共查询到20条相似文献,搜索用时 15 毫秒
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For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) − 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f c (z) = z(f (z)/z) c for a complex number c. We determine those values c for which the operator maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions. 相似文献
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Summary A family, E, consisting of normalised univalent functions with univalent derivatives is studied with regard to the zeros of
these functions.
Entrata in Redazione il 29 marzo 1978. 相似文献
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Ch. Pommerenke 《Mathematische Annalen》1978,236(3):199-208
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Zhongqiu Ye 《Proceedings of the American Mathematical Society》2005,133(11):3355-3360
The relative growth of successive coefficients of odd univalent functions is investigated. We prove that a conjecture of Hayman is true.
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V. V. Kozhevnikov 《Mathematical Notes》2007,81(1-2):213-221
In the paper, the construction of a variational method for univalent functions is suggested; this construction uses the factorization theorem. As a consequence, an analog of the Goluzin variational formula is obtained. 相似文献
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Let be the class of functions which are analytic in the unit disk . Let C(r) be the closed curve that is the image of the circle |z|=r < 1 under the mapping w = f(z), L(r) the length of C(r), and let A(r) be the area enclosed by the curve C(r). In 1968 D. K. Thomas shown that if , f is starlike with respect to the origin, and for 0≤r < 1, A(r) < A, an absolute constant, then Later, in 1969 Nunokawa has shown that if f is convex univalent, then This paper is devoted to obtaining a related correspondence between f(z) and L(r) for the case when f is univalent. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
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Henryka Siejka 《Israel Journal of Mathematics》1986,54(3):291-300
The class Σb is defined to consist of meromorphic univalent functionsH omitting a disc with the radiusb:H(z)=z+ Σ
0
∞
A
n
z
−n
,z>1,H(b)>b ∈ (0, 1). By aid of FitzGerald inequalities the inverse coefficients of odd Σb-functions are maximized. The result extends the corresponding estimation, due to Netanyahu and Schober, fromb=0 to the whole interval (0, 1).
The author wishes to express her gratitude to Professor O. Tammi for valuable discussions connected with the problem.
This work was supported by a grant from the Finnish Ministry of Education. 相似文献
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B. N. Rakhmanov 《Mathematical Notes》1978,23(4):324-328
We are considering a class S of functions F(z), F(0) = 0, F′(0) = 1 that are univalent and regular in the circle ¦z¦ < 1, and its subclasses s h * and K of starlike functions of order h and of convex functions respectively. Among others, we establish the following results: If F(z)εs and 0 < α < 1, then IfF (z) ε s (0 < a < 1) and $$\begin{gathered} 1 + \operatorname{Re} {{z_1 F^n \left( {z_1 } \right)} \mathord{\left/ {\vphantom {{z_1 F^n \left( {z_1 } \right)} {F'\left( {z_1 } \right)}}} \right. \kern-\nulldelimiterspace} {F'\left( {z_1 } \right)}} = \operatorname{Re} {{\alpha z_1 F''\left( {\alpha z_1 } \right)} \mathord{\left/ {\vphantom {{\alpha z_1 F''\left( {\alpha z_1 } \right)} {F'\left( {\alpha z_1 } \right)}}} \right. \kern-\nulldelimiterspace} {F'\left( {\alpha z_1 } \right)}} \hfill \\ \left( {2 - \sqrt 3< \left| {z_1 } \right| = r< 1} \right) \hfill \\ \end{gathered} $$ then we obtain the domain of values of the point αz1. 相似文献
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Gae sun Chung 《Journal of Applied Mathematics and Computing》1995,2(2):83-95
Given a closed subset of the familyS* (α) of functions starlike of order α, a continuous Fréchet differentiable functional,J, is constructed with this collection as the solution set to the extremal problem ReJ(f) overS* (α). The support points ofS* (α) is completely characterized and shown to coincide with the extreme points of its convex hulls. Given any finite collection of support points ofS* (α), a continuous linear functional,J, is constructed with this collection as the solution set to the extremal problem ReJ(f) overS* (α). 相似文献