The Schwarzian derivative of an analytic function is a basic tool in complex analysis. It appeared as early as 1873, when H. A. Schwarz sought to generalize the Schwarz-Christoffel formula to conformal mappings of polygons bounded by circular arcs. More recently, Nehari [5, 6, 7] and others have developed important criteria for global univalence in terms of the Schwarzian derivative, exploiting...

The class S of functions g(z) = z + c 2 z 2 + c 3 z 3 + ... analytic and univalent in the unit disk Izr < 1 has been thoroughly studied, and its properties are well known. Our purpose is to investigate another class of functions which, by contrast, seems to have been rather neglected. This is the class S o of functions f ( z ) = 1 + a 1 z + a 2 z Z + . . , analytic, univalent, and nonvanishing ...

In this paper, we obtain some subordination results for certian subclasses of univalent functions defined by convolution.

It is shown that an analytic function taking circles to ellipses must be a Möbius transformation. It then follows that a harmonic mapping taking circles to ellipses is a harmonic Möbius transformation. Analytic Möbius transformations take circles to circles. This is their most basic, most celebrated geometric property. We add the adjective ‘analytic’ because in a previous paper [1] we introduce...

Two classes of univalent harmonic functions on unit disc satisfying the conditions ∑∞ n=2(n−α)(|an|+|bn|) ≤ (1−α)(1−|b1|) and ∑∞ n=2 n(n−α)(|an|+|bn|) ≤ (1−α)(1−|b1|) are given. That the ranges of the functions belonging to these two classes are starlike and convex, respectively. Sharp coefficient relations and distortion theorems are given for these functions. Furthermore results concerning th...

The class of univalent harmonic functions on the unit disc satisfying the condition ∑∞ k=2 (k m − αk)(|ak|+ |bk|) ≤ (1−α)(1−|b1|) is given. Sharp coefficient relations and distortion theorems are given for these functions. In this paper we find that many results of Özturk and Yalcin [5] are incorrect. Some of the results of this paper correct the theorems and examples of [5]. Further, sharp coe...

It has been shown that univalent circle packings filling in the complex plane C are unique up to similarities of C. Here we prove that bounded degree branched circle packings properly covering C are uniquely determined, up to similarities of C, by their branch sets. In particular, when branch sets of the packings considered are empty we obtain the earlier result. We also establish a circle pack...

Virasoro algebra possesses a representation in the tangent space to the space of normalized univalent functions defined in the unit disk and smooth on its boundary. We discuss connections between representations of the Virasoro algebra and the Löwner-Kufarev equations in partial and ordinary derivatives. Virasoro generators turn to be conservative quantites for the Löwner-Kufarev ODE, and the L...