We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of " category " for which equality and equivalence of categories agree. Such categories satisfy a version of the Univalence Axiom, saying that the type of isomorphisms between any two objec...

Let S be the standard class of univalent functions in the unit disk, and let So be the class of nonvanishing univalent functions g with g(0) = 1. It is shown that the regions of variability {g(r): g £ So} and {(1 — r2)f'(r) : f £ S} are very closely related but are not quite identical. Let 5 be the class of functions / analytic and univalent in the unit disk D, with /(0) = 0 and /'(O) = 1. Let ...

By using generalized Salagean differential operator a newclass of univalent holomorphic functions with fixed finitely manycoefficients is defined. Coefficient estimates, extreme points,arithmetic mean, and weighted mean properties are investigated.

It is known that the complementarity problems and the variational inequality problems are reformulated equivalently as a vector equation by using the natural residual or Fischer-Burmeister function. In this short paper, we first study the global convergence of a sequential injective algorithm for weakly univalent vector equation. Then, we apply the convergence analysis to the regularized smooth...

The Univalent Foundations of Mathematics (UF) provide not only an entirely non-Cantorian conception of the basic objects of mathematics (“homotopy types” instead of “sets”) but also a novel account of how foundations ought to relate to mathematical practice. In this paper, I intend to answer the question: In what way is UF a new foundation of mathematics? I will begin by connecting UF to a prag...

Let F„ denote the wth coefficient region for this class of functions [S, §1.2]. Let F=Fia2, ä2, ■ • ■ , a„, a«) be a real-valued function. Write x,= (l/2)(a,+á,), y,= (l/2í)(a,-á,), F,= (l/2)(r3F/r)x, — idF/dy,). Let F satisfy the further conditions (a) F is defined in an open set O containing V„, (b) F and its derivatives F, are continuous in 0, (c) |grad F\ = ( E?-a | F,\ 2)1/2>0 in 0. Let/(z...