Abstract
The object of the present paper is to obtain a more general condition for univalence of meromorphic functions in the U*. The significant relationships and relevance with other results are also given. A number of known univalent conditions would follow upon specializing the parameters involved in our main results.
Results & Lemmas (7)
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Theorem 2.1.
Theorem 2.1. Let L(z, t) = a1(t)z + a2(t)z2 +... be analytic in Ur for all t ∈[0, ∞). Suppose that; (i) L(z, t) is a locally absolutely…
Theorem 2.1. Let L(z, t) = a1(t)z + a2(t)z2 + ... be analytic in Ur for all t ∈[0, ∞). Suppose that; (i) L(z, t) is a locally absolutely continuous function in the interval [0, ∞), and locally uni- formly with respect to Ur. (ii) a1(t) is a complex valued continuous function on [0, ∞) such that a1(t) ̸= 0, |a1(t)| →∞ for t →∞and L(z, t) a1(t) t∈[0,∞) forms a normal family of functions in Ur. (iii) There exists an analytic function p : U × [0, ∞) →C satisfying Re p(z, t) > 0 for all z ∈U, t ∈[
Theorem 3.1.
Theorem 3.1. Let f, g ∈P be locally univalent functions in U∗. If there exists an analytic function h such that Re h(ζ) ≥1 2 and h(ζ) = 1 +…
Theorem 3.1. Let f, g ∈P be locally univalent functions in U∗. If there exists an analytic function h such that Re h(ζ) ≥1 2 and h(ζ) = 1 + h2 ζ2 + ... for ζ ∈U∗, and for arbitrary α ∈C we have
Corollary 3.2.
Corollary 3.2. Let f ∈P be locally univalent function in U∗. If there exists an analytic function h with Re h(ζ) ≥1 2 in U∗and h(ζ) = 1 +…
Corollary 3.2. Let f ∈P be locally univalent function in U∗. If there exists an analytic function h with Re h(ζ) ≥1 2 in U∗and h(ζ) = 1 + h2 ζ2 + ... such that (3.13)
Corollary 3.3.
Corollary 3.3. Let f, g ∈P be locally univalent functions in U∗. If there exists an analytic function h with Re h(ζ) ≥1 2 in U∗and h(ζ) = 1…
Corollary 3.3. Let f, g ∈P be locally univalent functions in U∗. If there exists an analytic function h with Re h(ζ) ≥1 2 in U∗and h(ζ) = 1 + h2 ζ2 + ... such that
Corollary 3.4.
Corollary 3.4. Let f, g ∈P be locally univalent functions in U∗. If the following inequality (3.15)
Corollary 3.4. Let f, g ∈P be locally univalent functions in U∗. If the following inequality (3.15)
Corollary 3.5.
Corollary 3.5. Let f ∈P be locally univalent function in U∗. If the following inequality (3.16) (|ζ|2 −1)
Corollary 3.5. Let f ∈P be locally univalent function in U∗. If the following inequality (3.16) (|ζ|2 −1)
Corollary 3.6.
Corollary 3.6. Let f ∈P be locally univalent function in U∗. If the following inequality (3.17) |Sf(ζ)| ⩽ 2 (|ζ|2 −1)2 is satisfied for all…
Corollary 3.6. Let f ∈P be locally univalent function in U∗. If the following inequality (3.17) |Sf(ζ)| ⩽ 2 (|ζ|2 −1)2 is satisfied for all ζ ∈U∗, then f is univalent in U∗. Acknowledgement. The present investigation was supported by Atat¨urk University Rec- torship under BAP Project (The Scientific and Research Project of Atat¨urk University) Project No: 2010/28. References [1] J. Becker, Lownersche differentialgleichung und schlichtheitskriterien, Math. Ann. 202 (1973) 321-335. [2] E. Deniz, H. O
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