Coefficients and integral means of some classes of analytic functions
Author:
T. SheilSmall
Journal:
Proc. Amer. Math. Soc. 88 (1983), 275282
MSC:
Primary 30C45; Secondary 30C50
MathSciNet review:
695258
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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 .
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 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 Math. 524 (1972). MR 0322155 (48:519)
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 Albert Baernstein II, Integral means, univalent functions and circular symmetrization, Acta Math. 133 (1974), 139169. MR 0417406 (54:5456)
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 D. A. Brannan, On coefficient problems for certain power series (Proc. Sympos. Complex Analysis, Canterbury, 1973), London Math. Soc. Lecture Note Ser., No. 12, Cambridge Univ. Press, London, 1974, pp. 1727. MR 0412411 (54:537)
 [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 Math. 523 (1973). MR 0338343 (49:3108)
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 J. Clunie, On meromorphic schlicht functions, J. London Math. Soc. 34 (1959), 215216. MR 0107009 (21:5737)
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 Peter L. Duren, Theory of spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
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 Y. J. Leung, Integral means of the derivatives of some univalent functions, Bull. London Math. Soc. 11 (1979), 289294. MR 554395 (80m:30013)
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 Ch. Pommerenke, On meromorphic starlike functions, Pacific J. Math. 13 (1963), 221235. MR 0150279 (27:280)
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 St. Ruscheweyh, Some convexity and convolution theorems for analytic functions, Math. Ann. 238 (1978), 217228. MR 514429 (80i:30023)
 [10]
 T. SheilSmall, The Hadamard product and linear transformations of classes of analytic functions, J. Analyse Math. 34 (1978), 204239. MR 531276 (81b:30023)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198306952588
PII:
S 00029939(1983)06952588
Keywords:
Functions of bounded boundary rotation,
starlike functions,
closetoconvex functions
Article copyright:
© Copyright 1983
American Mathematical Society
