On functions of bounded boundary rotation
Ming-chit Liu
- Vol. 29, Iss: 2, pp 345-348
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TLDR
In this paper, the generalized area of the image of the set I z? r under some mapping f(z), L(r) the length of C(r), by A (r) we denote the integral rw2g rr fJ7F fJ 7F f f'(t) 12pdpdO (t pei0) which was defined by Nunokawa for the case k = 2.Abstract:
Let U = {z-reiO I r 2 the recent result of Nunokawa for the case k = 2. Let U be the unit disk, I z I 0. It is clear that V2 = K. Let C(r) denote the image of the circle z = r <1 under some mapping f(z), L(r) the length of C(r). By A (r) we denote the integral rw2g rr fJ7F f f'(t) 12pdpdO (t pei0) which is the generalized area of the image of the set I z ? r under the mappingf(z). Recently, Nunokawa [2, p. 332 ] obtained: Received by the editors September 15, 1970. AMS 1970 subject classifications. Primary 30A32; Secondary 30A04.read more
Citations
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Radius of Convexity and Radius of Starlikeness for Some Classes of Analytic Functions
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On Bazilevic functions
TL;DR: In this article, the authors generalized the class of Bazilevic functions by taking g to be a function of radius rotation at most kπ(k≥2), and solved the Archlength, difference of coefficient, Hankel determinant and some other problems for this generalized class.
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On Bounded Boundary and Bounded Radius Rotations
TL;DR: In this article, the relation between the functions of bounded boundary and bounded radius rotations was established by using three different techniques, and a well-known result was observed as a special case from the main result.
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Coefficients of univalent functions
TL;DR: A survey of univalent functions can be found in this paper, where a function is said to be univalent if it never takes the same value twice: f (z{) # f(z2) if zx #= z2.
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On a generalization of close-to-convexity
TL;DR: In this article, a class Tk of analytic functions in the unit disc is defined in which the concept of close-to-convexity is generalized, and a necessary condition for a function f to belong to Tk, raduis of convexity problem and a coefficient result are solved.
References
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Univalent functions $f(z)$ for which $zf^{\prime} (z)$ is spirallike.
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Coefficients of functions with bounded boundary rotation
TL;DR: For fixed k ≧ 2, Paatero and Renyi as discussed by the authors showed that the Bieberbach conjecture holds for functions convex in one direction in the Euclidean plane.