On Starlike and Convex Schlicht Functions

Abstract.For an analytic function f(z) = z + 2Zk0-2akzk m ^ unit disc E conditions are established such that all functions g(z) = z + 2£-2bkzk e ^«(Z)» i.e. X'k^2^\ak ~ * are m some class of univalent functions in E. For instance, we prove that every g e Nl/4(f) is starlike univalent in E if / is convex univalent.Let A denote the class of analytic functions/in the unit disc E = {z\ \z\ < 1} with /(0) = 0, /'(0) = 1. For /(z) = z -f-2k°.2akzk G A and S > 0 we define the neighborhood Ns(f) as follows.

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“…The following simple proof is due to J. Clunie and F. R. Keogh [1]. For g(z) = z + f bkzk G Nx(e) k = 2…”

mentioning

confidence: 99%

Neighborhoods of univalent functions

Ruscheweyh¹

1981

Proc. Amer. Math. Soc.

184

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“…However, we would like to point out that Lemma A gives generalizations of the some cases in Gromova and Vasil'ev [5], e.g. λ 2 = 0 in [5,Theorems 1,3,4]. We refer to Corollary 3.…”

Section: Preliminaries and The Main Resultsmentioning

confidence: 99%

On the problem of Gromova and Vasil'ev on integral means, and Yamashita's conjecture for spirallike functions

Ponnusamy¹,

Wirths²

2014

Ann. Acad. Sci. Fenn. Math.

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Abstract. In this article we consider some integral means problem for certain classes of univalent analytic functions, in particular for the class of the starlike functions of order β and for the class of α-spirallike functions of order β. Our investigation settles one of the open problems of Gromova and Vasil'ev. In addition, we solve another problem concerning area maximum property of α-spirallike functions of order β in the setting of Yamashita and hence, we find the solution to Yamashita's conjecture for certain Dirichlet-finite functions in a general form. Preliminaries and the main resultsIn this paper, we investigate some families of univalent functions for which the range of each of its member function has a specific geometric property. We will be particularly interested in determining estimates for certain integral means for functions with geometric property. In this connection one of the classical results of Rogosinski [13] for subordination is useful. Using this, we prove a general result and a particular case solves one of the open problems of Gromova and Vasil'ev [5] on the best estimate for a special integral means for starlike functions of order β. Also, we prove Yamashita's conjecture on area maximum property for α-spirallike functions of order β.Let D := {z ∈ C : |z| < 1} denote the open unit disk and H denote the class of all functions f analytic in D with the topology of uniform convergence of compact subsets of D and A = {f ∈ H : f (0) = 0 and f

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“…It is shown in [2] that if the area off(A) is finite, then also (na(n))elx. If, in addition, ¡i is absolutely continuous (relative to normalised Lebesgue measure m), then, as Pommerenke has shown in [5], lim na(n)=0.…”

Section: F(z) Jl -Zymentioning

confidence: 99%

“…Returning to (23) and noting that now ^=1, we infer that p, is concentrated on the set {£, lw, • ■ ■ , &"}, where con+x=l, cojíl. Since c(k)=0 ifl^k<n, p. has mass l/(«+l) at each point of this set, and a straightforward analysis of (2) shows that g reduces to (17). The proof is complete.…”

Section: Proof Of Theorem 2 Starting Withmentioning

confidence: 99%

On the coefficients of starlike functions

Holland¹

1972

Proc. Amer. Math. Soc.

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Abstract.Every probability measure p on the circle group generates a function / that is starlike univalent on the open unit disc A. In this note the relationship between (cn), the FourierStieltjes coefficients of ¡i, and (an), the Taylor coefficients of/, is examined. A number of theroems are presented which indicate (possibly in the presence of fairly mild restrictions) that the sequences (cn) and (nan) behave similarly. For example, it is shown that if/(A) is finite, then (c") converges to zero if, and only if, (nan) converges to zero, thereby completing a result of Pommerenke.

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On Starlike and Convex Schlicht Functions

Cited by 57 publications

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Neighborhoods of univalent functions

Neighborhoods of univalent functions

On the problem of Gromova and Vasil'ev on integral means, and Yamashita's conjecture for spirallike functions

On the coefficients of starlike functions

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