Extremal problems in the class of close-to-convex functions

Author:
Bernard Pinchuk

Journal:
Trans. Amer. Math. Soc. **129** (1967), 466-478

MSC:
Primary 30.42

DOI:
https://doi.org/10.1090/S0002-9947-1967-0217279-7

MathSciNet review:
0217279

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The class *K* of normalized close-to-convex functions in has a parametric representation involving two Stieltjes integrals. Using a variational method due to G. M. Goluzin [2] for classes of analytic functions defined by a Stieltjes integral, variational formulas are developed for *K*. With these variational formulas, two general extremal problems within *K* are solved.

The first problem is to maximize the functional over *K* where is a given entire function and *z* a given point in *D*. A special case of this is the rotation theorem for *K*. The second problem solved is a general coefficient problem. Both problems are solved by characterizing the measures which appear in the integral representation for the extremal functions.

The classes of convex univalent functions in *D* and functions whose derivative has a positive real part in *D* are proper subclasses of *K*. The methods used to solve the extremal problems in *K* can be used for these subclasses as well. Some of the results for the subclasses are known and are not presented here, even though the methods differ from those used previously.

It should be mentioned that Goluzin originally used these methods to solve extremal problems of the first type mentioned above within the classes of starlike and typically real functions.

**[1]**Ludwig Bieberbach,*Aufstellung und Beweis des Drehungssatzes für schlichte konforme Abbildungen*, Math. Z.**4**(1919), no. 3-4, 295–305 (German). MR**1544366**, https://doi.org/10.1007/BF01203017**[2]**G. M. Goluzin,*On a variational method in the theory of analytic functions*, Amer. Math. Soc. Transl. (2)**18**(1961), 1–14. MR**0124491****[3]**Maurice Heins,*Selected topics in the classical theory of functions of a complex variable*, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1962. MR**0162913****[4]**J. A. Hummel,*A variational method for starlike functions*, Proc. Amer. Math. Soc.**9**(1958), 82–87. MR**0095273**, https://doi.org/10.1090/S0002-9939-1958-0095273-7**[5]**Wilfred Kaplan,*Close-to-convex schlicht functions*, Michigan Math. J.**1**(1952), 169–185 (1953). MR**0054711****[6]**Zeev Nehari,*Conformal mapping*, McGraw-Hill Book Co., Inc., New York, Toronto, London, 1952. MR**0045823****[7]**Shôtarô Ogawa,*A note on close-to-convex functions. I*, J. Nara Gakugei Univ.**8**(1959), no. 2, 9–10. MR**0179340****[8]**M. S. Robertson,*Variational methods for functions with positive real part*, Trans. Amer. Math. Soc.**102**(1962), 82–93. MR**0133454**, https://doi.org/10.1090/S0002-9947-1962-0133454-X**[9]**W. E. Kirwan,*A note on extremal problems for certain classes of analytic functions*, Proc. Amer. Math. Soc.**17**(1966), 1028–1030. MR**0202995**, https://doi.org/10.1090/S0002-9939-1966-0202995-8**[10]**J. Krzyż,*Some remarks on close-to-convex functions*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**12**(1964), 25–28. MR**0161971**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30.42

Retrieve articles in all journals with MSC: 30.42

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1967-0217279-7

Article copyright:
© Copyright 1967
American Mathematical Society