## Convex hulls of some classical families of univalent functions

HTML articles powered by AMS MathViewer

- by L. Brickman, T. H. MacGregor and D. R. Wilken
- Trans. Amer. Math. Soc.
**156**(1971), 91-107 - DOI: https://doi.org/10.1090/S0002-9947-1971-0274734-2
- PDF | Request permission

## Abstract:

Let*S*denote the functions that are analytic and univalent in the open unit disk and satisfy $f(0) = 0$ and $f’(0) = 1$. Also, let

*K, St*, ${S_R}$, and

*C*be the subfamilies of

*S*consisting of convex, starlike, real, and close-to-convex mappings, respectively. The closed convex hull of each of these four families is determined as well as the extreme points for each. Moreover, integral formulas are obtained for each hull in terms of the probability measures over suitable sets. The extreme points for each family are particularly simple; for example, the Koebe functions $f(z) = z/{(1 - xz)^2},|x| = 1$ , are the extreme points of cl co

*St.*These results are applied to discuss linear extremal problems over each of the four families. A typical result is the following: Let

*J*be a “nontrivial” continuous linear functional on the functions analytic in the unit disk. The only functions in

*St.*that satisfy $\operatorname {Re} J(f) = \max \;\{ \operatorname {Re} \;J(g):g \in St\}$ are Koebe functions and there are only a finite number of them.

## References

- Louis Brickman,
*Extreme points of the set of univalent functions*, Bull. Amer. Math. Soc.**76**(1970), 372–374. MR**255788**, DOI 10.1090/S0002-9904-1970-12483-1 - Nelson Dunford and Jacob T. Schwartz,
*Linear Operators. I. General Theory*, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR**0117523** - Casper Goffman and George Pedrick,
*First course in functional analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR**0184064** - Wilfred Kaplan,
*Close-to-convex schlicht functions*, Michigan Math. J.**1**(1952), 169–185 (1953). MR**54711** - J. Krzyż,
*Some remarks on close-to-convex functions*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**12**(1964), 25–28. MR**161971**
A. Marx, - Zeev Nehari,
*Conformal mapping*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1952. MR**0045823** - Pasquale Porcelli,
*Linear spaces of analytic functions*, Rand McNally & Co., Chicago, Ill., 1966. MR**0259581** - Maxwell O. Reade,
*On close-to-convex univalent functions*, Michigan Math. J.**3**(1955), 59–62. MR**70715** - M. S. Robertson,
*On the coefficients of a typically-real function*, Bull. Amer. Math. Soc.**41**(1935), no. 8, 565–572. MR**1563142**, DOI 10.1090/S0002-9904-1935-06147-6 - Raphael M. Robinson,
*Univalent majorants*, Trans. Amer. Math. Soc.**61**(1947), 1–35. MR**19114**, DOI 10.1090/S0002-9947-1947-0019114-6 - Werner Rogosinski,
*Über positive harmonische Entwicklungen und typisch-reelle Potenzreihen*, Math. Z.**35**(1932), no. 1, 93–121 (German). MR**1545292**, DOI 10.1007/BF01186552 - George Springer,
*Extreme Punkte der konvexen Hülle schlichter Funktionen*, Math. Ann.**129**(1955), 230–232 (German). MR**69272**, DOI 10.1007/BF01362368 - Erich Strohhäcker,
*Beiträge zur Theorie der schlichten Funktionen*, Math. Z.**37**(1933), no. 1, 356–380 (German). MR**1545400**, DOI 10.1007/BF01474580 - Angus E. Taylor,
*Introduction to functional analysis*, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR**0098966** - Otto Toeplitz,
*Die linearen vollkommenen Räume der Funktionentheorie*, Comment. Math. Helv.**23**(1949), 222–242 (German). MR**32952**, DOI 10.1007/BF02565600

*Untersuchungen über schlichte Abbildungen*, Math. Ann.

**107**(1932/33), 40-67.

## Bibliographic Information

- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**156**(1971), 91-107 - MSC: Primary 30.42
- DOI: https://doi.org/10.1090/S0002-9947-1971-0274734-2
- MathSciNet review: 0274734