Extremal problems in the class of closetoconvex functions
Author:
Bernard Pinchuk
Journal:
Trans. Amer. Math. Soc. 129 (1967), 466478
MSC:
Primary 30.42
MathSciNet review:
0217279
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The class K of normalized closetoconvex 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. 34, 295–305 (German). MR
1544366, http://dx.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 (23 #A1803)
 [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
(29 #217)
 [4]
J.
A. Hummel, A variational method for starlike
functions, Proc. Amer. Math. Soc. 9 (1958), 82–87. MR 0095273
(20 #1779), http://dx.doi.org/10.1090/S00029939195800952737
 [5]
Wilfred
Kaplan, Closetoconvex schlicht functions, Michigan Math. J.
1 (1952), 169–185 (1953). MR 0054711
(14,966e)
 [6]
Zeev
Nehari, Conformal mapping, McGrawHill Book Co., Inc., New
York, Toronto, London, 1952. MR 0045823
(13,640h)
 [7]
Shôtarô
Ogawa, A note on closetoconvex functions. I, J. Nara Gakugei
Univ. 8 (1959), no. 2, 9–10. MR 0179340
(31 #3588)
 [8]
M.
S. Robertson, Variational methods for functions with
positive real part, Trans. Amer. Math. Soc.
102 (1962),
82–93. MR
0133454 (24 #A3288), http://dx.doi.org/10.1090/S0002994719620133454X
 [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
(34 #2854), http://dx.doi.org/10.1090/S00029939196602029958
 [10]
J.
Krzyż, Some remarks on closetoconvex functions, Bull.
Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.
12 (1964), 25–28. MR 0161971
(28 #5174)
 [1]
 L. Bieberbach, Aufstellung und Geweis des Drehungssatzes für schlichte konforme Abbildungen, Math. Z. 4 (1919), 295305. MR 1544366
 [2]
 G. M. Goluzin, On a variational method in the theory of analytic functions, Leningrad. Gos. Univ. Učen. Zap. 144 Ser. Mat. Nauk 23 (1952), 85101 ; Amer. Math. Soc. Transl. (2) 18 (1961), 114. MR 0124491 (23:A1803)
 [3]
 M. Heins, Selected topics in the classical theory of functions of a complex variable, Holt, Rinehart and Winston, New York, 1962. MR 0162913 (29:217)
 [4]
 J. A. Hummel, A variational method for starlike functions, Proc. Amer. Math. Soc. 9 (1958), 8287. MR 0095273 (20:1779)
 [5]
 W. Kaplan, Closetoconvex schlicht functions, Michigan Math. J. 1 (1952), 169185. MR 0054711 (14:966e)
 [6]
 Z. Nehari, Conformal mapping, McGrawHill, New York, 1952. MR 0045823 (13:640h)
 [7]
 S. Ogawa, A note on closetoconvex functions, J. Nara Gakugei Univ. 8 (1959), No. 2, 910. (Quoted from Math. Rev. 31 (1966), 646, #3588.) MR 0179340 (31:3588)
 [8]
 M. S. Robertson, Variational methods for functions with positive real part, Trans. Amer. Math. Soc. 101 (1962), 8293. MR 0133454 (24:A3288)
 [9]
 W. E. Kirwan, A note on extremal problems for certain classes of analytic functions, Proc. Amer. Math. Soc. 17 (1966), 10281031. MR 0202995 (34:2854)
 [10]
 J. Kryz, Some remarks on closetoconvex functions, Bull. Acad. Pol. Sci. 12 (1964), 2528. MR 0161971 (28:5174)
Similar Articles
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:
http://dx.doi.org/10.1090/S00029947196702172797
PII:
S 00029947(1967)02172797
Article copyright:
© Copyright 1967
American Mathematical Society
