Locally Univalent Functions and the Bloch and Dirichlet Norm
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Locally Univalent Functions and the Bloch and Dirichlet Norm Jochen Becker1 · Christian Pommerenke1
Received: 7 November 2014 / Accepted: 18 February 2015 © Springer-Verlag Berlin Heidelberg 2015
Abstract We consider analytic functions f satisfying f (z) = 0 in the unit disk D and norms of the Bloch and Dirichlet type, namely 1 2 f (z) 2 f (z) dxdy (∗) sup (1 − |z| ) , (∗∗) . f (z) D f (z) 2
In particular, we study the range of values of f for the norm (*) and the univalence of f for the norm (**). Keywords
Bloch function · Dirichlet norm · Value range · Univalence
Mathematics Subject Classification
30D45 · 30D55
1 Introduction In some aspects, the present paper is a continuation of the paper [4]. Section 3 revolves around the well-known expression (1 − |z|2 )| f (z)/ f (z)|.
Communicated by Stephan Ruscheweyh.
B
Christian Pommerenke [email protected] Jochen Becker [email protected]
1
Institute of Mathematics, Technical University Berlin, Berlin, Germany
123
J. Becker, C. Pommerenke
What can we derive from the knowledge of bounds for this expression? Section 4 deals with the area integral in (1.4). What bounds on this integral imply that f is univalent? For a function g analytic in the unit disk D, the Bloch norm is defined by gB := sup(1 − |z|2 )|g (z)|
(1.1)
z∈D
and the Dirichlet norm by 1 gD := π
2
D
|g (z)| dxdy = 2
∞
n|bn |2
(1.2)
n=1
n where we have written g(z) = ∞ n=0 bn z , see e.g. [5, p. 447], [6]. Our usage of the word “norm” is not quite correct, it should be “seminorm” because, for instance in (1.1), the additive term |g(0)| is missing. The analytic function f is called locally univalent in the unit disk D if f (z) = 0 for z ∈ D. We will mainly consider the case that g = log f for locally univalent functions f . Then, (1.1) and (1.2) become f (z) (1.3) log f B = sup (1 − |z|2 ) , f (z) z∈D
log f D =
1 π
1 2 2 f (z) . f (z) dxdy D
(1.4)
2 Some Basic Relations The facts in this section are well known with exception perhaps of Lemma 2.3. Lemma 2.1 Let f and ϕ be locally univalent in D and let ϕ(D) ⊂ D. Then, f (ϕ(z)) + log ϕ B log( f ◦ ϕ) B ≤ sup (1 − |ϕ(z)| ) f (ϕ(z)) z∈D
2
≤ log f B + log ϕ B .
(2.1)
Proof Since (1 − |z|2 )|ϕ (z)| ≤ 1 − |ϕ(z)|2 , we have (1 − |z|2 )|
f (ϕ d + (1 − |z|2 ) ϕ log( f ◦ ϕ) | ≤ (1 − |z|2 ) ϕ ϕ dz f (ϕ) f (ϕ) + (1 − |z|2 ) ϕ , ≤ (1 − |ϕ|2 ) ϕ f (ϕ)
which implies (2.1) because of (1.3).
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Locally Univalent Functions…
Lemma 2.2 If f is locally univalent in D then log f B ≤ log f D .
(2.2)
Proof Again we write g = log f and obtain from (1.2) and the Schwarz inequality that 2 2 ∞ ∞ ∞ f (z) n−1 2 = nb z ≤ n|b | n|z|2(n−1) n n f (z) n=1
n=1
n=1
= log f 2D (1 − |z|2 )−2 .
(2.3)
Hence, (2.2) follows fro
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