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Transactions of the American Mathematical Society

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A general extremal problem for the class of close-to-convex functions


Author: John G. Milcetich
Journal: Trans. Amer. Math. Soc. 225 (1977), 307-323
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9947-1977-0507714-5
MathSciNet review: 0507714
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Abstract: For $ \beta \geqslant 0,{K_\beta }$ denotes the set of functions $ f(z) = z + {a_2}{z^2} + \cdots $ defined on the unit disc U with the representation $ f'(z) = a{p^\beta }(z)s(z)/z$, where $ a \in C$, p is an analytic function with positive real part in U, and s is a normalized starlike function. If $ 0 \leqslant \beta \leqslant 1$, and $ \zeta \in U$, let $ F(u,v)$ be analytic in a neighborhood of $ \{ (f(\zeta ),\zeta ):f \in {K_\beta }\} $. Then $ \max \{ \operatorname{Re} F(f(\zeta ),\zeta ):f \in {K_\beta }\} $ occurs for a function of the form

$\displaystyle f(z) = {(\beta + 1)^{ - 1}}{(x - y)^{ - 1}}[{(1 + xz)^{\beta + 1}}{(1 + yz)^{ - \beta - 1}} - 1],$

where $ \vert x\vert = \vert y\vert = 1$ and $ x \ne y$. If $ 0 < \beta < 1$ these are the only extremal functions. A consequence of this result is the determination of the value region $ \{ f(\zeta )/\zeta :f \in {K_\beta }\} $ as $ \{ {(\beta + 1)^{ - 1}}{(s - t)^{ - 1}}[{(1 + s)^{\beta + 1}}{(1 + t)^{ - \beta - 1}} - 1]:\vert s\vert,\vert t\vert \leqslant \vert\zeta \vert\} $.

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  • [1] R. W. Barnard, Extremal problems for univalent functions whose ranges contain a fixed disc, Ph. D. Dissertation, Univ. of Maryland, 1971.
  • [2] M. Biernacki, Sur la représentation conforme des domaines linéairement accessibles, Prace Mat.-Fiz. 44 (1936), 293-314.
  • [3] 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. AI No. 523 (1973), 18 pp. MR 0338343 (49:3108)
  • [4] 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)
  • [5] G. Julia, Sur une equation aux dérivées fonctionelles liée à la représentation conforme, Ann. Sci. École Norm. Sup. 39 (1922), 1-28. MR 1509240
  • [6] W. K. Kaplan, Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), 169-185 (1953). MR 14, 966. MR 0054711 (14:966e)
  • [7] W. E. Kirwan, A note on extremal problems for certain classes of analytic functions, Proc. Amer. Math. Soc. 17 (1966), 1028-1030. MR 34 #2854. MR 0202995 (34:2854)
  • [8] -, Extremal problems for functions with bounded boundary rotation, Ann. Acad. Sci. Fenn. No. 595 (1975), 19 pp. MR 0393461 (52:14271)
  • [9] Z. Lewandowski, Sur l'identité de certaines classes de fonctions univalentes. II, Ann. Univ. Mariae Curie-Skłodowska Sect. A 14 (1960), 19-46. MR 28 #200. MR 0156958 (28:200)
  • [10] V. Paatero, Über die konforme Abbildung von Gebieten, deren Ränder von beschränkter Drehung sind, Ann. Acad. Sci. Fenn. Ser. A 33 (1931), 1-79.
  • [11] Ch. Pommerenke, On close-to-convex analytic functions, Trans. Amer. Math. Soc. 114 (1965), 176-186. MR 30 #4920. MR 0174720 (30:4920)
  • [12] M. S. Robertson, Variational methods for functions with positive real part, Trans. Amer. Math. Soc. 102 (1962), 82-93. MR 24 #A3288. MR 0133454 (24:A3288)
  • [13] M. M. Schiffer and O. Tammi, A method of variations for functions with bounded boundary rotation, J. Analyse Math. 17 (1966), 109-144. MR 35 #5601. MR 0214752 (35:5601)
  • [14] E. Strohhäcker, Beiträge zur Theorie der schlichten Funktionen, Math. Z. 37 (1933), 356-380. MR 1545400

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0507714-5
Keywords: Close-to-convex functions, extremal problems, variational methods
Article copyright: © Copyright 1977 American Mathematical Society

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