Univalent functions $f(z)$ for which $zf^{\prime} (z)$ is spirallike.

Robertson, M. S.

doi:10.1307/mmj/1029000208

Cited by 56 publications

(33 citation statements)

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“…Putting A = 0 in Theorem 1, we arrive at the following result which is an improvement of earlier results due to Robertson [4] and Libera and Ziegler [2]. …”

Section: Introduction a Function /(Z) = Z + A Z + • • • Regular In Tsupporting

confidence: 71%

“…Recently Robertson [4] introduced the class of functions /(z) regular in E and satisfying the condition that zf (z) is a-spiral-like in E. We denote this class of functions by J .It is well known that J" is the class of con-J a 0 vex functions in E. Robertson proved that /(z) £ j is univalent in E if 0 < cos a < 0.2315-.--Later Libera and Ziegler [2] gave an improvement on the range of a for which /(z) is univalent in E. They showed that /(z) £ J is univalent in E if 0 < cos a < 0.2564.... In this paper a slight improvement of this result is given.…”

Section: Introduction a Function /(Z) = Z + A Z + • • • Regular In Tmentioning

confidence: 99%

See 1 more Smart Citation

Regular functions 𝑓(𝑧) for which 𝑧𝑓^{’}(𝑧) is 𝛼-spiral-like

Chichra¹

1975

Proc. Amer. Math. Soc.

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Let F α λ \mathfrak {F}_\alpha ^\lambda be the class of functions f ( z ) = z + a 2 z 2 + ⋯ f(z) = z + {a_2}{z^2} + \cdots which are regular in E = { z / | z | > 1 } E = \{ z/|z| > 1\} and satisfy \[ Re ⁡ { e i α ( 1 + z f ( z ) / f ′ ( z ) ) } > λ cos ⁡ α \operatorname {Re} \{ {e^{i\alpha }}(1 + zf(z)/f’(z))\} > \lambda \cos \alpha \] for some α , | α | > π / 2 \alpha ,|\alpha | > \pi /2 , and for some λ , 0 ≤ λ > 1 \lambda ,0 \leq \lambda > 1 . The author finds a range on α \alpha for which f ( z ) f(z) in F α λ \mathfrak {F}_\alpha ^\lambda is univalent in E E . In particular, the author improves upon the range on a for which f ( z ) ∈ F α 0 f(z) \in \mathfrak {F}_\alpha ^0 is known to be univalent in E E . Also a corresponding result is obtained for those functions f ( z ) f(z) in F α λ \mathfrak {F}_\alpha ^\lambda for which f ( 0 ) = 0 f(0) = 0 .

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“…Putting A = 0 in Theorem 1, we arrive at the following result which is an improvement of earlier results due to Robertson [4] and Libera and Ziegler [2]. …”

Section: Introduction a Function /(Z) = Z + A Z + • • • Regular In Tsupporting

confidence: 71%

Section: Introduction a Function /(Z) = Z + A Z + • • • Regular In Tmentioning

confidence: 99%

Regular functions 𝑓(𝑧) for which 𝑧𝑓^{’}(𝑧) is 𝛼-spiral-like

Chichra¹

1975

Proc. Amer. Math. Soc.

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“…This result was obtained independently and using different methods by both Robertson [9] and Libera [6]. Minimizing either (2.6) or (2.10) as a function of a we obtain the following which appears also in [9].…”

mentioning

confidence: 62%

Regular functions 𝑓(𝑧) for which 𝑧𝑓’(𝑧) is 𝛼-spiral

Libera¹,

Ziegler²

1972

Trans. Amer. Math. Soc.

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Abstract. A function /(z) = z+2°=2 onzn regular in the open unit disk A = {z : |z| 0 for z in A; in this case we write f(z)e&a.A fundamental result of this paper shows that the transformation f (z\ = azf(iz + a)lil+âz))efines a function in &a whenever/(z) is in &a and a is in A. If g(z) is regular in A, g(0) = 0 and g'(0) = 1, then g(z) is in 0O if and only if zg'(z) is in &". The main result of the paper is the derivation of the sharp radius of close-toconvexity for each class ^"; it is given as the solution of an equation in r which is dependent only on a. (Approximate solutions of this equation were made by computer and these suggest that the radius of close-to-convexity of the class <& = \Ja &a is approximately .99097+ .) Additional results are also obtained such as the radius of convexity of 0O, a range of a for which g(z) in ^" is always univalent is given, etc. These conclusions all depend heavily on the transformation cited above and its analogue for @a.

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“…[3,4]). Therefore, S p (α, β) defined by (1.2) is the reciprocal class of p-valently spirallike functions in U, and C p (α, β) defined by (1.3) is the reciprocal class of p-valently Robertson functions in U.…”

Section: Introductionmentioning

confidence: 99%