Abstract We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied Ahrens, Kapulkin, and Shulman, we define study bicategories. To construct examples bicategories a modular fashion, displayed , an analog 1-categories introduced Ahrens Lumsdaine. demonstrate applicability this prove that several interest are univalent. Among these catego...

A necessary and sufficient coefficient is given for functions in a class of complexvalued harmonic univalent functions using the Dziok-Srivastava operator. Distortion bounds, extreme points, an integral operator, and a neighborhood of such functions are considered.

We classify Baker domains U for entire maps with fj U univalent into three diierent types, giving several criteria which characterize them. Some new examples of such domains are presented, including a domain with disconnected boundary in C and a domain which spirals towards innnity.

The purpose of the present paper is to study a certain subclass of harmonic univalent functions associated with Dziok-Srivastava operator. We obtain coefficient conditions, distortion bounds, and extreme points for the above class of harmonic univalent functions belonging to this class and discuss a class preserving integral operator. We also show that class studied in this paper is closed unde...

In vitro killing of the human Daudi cell line by either univalent [F(ab')] or divalent (IgG) forms of rabbit anti-human Ig (RAHIg) coupled to ricin A chain can be specifically potentiated by a "piggyback" treatment with ricin B chain coupled to goat anti-rabbit Ig (GARIg). When cells are treated with univalent immunotoxin (IT) [F(ab') RAHIg-A] and then cultured, IT can be detected on the cell s...

Succinyl CoA synthetase from Nicotiana tabacum exhibited a requirement for univalent and divalent cations. Mn(2+) replaced Mg(2+) in the assay medium and Co(2+) and Ca(2+) partially replaced Mg(2+). Addition of Zn(2+) resulted in no enzyme activity. The enzyme was activated by univalent cations K(+), Rb(+), NH(4) (+), and Na(+); Li(+) showed little or no activation. Maximum enzyme activity vari...

A normalized univalent function f is called Ma–Minda starlike or convex if zf ðzÞ=f ðzÞ uðzÞ or 1þ zf ðzÞ=f 0ðzÞ uðzÞ where u is a convex univalent function with uð0Þ 1⁄4 1. The class of Ma–Minda convex functions is shown to be closed under certain operators that are generalizations of previously studied operators. Analogous inclusion results are also obtained for subclasses of starlike and clo...

A model theory in the framework of Univalent Foundations requires a logic that allows us to define structures on homotopy (n-)types, similar to how first-order logic can define structures on sets. We define such an “n-level” logic for finite n. The syntax is based on a generalization of Makkai’s FOLDS, obtained by an operation that allows us to add equality sorts to FOLDS-signatures. We then gi...

The well-known Schwarz-Pick lemma states that any analytic mapping φ of the unit disk U into itself satisfies the inequality |φ ′ (z)| ≤ 1 − |φ(z)| 2 1 − |z| 2 , z ∈ U. This estimate remains the same if we restrict ourselves to univalent map-pings. The lower estimate is |φ ′ (z)| ≥ 0 generally or |φ ′ (z)| > 0 for univa-lent functions. To make the lower estimate non-trivial we consider univalen...

For the standard class S of normalized univalent functions f analytic in the unit disk U , we consider a problem on the minimal area of the image f(U) concentrated in any given half-plane. This question is related to a well-known problem posed by A. W. Goodman in 1949 that regards minimizing area covered by analytic univalent functions under certain geometric constraints. An interesting aspect ...