Hull subordination and extremal problems for starlike and spirallike mappings
Author:
Thomas H. MacGregor
Journal:
Trans. Amer. Math. Soc. 183 (1973), 499510
MSC:
Primary 30A32
MathSciNet review:
0338339
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Abstract: Let be a compact subset of the family of functions analytic in , and let be a continuous linear operator of order zero on . We show that if the extreme points of the closed convex hull of is the set , then is hull subordinate to in . This generalizes results of R. M. Robinson corresponding to families of functions that are subordinate to or to . Families to which this theorem applies are discussed and we identify each such operator with a suitable sequence of complex numbers. Suppose that is a nonconstant entire function and that . We show that the maximum of over the class of starlike functions of order a is attained only by the functions . A similar result is obtained for spirallike mappings. Both results generalize a theorem of G. M. Golusin corresponding to the family of starlike mappings.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197303383399
PII:
S 00029947(1973)03383399
Keywords:
Subordination,
hull subordination,
continuous linear operator,
functions with positive real part,
Herglotz representation formula,
extreme points,
closed convex hull,
univalent functions,
starlike mapping,
starlike functions of order a,
spirallike functions,
extremal problems,
KreĭnMilman theorem,
probability measure,
convex mapping,
convex functions of order a,
continuous linear functional
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© Copyright 1973 American Mathematical Society
