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Journal of the London Mathematical Society 1997 55(3):489-498; doi:10.1112/S0024610797005188
© 1997 by London Mathematical Society
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© The London Mathematical Society

Conformal Mappings of Close-to-Convex Domains

Tom Carroll and J. B. Twomey

Department of Mathematics, University College Cork, Ireland

Received 14 September 1994. Revision received 26 April 1995.

Let f be analytic and univalent in {Delta}={z:|z|<1}. We write


Formula

for 0≤r<1 and {theta}isin[0, 2{pi}], and remark that L(r, {theta}) is the length of the image in f({Delta}) of the ray joining 0 to rei{theta} in {Delta}. We shall be concerned with the growth of L(r, {theta}), as a function of r, for univalent functions in {Delta} and, more particularly, for functions in the subclass of close-to-convex univalent functions in {Delta}.


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