${\mathrm{T}}_g$, the radii of local univalence, of starlikeness and of univalence of ${\mathrm{T}}_g$.

This paper studies the unicity of meromorphic(resp. entire) functions of the form $f^n{f}^{\text{'}}$ and obtains the following main result: Let $f$ and $g$ be two non-constant meromorphic (resp. entire) functions, and let $a\in ℂ\setminus \left\{0\right\}$ be a non-zero finite value. Then, the condition that $E_{3)}(a,f^n{f}^{\text{'}})=E_{3)}(a,g^n{g}^{\text{'}})$ implies that either $f=dg$ for some $(n+1)$-th root of unity $d$, or $f=c_1{e}^{cz}$ and $g=c_2{e}^{-cz}$ for three non-zero constants $c$, $c_1$ and $c_2$ with ${\left(c_1{c}_2\right)}^{n+1}{c}^2=-a^2$ provided that $n\ge 11$ (resp. $n\ge 6$). It improves a result of C. C. Yang and X. H. Hua. Also, some other related problems are discussed.

We introduce a new class of normalized functions regular and univalent in the unit disk. These functions, called uniformly convex functions, are defined by a purely geometric property. We obtain a few theorems about this new class and we point out a number of open problems.

We study the uniqueness of meromorphic functions when two nonlinear differential polynomials generated by two meromorphic functions share the same 1-points. Our results improve results of Fang-Fang and Lin-Yi and supplement a recent result of Lahiri-Pal.

We investigate the uniqueness of transcendental algebroid functions with shared values in some angular domains instead of the whole complex plane ℂ. We obtain two theorems which are counterparts of results for meromorphic functions obtained by Zheng.

We prove the uniqueness of meromorphic functions sharing some three sets with finite weights.

We deal with a uniqueness theorem of two meromorphic functions that share three values with weights and also share a set consisting of two small meromorphic functions. Our results improve those by G. Brosch, I. Lahiri & P. Sahoo, T. C. Alzahary & H. X. Yi, P. Li & C. C. Yang, and others.

We consider the problem of univalence of the integral operator [...] Imposing on functions f(z), g(z) various conditions and making use of a close-to-convexity property of the operator, we establish many suffcient conditions for univalence. Our results extend earlier ones. Some questions remain open.