Applications of the subordination principle to univalent functions

Robertson, M. S.

doi:10.2140/pjm.1961.11.315

Cited by 31 publications

(16 citation statements)

References 4 publications

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“…In the next theorem we will use the following result of Robertson [6] to determine a sufficient condition for a univalent function to be in K(a). Lemma 3.…”

Section: Proofmentioning

confidence: 99%

Subordination by Univalent Functions

Singh¹,

Singh²

1981

Proceedings of the American Mathematical Society

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Abstract.Let A' be the class of functions f(z) ~ z + a2z2 + • • • , which are regular and univalently convex in \z\ < 1. In this paper we establish certain subordination relations between an arbitrary member / of K, its partial sums and the functions (A/z)/£/(/)<* and y. f'Q t~lf(t)dt. The well-known result that z/2 is subordinate to f(z) in |z| < 1 for every/belonging to K follows as a particular case from our results. We also improve certain results of Robinson regarding subordination by univalent functions. A sufficient condition for a univalent function to be convex of order o is also given.Introduction. Let A denote the class of functions /(z) = z + a2z2 + • • • which are regular in |z| < 1. We denote by 5 the subclass of A consisting of functions/ which are univalent in |z| < 1; S* and K will stand for the usual subclasses of S whose members are, respectively, starlike (w.r.t. the origin) and convex in |z| < 1. A function/belonging to A is said to be convex of order a, 0 < a < 1, in |z| < 1 if

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“…In the next theorem we will use the following result of Robertson [6] to determine a sufficient condition for a univalent function to be in K(a). Lemma 3.…”

Section: Proofmentioning

confidence: 99%

Subordination by Univalent Functions

Singh¹,

Singh²

1981

Proceedings of the American Mathematical Society

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“…We note from (2.14) that log{zl-ef'(z)/f(z) I-} logh(z) + a log{g(z)/z} (2.15) Since, for a fixed z E U, the range of logh(z), as h(z) ranges over the class P (8) LEMMA 4 (Robertson [14]). Let f(z) e S. For each 0 < t < i let F(z,t) be regular in the unit disk U, let F(z,0)__ f(z) and F(0,t) 0.…”

mentioning

confidence: 99%

Certain subclasses of Bazilevič functions of type α

Owa

Obradović²

1986

International Journal of Mathematics and Mathematical Sciences

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“…In one approach to extending these results to C n , n ≥ 2, Suffridge [11] generalizes some of Robertson's results [9], which use the principle of subordination in the plane, to higher dimensions.…”

Section: Example 1 Let B Be the Euclidean Ball In Cmentioning

confidence: 87%

Convexity properties of holomorphic mappings in ℂⁿ

Roper

Suffridge

1999

Trans. Amer. Math. Soc.

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Abstract. Not many convex mappings on the unit ball in C n for n > 1 are known. We introduce two families of mappings, which we believe are actually identical, that both contain the convex mappings. These families which we have named the "Quasi-Convex Mappings, Types A and B" seem to be natural generalizations of the convex mappings in the plane. It is much easier to check whether a function is in one of these classes than to check for convexity. We show that the upper and lower bounds on the growth rate of such mappings is the same as for the convex mappings.

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Applications of the subordination principle to univalent functions

Cited by 31 publications

References 4 publications

Subordination by Univalent Functions

Subordination by Univalent Functions

Certain subclasses of Bazilevič functions of type α

Convexity properties of holomorphic mappings in ℂⁿ

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