In this paper some simple conditions on f ′(z) and f ′′(z) which lead to some subclasses of univalent functions will be considered.

In this short note we give a glimpse of homotopy type theory, a new field of mathematics at the intersection of algebraic topology and mathematical logic, and we explain Vladimir Voevodsky’s univalent interpretation of it. This interpretation has given rise to the univalent foundations program, which is the topic of the current special year at the Institute for Advanced Study. The Institute for...

In this note we show that Voevodsky’s univalence axiom holds in the model of type theory based on cubical sets as described in [2, 6]. We will also discuss Swan’s construction of the identity type in this variation of cubical sets. This proves that we have a model of type theory supporting dependent products, dependent sums, univalent universes, and identity types with the usual judgmental equa...

It is known that the condition $mathfrak {Re} left{zf'(z)/f(z)right}>0$, $|z|

Since in many particular cases checking directly the conditions from definitions of starlikeness or convexity a function can be difficult, this paper we use theory differential subordination and method admissible functions order to determine for confluent (Kummer) hypergeometric first kind. Having mind results obtained by Miller Mocanu 1990 who used a,c?R, function, investigation c complex numb...

For the analytic functions ?(?)=?+?k=n??k?k in unit disk O, we calculate values of n and ?, where condition ?1+???(?)/??(?)>?? or ?1+???(?)/??(?)<1+?/2 yields univalence starlikeness. Conditions imply ? class all normalized U, with ?/?(?)2??(?)?1<1 are obtained. Recent findings gained, unique cases demonstrated. The generalization Jack lemma serves proof main result that our methodolog...