Coefficients and integral means of some classes of analytic functions

Author:
T. Sheil-Small

Journal:
Proc. Amer. Math. Soc. **88** (1983), 275-282

MSC:
Primary 30C45; Secondary 30C50

MathSciNet review:
695258

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The sharp coefficient bounds for the classes of functions of bounded boundary rotation are obtained by a short and elementary argument. Elementary methods are also applied for the coefficients of related classes characterised by a generalised Kaplan condition. The result is proved simply. It is further shown that the functions are extremal for the th means ( an arbitrary real) of all Kaplan classes .

**[1]**D. Aharonov and S. Friedland,*On an inequality connected with the coefficient conjecture for functions of bounded boundary rotation*, Ann. Acad. Sci. Fenn. Ser. A I**524**(1972), 14. MR**0322155****[2]**Albert Baernstein II,*Integral means, univalent functions and circular symmetrization*, Acta Math.**133**(1974), 139–169. MR**0417406****[3]**D. A. Brannan,*On coefficient problems for certain power series*, Proceedings of the Symposium on Complex Analysis (Univ. Kent, Canterbury, 1973) Cambridge Univ. Press, London, 1974, pp. 17–27. London Math. Soc. Lecture Note Ser., No. 12. MR**0412411****[4]**D. A. Brannan, J. G. Clunie, and W. E. Kirwan,*On the coefficient problem for functions of bounded boundary rotation*, Ann. Acad. Sci. Fenn. Ser. A I**523**(1973), 18. MR**0338343****[5]**J. Clunie,*On meromorphic schlicht functions*, J. London Math. Soc.**34**(1959), 215–216. MR**0107009****[6]**Peter L. Duren,*Theory of 𝐻^{𝑝} spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655****[7]**Y. J. Leung,*Integral means of the derivatives of some univalent functions*, Bull. London Math. Soc.**11**(1979), no. 3, 289–294. MR**554395**, 10.1112/blms/11.3.289**[8]**Ch. Pommerenke,*On meromorphic starlike functions*, Pacific J. Math.**13**(1963), 221–235. MR**0150279****[9]**Stephan Ruscheweyh,*Some convexity and convolution theorems for analytic functions*, Math. Ann.**238**(1978), no. 3, 217–228. MR**514429**, 10.1007/BF01420249**[10]**T. Sheil-Small,*The Hadamard product and linear transformations of classes of analytic functions*, J. Analyse Math.**34**(1978), 204–239 (1979). MR**531276**, 10.1007/BF02790013

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
30C45,
30C50

Retrieve articles in all journals with MSC: 30C45, 30C50

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1983-0695258-8

Keywords:
Functions of bounded boundary rotation,
starlike functions,
close-to-convex functions

Article copyright:
© Copyright 1983
American Mathematical Society