Abstract
The main result shows a small perturbation of a univalent function is again a univalent function, hence a univalent function has a neighborhood consisting entirely of univalent functions.
For the particular choice of a linear function in the hypothesis of the main theorem, we obtain a corollary which is equivalent to the classical Noshiro-Warschawski-Wolff univalence criterion.
We also present an application of the main result in terms of Taylor series, and we show that the hypothesis of our
Results & Lemmas (7)
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Theorem 1.1.
Theorem 1.1. If f: D →C is analytic in the convex domain D and Re f ′ (z) > 0, z ∈D, then f is univalent in D. In the present paper we…
Theorem 1.1. If f : D →C is analytic in the convex domain D and Re f ′ (z) > 0, z ∈D, then f is univalent in D. In the present paper we introduce the constant K (f, D) associated with a func- tion f : D →C analytic in a domain D, which is a measure of the ”degree of univalence” of f (see Proposition 2.1 and the remark following it). Using the constant K (f, D) thus introduced, in Theorem 2.4 we obtain a suffi- cient condition for univalence, which shows that a small perturbation of a univalent fun
Proposition 2.1.
Proposition 2.1. Let f: D →C be an analytic function in the domain D. If K (f, D) > 0 then f is univalent in D. Conversely, if f is…
Proposition 2.1. Let f : D →C be an analytic function in the domain D. If K (f, D) > 0 then f is univalent in D. Conversely, if f is univalent in D and Ω⊂Ω⊂D is a domain strictly contained in D, then K (f, Ω) > 0.
Theorem 2.4.
Theorem 2.4. Let f: D →C be a non-constant analytic function in the convex domain D. If there exists an analytic function g: D →C univalent…
Theorem 2.4. Let f : D →C be a non-constant analytic function in the convex domain D. If there exists an analytic function g : D →C univalent in D such that (2.2) |f ′ (z) −g′ (z)| ≤K (g, D) , z ∈D, then the function f is also univalent in D.
Theorem 2.5.
Theorem 2.5. Let f: U →C be a non-constant analytic function in the unit disk. If there exists an analytic function g: U →C univalent in U…
Theorem 2.5. Let f : U →C be a non-constant analytic function in the unit disk. If there exists an analytic function g : U →C univalent in U such that (2.8) |f ′ (z) −g′ (z)| ≤K (g, U) , z ∈U, then the function f is also univalent in U. As a corollary of Theorem 2.4 we obtain immediately the following:
Corollary 2.6.
Corollary 2.6. If f: D →C is non-constant and analytic in the convex domain D and there exists c > 0 such that (2.9) |f ′ (z) −c| ≤c, z ∈D,
Corollary 2.6. If f : D →C is non-constant and analytic in the convex domain D and there exists c > 0 such that (2.9) |f ′ (z) −c| ≤c, z ∈D,
Theorem 2.11.
Theorem 2.11. Let g: U →C be an analytic univalent function with Taylor series representation (2.11) g (z) = ∞ X n=0 bnzn, z ∈U. If the…
Theorem 2.11. Let g : U →C be an analytic univalent function with Taylor series representation (2.11) g (z) = ∞ X n=0 bnzn, z ∈U. If the coefficients a0, a1, . . . ∈C satisfy the inequality (2.12) ∞ X n=1 n |an −bn| < K (g, U)
Corollary 2.12.
Corollary 2.12. Let g: U →C be an analytic univalent function with Taylor series representation (2.14) g (z) = ∞ X n=0 bnzn, z ∈U. If the…
Corollary 2.12. Let g : U →C be an analytic univalent function with Taylor series representation (2.14) g (z) = ∞ X n=0 bnzn, z ∈U. If the coefficients a0, a1, . . . ∈C satisfy the inequality (2.15) |an −bn| < K (g, U) ζ (p) np+1 , n = 1, 2, . . . , for some p > 1 (ζ denotes the Riemann zeta function), then the function f : U →C
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