We show that the type T Z of -torsors has dependent universal property circle, which characterizes it up to a unique homotopy equivalence. The construction uses Voevodsky's Univalence Axiom and propositional truncation, yielding stand-alone circle not using higher inductive types.

Let A denote the class of functions f (z) which are analytic in the open unit disk and normalized by f (0) = f (0) − 1 = 0. In this paper the expression f (z)−1 f (z)/z is studied using differential subordinations and different properties of f (z) z , as well as sufficient conditions for starlikeness and univalence of f (z) ∈ A, are obtained. Also, several open problems are posed. © 2011 Elsevi...

In their usual form, representation independence metatheorems provide an external guarantee that two implementations of abstract interface are interchangeable when they related by operation-preserving correspondence. If our programming language is dependently-typed, however, we would like to appeal such invariance results within the itself, in order obtain correctness theorems for complex trans...

Combining the definition of Schwarzian derivative for conformal mappings between Riemannian manifolds given by Osgood and Stowe with that for parametrized curves in Euclidean space given by Ahlfors, we establish injectivity criteria for holomorphic curves φ : D → C. The result can be considered a generalization of a classical condition for univalence of Nehari.

The object of the present paper is to derive certain sufficient conditions for univalence, p-valently starlikeness and p-valently close-to-convexity of analytic functions in the unit disk. Our results extend and improve some results due to Owa [3], Frasin and Darus [2] and Chen [1].