Aharonov invariants revisited
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Aharonov invariants revisited Toshiyuki Sugawa1 Received: 15 April 2019 / Revised: 6 December 2019 / Accepted: 28 May 2020 © Springer Nature Switzerland AG 2020
Abstract In this note, we reformulate Aharonov’s univalence criterion in terms of projective Schwarzian derivatives of higher order. We also mention some relations between quasiconformal extendibility and the Aharonov criterion. Keywords Schwarzian derivative · Univalence criterion · Quasiconformal extension Mathematics Subject Classification Primary 30F45; Secondary 30C55
1 Introduction The pre-Schwarzian derivative Tf =
f f
and the Schwarzian derivative 1 S f = T f − (T f )2 2 play an important role in Geometric Function Theory. See, for instance Nehari’s textbook [11] on conformal mappings and Lehto’s book [9] on Teichmüller spaces. One of the most remarkable properties of the (pre-)Schwarzian derivative is a close connection with univalence criteria of holomorphic functions. For instance, the following is due to Nehari [10] (the former part was proved by Kraus [6] but it was overlooked for a long time. See also [9]).
Dedicated to Professor Dov Aharonov on the occasion of his 80th birthday.
B 1
Toshiyuki Sugawa [email protected] Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan 0123456789().: V,-vol
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T. Sugawa
Theorem A Let f be a non-constant meromorphic function on the unit disk D = {z ∈ C : |z| < 1}. If f is univalent on D, then (1 − |z|2 )2 |S f (z)| ≤ 6 (|z| < 1).
(1.1)
Conversely, if (1 − |z|2 )2 |S f (z)| ≤ 2 (|z| < 1), then f is univalent on the unit disk. Both constants 2 and 6 are known to be sharp. Aharonov [1] introduced a sequence of quantities which can be regarded as an extension of the Schwarzian derivative. The following definition, which is slightly modified from the original one in [1], is due to Harmelin [2]: For a non-constant analytic function f , we consider the Laurent expansion ∞
1 f (z) = − ψn [ f ](z)w n−1 f (z + w) − f (z) w
G=
n=1
about the point z for small enough w. The quantity ψn [ f ](z) is called the Aharonov invariant of order n + 1 (as a differential operator). We observe that ψ1 [ f ] =
Tf Sf (S f ) , ψ2 [ f ] = , ψ3 [ f ] = . 2! 3! 4!
It is often more convenient to consider the quantities n [ f ] = (n + 1)!ψn [ f ] (n = 1, 2, . . .) so that 1 [ f ] = T f , 2 [ f ] = S f . The Aharonov invariants are invariant under the post-composition of Möbius transformations. That is, for a Möbius transformation g, n [g ◦ f ] = n [ f ], n ≥ 2.
(1.2)
Aharonov [1] found the following necessary and sufficient condition for univalence. Theorem B Let f be a non-constant meromorphic function on the unit disk D. Then f is univalent if and only if n 2 ∞ n − 1 n−k 2 k+1 (−¯z ) (1 − |z| ) ψk+1 [ f ](z) ≤ 1 n B( f , z) := k−1 n=1
k=1
Aharonov invariants revisited
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for all z ∈ D. Moreover, for a univalent f , B( f , z) ≡ 1 if and only if f (D) is of full measure; namely, C \ f (D) is of area zero
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