Support points of families of analytic functions described by subordination

Authors:
D. J. Hallenbeck and T. H. MacGregor

Journal:
Trans. Amer. Math. Soc. **278** (1983), 523-546

MSC:
Primary 30C45; Secondary 30C80

MathSciNet review:
701509

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Abstract: We determine the set of support points for several families of functions analytic in the open unit disc and which are generally defined in terms of subordination. The families we study include the functions with a positive real part, the typically-real functions, and the functions which are subordinate to a given majorant. If the majorant is univalent then each support point has the form , where is a finite Blaschke product and . This completely characterizes the set of support points when is convex. The set of support points is found for some specific majorants, including where . Let and denote the set of normalized convex and starlike mappings, respectively. We find the support points of the families and defined by the property of being subordinate to some member of or , respectively.

**[1]**Yusuf Abu-Muhanna and Thomas H. MacGregor,*Extreme points of families of analytic functions subordinate to convex mappings*, Math. Z.**176**(1981), no. 4, 511–519. MR**611640**, 10.1007/BF01214761**[2]**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****[3]**L. Brickman, T. H. MacGregor, and D. R. Wilken,*Convex hulls of some classical families of univalent functions*, Trans. Amer. Math. Soc.**156**(1971), 91–107. MR**0274734**, 10.1090/S0002-9947-1971-0274734-2**[4]**Paul Cochrane and Thomas H. MacGregor,*Fréchet differentiable functionals and support points for families of analytic functions*, Trans. Amer. Math. Soc.**236**(1978), 75–92. MR**0460611**, 10.1090/S0002-9947-1978-0460611-7**[5]**J. Feng,*Extreme points and integral means for classes of analytic functions*, Ph.D. dissertation, SUNY at Albany, 1974.**[6]**David J. Hallenbeck,*Convex hulls and extreme points of families of starlike and close-to-convex mappings*, Pacific J. Math.**57**(1975), no. 1, 167–176. MR**0379820****[7]**D. J. Hallenbeck and T. H. MacGregor,*Subordination and extreme-point theory*, Pacific J. Math.**50**(1974), 455–468. MR**0361035****[8]**Kenneth Hoffman,*Banach spaces of analytic functions*, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR**0133008****[9]**Thomas H. MacGregor,*Applications of extreme-point theory to univalent functions*, Michigan Math. J.**19**(1972), 361–376. MR**0311885****[10]**Werner Rogosinski,*On the coefficients of subordinate functions*, Proc. London Math. Soc. (2)**48**(1943), 48–82. MR**0008625****[11]**Glenn Schober,*Univalent functions—selected topics*, Lecture Notes in Mathematics, Vol. 478, Springer-Verlag, Berlin-New York, 1975. MR**0507770****[12]**T. J. Suffridge,*Some remarks on convex maps of the unit disk*, Duke Math. J.**37**(1970), 775–777. MR**0269827****[13]**Otto Toeplitz,*Die linearen vollkommenen Räume der Funktionentheorie*, Comment. Math. Helv.**23**(1949), 222–242 (German). MR**0032952**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1983-0701509-8

Keywords:
Analytic function,
continuous linear functional,
support point,
extreme point,
closed convex hull,
subordination,
univalent function,
convex mapping,
starlike mapping,
function with a positive real part,
bounded function,
Herglotz formula,
probability measure,
finite Blaschke product

Article copyright:
© Copyright 1983
American Mathematical Society