Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean n-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. Väisälä published in the early 1960’s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value p = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.

Social capital and health : variations, associations and challenges

Although social capital and health have been extensively studied during the last decade, there are still open issues in current empirical research. These concern for instance the measurement of the concept in different contexts, as well as the association between different types of social capital and different dimensions of health. The present thesis addressed these questions. The general aim was to promote the understanding of social capital and health by investigating the oldest old and the two major language groups in Finland, Swedish- and Finnish-speakers. Another aim was to contribute to the discussion on methodological issues in social capital and health research. The present thesis investigated two empirical data sets, Umeå 85+ and Health 2000. The Umeå 85+ study was a cross-sectional study of 163 individuals aged 85, 90, and 95 or older, living in the municipality of Umeå, Sweden, in the year of 2000. The Health 2000 survey was a national study of 8,028 persons aged 30 or above carried out in Finland in 2000-2001. Different indicators of structural (e.g. social contacts) and cognitive (e.g. trust) social capital, as well as health indicators were used as variables in the analyses. The Umeå 85+ data set was analyzed with factor analysis, as well as univariate and multivariate analysis of variance. The Health 2000 data was analyzed with logistic regression techniques. The results showed that the Swedish-speakers in the Finnish data set Health 2000 had consistently higher prevalence of social capital compared to the Finnish-speakers even after controlling for central sociodemographic variables. The results further showed that even if the language group differences in health were small, the Swedishspeakers experienced in general better self-reported health compared with the Finnish-speakers. Common sociodemographic variables could not explain these observed differences in health. The results imply that social capital is often, but not always, associated with health. This was clearly seen in the Umeå 85+ data set where only one health indicator (depressive symptoms) was associated with structural social capital among the oldest old. The results based on the analysis of the Health 2000 survey demonstrated that the cognitive component of social capital was associated with self-rated health and psychological health rather than with participation in social activities and social contacts. In addition, social capital statistically reduced the health advantage especially for Swedish-speaking men, indicating that high prevalence of social capital may promote health. Finally, the present thesis also discussed the issue of methodological challenges faced with when analyzing social capital and health. It was suggested that certain components of social capital such as bonding and bridging social capital may be more relevant than structural and cognitive components when investigating social capital among the two language groups in Finland. The results concerning the oldest old indicated that the structural aspects of social capital probably reflect current living conditions, whereas cognitive social capital reflects attitudes and traits often acquired decades earlier. This is interpreted as an indication of the fact that structural and cognitive social capital are closely related yet empirically two distinctive concepts. Taken together, some components of social capital may be more relevant to study than others depending on which population group and age group is under study. The results also implied that the choice of cut-off point of dichotomization of selfrated health has an impact on the estimated effects of the explanatory variables. When the whole age interval, 35-64 years, was analyzed with logistic regression techniques the choice of cut-off point did not matter for the estimated effects of marital status and educational level. The results changed, however, when the age interval was divided into three shorter intervals. If self-rated health is explored using wide age intervals that do not account for age-dependent covariates there is a risk of drawing misleading conclusions. In conclusion, the results presented in the thesis suggest that the uneven distribution of social capital observed between the two language groups in Finland are of importance when trying to further understand health inequalities that exist between Swedish- and Finnish-speakers in Finland. Although social capital seemed to be relevant to the understanding of health among the oldest old, the meaning of social capital is probably different compared to a less vulnerable age group. This should be noticed in future empirical research. In the present thesis, it was shown that the relationship between social capital and health is complex and multidimensional. Different aspects of social capital seem to be important for different aspects of health. This reduces the possibility to generalize the results and to recommend general policy implementations in this area. An increased methodological awareness regarding social capital as well as health are called for in order to further understand the cfomplex association between them. However, based on the present data and findings social capital is associated with health. To understand individual health one must also consider social aspects of the individuals’ environment such as social capital.

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Effects of Socioeconomic Status and Sociodemographic Features on Cardiovascular Disease. Mortality and Morbidity in Finland

Despite declining trends in morbidity and mortality, cardiovascular diseases have a considerable impact on Finnish public health. A goal in Finnish health policy is to reduce inequalities in health and mortality among population groups. The aim of this study was to assess inequalities in cardiovascular diseases according to socioeconomic status (SES), language groups and other sociodemographic characteristics. The main data source was generated from events in 35-99 year-old men and women registered in the population-based FINMONICA and FINAMI myocardial infarction registers during the years ranging from 1988-2002. Information on population group characteristics was obtained from Statistics Finland. Additional data were derived from the FINMONICA and FINSTROKE stroke registers and the FINRISK Study. SES, measured by income level, was a major determinant of acute coronary syndrome (ACS) mortality. Among middle-aged men, the 28-day mortality rate of the lowest group of six income groups was 5.2 times and incidence 2.7 times as high when compared to the highest income group. Among women, the differences were even larger. Among the unmarried, the incidence of ACS was approximately 1.6 times as high and their prognosis was significantly worse than among married persons - both in men and women and independent of age. Higher age-standardized attack rates of ACS and stroke were found among Finnish-speaking compared to Swedish-speaking men in Turku and these differences could not be completely explained by SES. In these language groups, modest differences were found in traditional risk factor levels possibly explaining part of the found morbidity and mortality inequality. In conclusion, there are considerable differences in the morbidity and mortality of ACS and stroke between socioeconomic and sociodemographic groups, in Finland. Focusing measures to reduce the excess morbidity and mortality, in groups at high risk, could decrease the economic burden of cardiovascular diseases and thus be an important public health goal in Finland.

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.