On certain convex sets in the space of locally schlicht functions
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- by Y. J. Kim and E. P. Merkes
- Trans. Amer. Math. Soc. 196 (1974), 217-224
- DOI: https://doi.org/10.1090/S0002-9947-1974-0349981-4
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Abstract:
Let $H = H{(^ \ast },[ + ])$ denote the real linear space of locally schlicht normalized functions in $|z| < 1$ as defined by Hornich. Let K and C respectively be the classes of convex functions and the close-to-convex functions. If $M \subset H$ there is a closed nonempty convex set in the $\alpha \beta$-plane such that for $(\alpha ,\beta )$ in this set ${\alpha ^ \ast }f[ + ]{\beta ^ \ast }g \in C$ (in K) whenever f, $g \in M$. This planar convex set is explicitly given when M is the class K, the class C, and for other classes. Some consequences of these results are that K and C are convex sets in H and that the extreme points of C are not in K.References
- W. M. Causey, The close-to-convexity and univalence of an integral, Math. Z. 99 (1967), 207–212. MR 215972, DOI 10.1007/BF01112451
- J. A. Cima and J. A. Pfaltzgraff, A Banach space of locally univalent functions, Michigan Math. J. 17 (1970), 321–334. MR 278052
- Hans Hornich, Ein Banachraum analytischer Funktionen in Zusammenhang mit den schlichten Funktionen, Monatsh. Math. 73 (1969), 36–45 (German). MR 243087, DOI 10.1007/BF01297700
- Wilfred Kaplan, Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), 169–185 (1953). MR 54711
- E. P. Merkes and D. J. Wright, On the univalence of a certain integral, Proc. Amer. Math. Soc. 27 (1971), 97–100. MR 269825, DOI 10.1090/S0002-9939-1971-0269825-1
- Mamoru Nunokawa, On the univalence of a certain integral, Trans. Amer. Math. Soc. 146 (1969), 439–446. MR 251205, DOI 10.1090/S0002-9947-1969-0251205-1
- W. C. Royster, On the univalence of a certain integral, Michigan Math. J. 12 (1965), 385–387. MR 183866, DOI 10.1307/mmj/1028999421
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 196 (1974), 217-224
- MSC: Primary 30A32; Secondary 30A98
- DOI: https://doi.org/10.1090/S0002-9947-1974-0349981-4
- MathSciNet review: 0349981