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# Some Properties of a Class of Regular Functions

F. Holland
Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences
Vol. 69 (1970), pp. 85-95
Stable URL: http://www.jstor.org/stable/20488687
Page Count: 11
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## Abstract

Given a complex Borel measure μ on the unit circle C the formula $f(z)=z\ \text{exp}-2\int_{c}\text{Log}(1-z\overline{t})d\mu (t)\ (|z|<1)$ defines a function f that is regular on the unit disc. The positive measures generate the well-known class of starlike univalent functions, in the study of which the positivity of μ plays a major role. It is the aim of this paper to investigate the growth behaviour of those functions f that admit of the above representation in terms of a real measure. A series of theorems demonstrates that these functions enjoy a number of properties in common with the starlike ones, especially if they are, in addition, univalent.

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