Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Fréchet differentiable functionals and support points for families of analytic functions


Authors: Paul Cochrane and Thomas H. MacGregor
Journal: Trans. Amer. Math. Soc. 236 (1978), 75-92
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9947-1978-0460611-7
MathSciNet review: 0460611
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given a closed subset of the family $ {S^\ast}(\alpha )$ of functions starlike of order $ \alpha $ of a particular form, a continuous Fréchet differentiable functional, J, is constructed with this collection as the solution set to the extremal problem $ \max \operatorname{Re} J(f)$ over $ {S^\ast}(\alpha )$. Similar results are proved for families which can be put into one-to-one correspondence with $ {S^\ast}(\alpha )$.

The support points of $ {S^\ast}(\alpha )$ and $ K(\alpha )$, the functions convex of order $ \alpha $, are completely characterized and shown to coincide with the extreme points of their respective convex hulls. Given any finite collection of support points of $ {S^\ast}(\alpha )$ (or $ K(\alpha )$), a continuous linear functional, J, is constructed with this collection as the solution set to the extremal problem $ \max \operatorname{Re} J(f)$ over $ {S^\ast}(\alpha )$ (or $ K(\alpha )$).


References [Enhancements On Off] (What's this?)

  • [1] N. I. Akhiezer, The classical moment problem and some related questions in analysis, Fizmatgiz, Moscow, 1961; English transl., Hafner, New York, 1965. MR 27 #4028; 32 #1518. MR 0184042 (32:1518)
  • [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. A I No. 523 (1973), 18 pp. MR 49 #3108. MR 0338343 (49:3108)
  • [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 43 #494. MR 0274734 (43:494)
  • [4] L. Brickman, D. J. Hallenbeck, T. H. MacGregor and D. R. Wilken, Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc. 185 (1973), 413-428 (1974).MR 49 #3102. MR 0338337 (49:3102)
  • [5] C. Carathéodory, Über den Variabilitätsbereich der Fourier'schen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32 (1911), 193-217.
  • [6] -, Funktionentheorie, Bd. 2, Birkhäuser, Basel, 1950; English transl., Theory of functions of a complex variable, Vol. 2, Chelsea, New York, 1954. MR 12, 248; 16 346.
  • [7] P. Dienes, The Taylor series, Dover, New York, 1957. MR 19,735.
  • [8] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Interscience, New York and London 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [9] P. L. Duren, Theory of $ {H^p}$ spaces, Academic Press, New York, 1970. MR 42 #3552. MR 0268655 (42:3552)
  • [10] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York, 1973. MR 49 #6269. MR 0341518 (49:6269)
  • [11] G. M. Goluzin, Geometric theory of functions of a complex variable, 2nd ed., ``Nauka", Moscow, 1966; English transl., Transl. Math. Monographs, vol. 26, Amer. Math. Soc., Providence, R. I., 1969. MR 36 #2793; 40 #308. MR 0247039 (40:308)
  • [12] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 31, rev. ed., Amer. Math. Soc., Providence, R. I., 1957. MR 19, 664. MR 0089373 (19:664d)
  • [13] J. A. Hummel, Extremal problems in the class of starlike functions, Proc. Amer. Math. Soc. 11 (1960), 741-749. MR 22 # 11133. MR 0120379 (22:11133)
  • [14] W. E. Kirwan and G. Schober, On extreme points and support points for some families of univalent functions, Duke Math. J. (to appear). MR 0367174 (51:3416)
  • [15] T. H. MacGregor, Applications of extreme-point theory to univalent functions, Michigan Math. J. 19 (1972), 361-376. MR 47 #447. MR 0311885 (47:447)
  • [16] -, Rotations of the range of an analytic function, Math. Ann. 201 (1973), 113-126. MR 48 #6390. MR 0328048 (48:6390)
  • [17] -, Hull subordination and extremal problems for starlike and spirallike mappings, Trans. Amer. Math. Soc. 183 (1973), 499-510. MR 49 #3104. MR 0338339 (49:3104)
  • [18] A. J. Macintyre and W. W. Rogosinski, Extremum problems in the theory of analytic functions, Acta Math. 82 (1950), 275-325. MR 12, 89. MR 0036314 (12:89e)
  • [19] Z. Nehari, Conformal mapping, McGraw-Hill, New York, 1952. MR 13, 640. MR 0045823 (13:640h)
  • [20] Ch. Pommerenke, On starlike and convex functions, J. London Math. Soc. 37 (1962), 209-224. MR 25 #1279. MR 0137830 (25:1279)
  • [21] P. Porcelli, Linear spaces of analytic functions, Rand McNally, Chicago, 1966. MR 41 #4219. MR 0259581 (41:4219)
  • [22] M. S. Robertson, On the theory of univalent functions, Ann. of Math. (2) 37 (1936), 374-408. MR 1503286
  • [23] W. W. Rogosinski, Über positive harmonische Entwicklungen und typisch-reelle Potenzreihen, Math. Z. 35 (1932), 93-121. MR 1545292
  • [24] J. Schur, Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, J. Math. 147 (1917), 205-232.
  • [25] -, Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, J. Math. 148 (1918), 122-145.
  • [26] A. E. Taylor, Introduction to functional analysis, Wiley, New York; Chapman and Hall, London, 1958. MR 20 #5411. MR 0098966 (20:5411)
  • [27] O. Toeplitz, Die linearen vollkommenen Räume der Funktionentheorie, Comment. Math. Helv. 23 (1949), 222-242. MR 11, 372. MR 0032952 (11:372f)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30A32

Retrieve articles in all journals with MSC: 30A32


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0460611-7
Keywords: Analytic function, continuous linear functional, Fréchet differentiable functional, univalent function, starlike mapping, starlike function of order $ \alpha $, convex mapping, convex function of order $ \alpha $, variations of functions, bounded functions, support point, extreme point, convex hull, coefficient region, function with a positive real part, subordination
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society