Hull subordination and extremal problems for starlike and spirallike mappings
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- by Thomas H. MacGregor PDF
- Trans. Amer. Math. Soc. 183 (1973), 499-510 Request permission
Abstract:
Let $\mathfrak {F}$ be a compact subset of the family $\mathcal {A}$ of functions analytic in $\Delta = \{ z:\;|z| < 1\}$, and let $\mathcal {L}$ be a continuous linear operator of order zero on $\mathcal {A}$. We show that if the extreme points of the closed convex hull of $\mathcal {F}$ is the set $\{ {f_0}(xz)\} (|x| = 1)$, then $\mathcal {L}(f)$ is hull subordinate to $\mathcal {L}({f_0})$ in $\Delta$. This generalizes results of R. M. Robinson corresponding to families $\mathcal {F}$ of functions that are subordinate to $(1 + z)/(1 - z)$ or to $1/{(1 - z)^2}$. Families $\mathcal {F}$ to which this theorem applies are discussed and we identify each such operator $\mathcal {L}$ with a suitable sequence of complex numbers. Suppose that $\Phi$ is a nonconstant entire function and that $0 < |{z_0}| < 1$. We show that the maximum of $\operatorname {Re} \{ \Phi [\log (f({z_0})/{z_0})]\}$ over the class of starlike functions of order a is attained only by the functions $f(z) = z/{(1 - xz)^{2 - 2\alpha }},\;|x| = 1$. A similar result is obtained for spirallike mappings. Both results generalize a theorem of G. M. Golusin corresponding to the family of starlike mappings.References
- 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
- 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 338337, DOI 10.1090/S0002-9947-1973-0338337-5
- 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
- G. M. Goluzin, On a variational method in the theory of analytic functions, Amer. Math. Soc. Transl. (2) 18 (1961), 1–14. MR 0124491, DOI 10.1090/trans2/018/01 D. J. Hallenbeck, Convex hulls and extreme points of some families of univalent functions, Ph.D. Dissertation, State University of New York, Albany, N. Y., 1972.
- D. J. Hallenbeck and T. H. MacGregor, Subordination and extreme-point theory, Pacific J. Math. 50 (1974), 455–468. MR 361035, DOI 10.2140/pjm.1974.50.455
- D. J. Hallenbeck, Convex hulls and extreme points of some families of univalent functions, Trans. Amer. Math. Soc. 192 (1974), 285–292. MR 338338, DOI 10.1090/S0002-9947-1974-0338338-8
- I. S. Jack, Functions starlike and convex of order $\alpha$, J. London Math. Soc. (2) 3 (1971), 469–474. MR 281897, DOI 10.1112/jlms/s2-3.3.469
- W. E. Kirwan, A note on extremal problems for certain classes of analytic functions, Proc. Amer. Math. Soc. 17 (1966), 1028–1030. MR 202995, DOI 10.1090/S0002-9939-1966-0202995-8
- Zeev Nehari, Conformal mapping, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1952. MR 0045823
- Malcolm I. S. Robertson, On the theory of univalent functions, Ann. of Math. (2) 37 (1936), no. 2, 374–408. MR 1503286, DOI 10.2307/1968451
- M. S. Robertson, An extremal problem for functions with positive real part, Michigan Math. J. 11 (1964), 327–335. MR 170002, DOI 10.1307/mmj/1028999185
- Raphael M. Robinson, Univalent majorants, Trans. Amer. Math. Soc. 61 (1947), 1–35. MR 19114, DOI 10.1090/S0002-9947-1947-0019114-6 L. Špaček, Contribution à la théorie des fonctions univalentes, Časopis Pěst. Mat. 62 (1932), 12-19.
- Erich Strohhäcker, Beiträge zur Theorie der schlichten Funktionen, Math. Z. 37 (1933), no. 1, 356–380 (German). MR 1545400, DOI 10.1007/BF01474580
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 183 (1973), 499-510
- MSC: Primary 30A32
- DOI: https://doi.org/10.1090/S0002-9947-1973-0338339-9
- MathSciNet review: 0338339