NEIGHBOURHOODS OF UNIVALENT FUNCTIONS

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. The authors kindly acknowledge the support from CNCSIS research grant PNII-IDEI 209. c⃝2010 Australian Mathematical Publishing Association Inc. 0004-9727/2010 $16.00 210 https://doi.org/10.1017/S0004972710000468 Published online by Cambridge University Press$

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  ⊂ ⊂D is a domain strictly contained in D, then K( f, ) > 0. PROOF. The first statement follows from the inequality | f (a) −f (b)| ≥|a −b|K( f, D) > 0, for any distinct points a, b ∈D. To prove the converse, note that K( f, ) ≥inf a,b∈ |F(a, b)|, https://doi.org/10.1017/S0004972710000468 Published online by Cambridge University Press

THEOREM 2.4. Let f : D →C be a nonconstant analytic function in the convex domain D. If there exists an analytic function g : D →C univalent in D such that | f ′(z) −g′(z)| ≤K(g, D), z ∈D, (2.2) then the function f is also univalent in D. PROOF. Assuming that f is not univalent in D, there exist distinct points z1,2 ∈D such that f (z1) = f (z2). Integrating the derivative of f −g along the line segment [z1, z2] ⊂D and using the hypothesis (2.2) we obtain |g(z2) −g(z1)| = |( f (z2) −g(z2)) −( f (

THEOREM 2.5. Let f : U →C be a nonconstant analytic function in the unit disk. If there exists an analytic function g : U →C univalent in U such that | f ′(z) −g′(z)| ≤K(g, U), z ∈U, (2.8) then the function f is also univalent in U. As a corollary of Theorem 2.4 we obtain immediately the following result.

COROLLARY 2.6. If f : D →C is nonconstant and analytic in the convex domain D and there exists c > 0 such that | f ′(z) −c| ≤c, z ∈D, (2.9) then f is univalent in D. PROOF. Considering the univalent function g : D →C defined by g(z) = cz, we have g′(z) = c for z ∈D and K(g, D) = inf a,b∈D a̸=b

Theorem 2.5 in situations when the Noshiro–Warschawski–Wolff univalence criterion cannot be applied. EXAMPLE 2.10. Consider the linear map g : U →C defined by g(z) = z/(1 −z). The function g is univalent in U and K(g, U) = inf a,b∈U a̸=b

THEOREM 2.11. Let g : U →C be an analytic univalent function with Taylor series representation g(z) = ∞ X n=0 bnzn, z ∈U. (2.11) If the coefficients a0, a1, . . . ∈C satisfy the inequality ∞ X n=1 n|an −bn| < K(g, U), (2.12)

COROLLARY 2.12. Let g : U →C be an analytic univalent function with Taylor series representation g(z) = ∞ X n=0 bnzn, z ∈U. (2.14) If the coefficients a0, a1, . . . ∈C satisfy the inequality |an −bn| < K(g, U) ζ(p) n p+1 , n = 1, 2, . . . , (2.15) for some p > 1 (ζ denotes the Riemann zeta function), then the function f : U →C