Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the coefficients of functions with bounded boundary rotation


Author: D. K. Thomas
Journal: Proc. Amer. Math. Soc. 36 (1972), 123-129
MSC: Primary 30A34
DOI: https://doi.org/10.1090/S0002-9939-1972-0308384-2
MathSciNet review: 0308384
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {V_k}$ be the class of normalised functions of bounded boundary rotation. For $ f \in {V_k}$ define

$\displaystyle M(r,f) = \mathop {\max }\limits_{\vert z\vert = r} \vert f(z)\vert,$

and let $ L(r,f)$ denote the length of $ f(\vert z\vert = r)$. Then if $ f(z) = z + \sum\nolimits_{n = 2}^\infty {{a_n}{z^n}} $, it is shown that (i) $ 2M(r,f) < L(r,f) \leqq k\pi M(r,f)$, and (ii) $ {n^2}\vert{a_n}\vert \leqq (3k/{r^{n - 1}})M(r,f'),n \geqq 2$. The class $ {\Lambda _k}$ of meromorphic functions of boundary rotation is also studied and estimates for the coefficients are given.

References [Enhancements On Off] (What's this?)

  • [1] D. A. Brannan, On functions of bounded boundary rotation. I, Proc. Edinburgh Math. Soc. (2) 16 (1968/69), 339-347. MR 41 #8642. MR 0264045 (41:8642)
  • [2] D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions, J. London Math. Soc. (2) 1 (1969), 431-443. MR 40 #4439. MR 0251208 (40:4439)
  • [3] J. Clunie and Ch. Pommerenke, On the coefficients of close-to-convex univalent functions, J. London Math. Soc. 41 (1966), 161-165. MR 32 #7734. MR 0190321 (32:7734)
  • [4] W. E. Kirwan, On the coefficients of functions with bounded boundary rotation, Michigan Math. J. 15 (1968), 277-282. MR 38 #1250. MR 0232927 (38:1250)
  • [5] O. Lehto, On the distortion of conformal mappings with bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. AI Math.-Phys. No. 124 (1952). MR 14, 743. MR 0053241 (14:743a)
  • [6] J. W. Noonan, Meromorphic functions of bounded boundary rotation, Michigan Math. J. 18 (1971), 343-352. MR 0293075 (45:2154)
  • [7] J. Pfaltzgraff and B. Pinchuk, A variational method for classes of meromorphic functions (to appear). MR 0281899 (43:7613)
  • [8] M. S. Robertson, Coefficients of functions with bounded boundary rotation, Canad. J. Math. 21 (1969), 1477-1482. MR 41 #458. MR 0255798 (41:458)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30A34

Retrieve articles in all journals with MSC: 30A34


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0308384-2
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society