Hyperbolic type metrics and distortion of quasiconformal map pings

This Ph.D. thesis consists of four original papers. The papers cover several topics from geometric function theory, more specifically, hyperbolic type metrics, conformal invariants, and the distortion properties of quasiconformal mappings. The first paper deals mostly with the quasihyperbolic metric. The main result gives the optimal bilipschitz constant with respect to the quasihyperbolic metric for the M¨obius self-mappings of the unit ball. A quasiinvariance property, sharp in a local sense, of the quasihyperbolic metric under quasiconformal mappings is also proved. The second paper studies some distortion estimates for the class of quasiconformal self-mappings fixing the boundary values of the unit ball or convex domains. The distortion is measured by the hyperbolic metric or hyperbolic type metrics. The results provide explicit, asymptotically sharp inequalities when the maximal dilatation of quasiconformal mappings tends to 1. These explicit estimates involve special functions which have a crucial role in this study. In the third paper, we investigate the notion of the quasihyperbolic volume and find the growth estimates for the quasihyperbolic volume of balls in a domain in terms of the radius. It turns out that in the case of domains with Ahlfors regular boundaries, the rate of growth depends not merely on the radius but also on the metric structure of the boundary. The topic of the fourth paper is complete elliptic integrals and inequalities. We derive some functional inequalities and elementary estimates for these special functions. As applications, some functional inequalities and the growth of the exterior modulus of a rectangle are studied.

Helsingin matkailijakartta = Turistkarta över Helsingfors = Helsinki : Touristenkarte = Helsinki : tourist map = Helsinki : plan touristique

Kartta kääntöpuolella

Map of Helsinki-Helsingfors

Kääntöpuolella kannessa: Finland Travel Bureau Ltd Esplanade 19, Branch Office Stockmanns Department Stores

Helsingin matkailijakartta = Turistkarta över Helsingfors = Helsinki : Touristenkarte = Helsinki : tourist map = Helsinki : plan touristique

Kartta kääntöpuolella

Nieuwe paskaart voor een gedeelte van de Oost-Zee = Nouvelle carte hydrographique d'une partie de la Mer Baltique = Nya pass cort öfwer yen dehl af Oster-Siöön = The new sea map of the second of the East-Sea

Sisältää myös kartakkeet: Pastkaartje van de Noord Bodem, Suomenlahden itäosa ja Laatokka

Nieuwe paskaart voor een gedeelte van de Oost-Zee = Nouvelle carte hydrographique d'une partie de la Mer Baltique = Nya pass cort öfwer yen dehl af Oster-Siöön = The new sea map of the second of the East-Sea

Sisältää myös kartakkeet: Pastkaartje van de Noord Bodem, Suomenlahden itäosa ja Laatokka

A complete map of the southern continent

Nimeketiedot nimiönkehyksissä

A complete map of the southern continent

Nimeketiedot nimiönkehyksissä

A map of Marco Polo's voyages & travels in the 13th century through a great part of Asia, all Tartary, the East India islands & part of Africa

Nimeketiedot nimiönkehyksissä

A map of Russia

Kartta kuuluu A. E. Nordenskiöldin kokoelmaan

A map of the western part of the Roman Empire

Kartta kuuluu A. E. Nordenskiöldin kokoelmaan

A map of the eastern part of the Roman Empire

Kartta kuuluu A. E. Nordenskiöldin kokoelmaan

An account of a copy from the 15th century of a map of the world engraved on metal, which is preserved in cardinal Stephan Borgia's museum at Velletri

Erip.: Ymer 1891.

Sketch map of part of Spitzbergen

Kartta kuuluu A. E. Nordenskiöldin kokoelmaan

Sketch map of part of Spitzbergen

Kartta kuuluu A. E. Nordenskiöldin kokoelmaan