## Support points of families of analytic functions described by subordination

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- by D. J. Hallenbeck and T. H. MacGregor PDF
- Trans. Amer. Math. Soc.
**278**(1983), 523-546 Request permission

## 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 $F$ is univalent then each support point has the form $F \circ \;\phi$, where $\phi$ is a finite Blaschke product and $\phi (0) = 0$. This completely characterizes the set of support points when $F$ is convex. The set of support points is found for some specific majorants, including $F(z) = {((1 + z)/(1 - z))^p}$ where $p > 1$. Let $K$ and ${\text {St}}$ denote the set of normalized convex and starlike mappings, respectively. We find the support points of the families ${K^{\ast } }$ and ${\text {St}}^{\ast }$ defined by the property of being subordinate to some member of $K$ or ${\text {St}}$, respectively.## References

- 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**, DOI 10.1007/BF01214761 - 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**338343** - 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**274734**, DOI 10.1090/S0002-9947-1971-0274734-2 - 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**460611**, DOI 10.1090/S0002-9947-1978-0460611-7
J. Feng, - 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**379820** - D. J. Hallenbeck and T. H. MacGregor,
*Subordination and extreme-point theory*, Pacific J. Math.**50**(1974), 455–468. MR**361035** - Kenneth Hoffman,
*Banach spaces of analytic functions*, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR**0133008** - Thomas H. MacGregor,
*Applications of extreme-point theory to univalent functions*, Michigan Math. J.**19**(1972), 361–376. MR**311885** - Werner Rogosinski,
*On the coefficients of subordinate functions*, Proc. London Math. Soc. (2)**48**(1943), 48–82. MR**8625**, DOI 10.1112/plms/s2-48.1.48 - Glenn Schober,
*Univalent functions—selected topics*, Lecture Notes in Mathematics, Vol. 478, Springer-Verlag, Berlin-New York, 1975. MR**0507770** - T. J. Suffridge,
*Some remarks on convex maps of the unit disk*, Duke Math. J.**37**(1970), 775–777. MR**269827** - Otto Toeplitz,
*Die linearen vollkommenen Räume der Funktionentheorie*, Comment. Math. Helv.**23**(1949), 222–242 (German). MR**32952**, DOI 10.1007/BF02565600

*Extreme points and integral means for classes of analytic functions*, Ph.D. dissertation, SUNY at Albany, 1974.

## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**278**(1983), 523-546 - MSC: Primary 30C45; Secondary 30C80
- DOI: https://doi.org/10.1090/S0002-9947-1983-0701509-8
- MathSciNet review: 701509