AN EXTENSION OF THE AREA THEOREM ´ MIODRAG MATELJEVIC
Research partially supported by MNTRS, Serbia, Grant No. 174 032 The area theorem is an important tool in theory of univalent functions. By D we denote the unit disk and E = {z : |z| > 1}. We say F ∈ Σ if it is univalent on E = {z : |z| > 1} and (2)
F (z) = z +
b1 bk + · · · + k + · · ·. z z
∗ Theorem P∞ 0.1 2(the area theorem). If F ∈ Σ and E = C \ F (E), then a) 1 k |bk | ≤ 1 P ∞ b) area(E∗ ) = π (1 − 1 k |bk |2 ).
Theorem 0.2 ([5]). Let w = f (z) = λz + az1 +· · ·+ aznn +· · · be an analytic function on E = {z : |z| > 1} and let G = C\f (E) be the omitted set. Then a) ∞ X π |λ|2 − k|ak |2 ≤ area(G). k=1
Equality holds if and only if f is a univalent function on E. b) area(G) ≤ π |λ|2 . Equality holds in (b) iff f (z) = λz. Hayman proved (see [2, 4], Chapther VIII) the following: Let F be meromorphic in E, and b1 bk F (z) = λ z + +···+ k +··· (4) z z near ∞, λ 6= 0. If E is the omitted set of F , then cap(E) ≤ |λ| . Proof: Let Kρ be the circle |z| = ρ with positive orientation and let γρ be the curve defined by the equation w = fρ (eit ) = f (ρeit ), 0 ≤ t ≤ 2π. For given w 6= ∞ let n(w) be the number of roots of f (z) = w in |z| > ρ. Assume that f 6= w on Kρ and λ 6= 0. Since f has a pole of order 1 at ∞, we have f (z) 6= w in |z| ≥ r for a large r and consequently, by the argument principle, Z 1 f 0 (z) (0.1) n(w) = dz = 1 − χ(γρ , w), 2πi Kr −Kρ f (z) − w where χ = χ(γρ , w) is the winding number (or index) of the curve γρ with respect to the point w. By the analytic Green’s theorem (see, for example [6]), the area Z Z Z 1 1 (0.2) Iρ = wdw ¯ = χ(γρ , w)dudv. 2πi γρ π R2 1
´ MIODRAG MATELJEVIC
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Let Gρ be the set omitted by f on Eρ = {|z| > ρ}. By (0.1) w ∈ Gρ if and only if χ(γρ , w) = 1. Also, it follows from (0.1) that χ(w) = 1 − n(w), and therefore ¯ χ(γρ , w) is an R integer less R than or equal to zero if w 6∈ Gρ . This together with (0.2) gives πIρ = G dudv + Gc dudv and therefore (0.3)
πIρ ≤ area(Gρ ).
By the isoperimetric inequality area(G) ≤ πcap2 (G). cap(E) ≤ |λ| and therefore (b).
By Hayman’s result
1. Appendix The area theorem is an important tool in theory of univalent functions. Theorem 1.1. Let w = f (z) = λz + az1 + · · · + aznn + · · · be an analytic function on E = {z : |z| > 1} and let G = C\f (E) be the omitted set. Then ∞ X π |λ|2 − k|ak |2 ≤ area(G). k=1
Equality holds if and only if f is a univalent function on E. Proof:Let Kρ be the circle |z| = ρ with positive orientation and let γρ be the curve defined by the equation w = fρ (eit ) = f (ρeit ), 0 ≤ t ≤ 2π. For given w 6= ∞ let n(w) be the number of roots of f (z) = w in |z| > ρ. Assume that f 6= w on Kρ and λ 6= 0.Since f has a pole of order 1 at ∞, we have f (z) 6= w in |z| ≥ r for a large r and consequently, by the argument principle, Z f 0 (z) 1 dz = 1 − χ(γρ , w), (1.1) n(w) = 2πi Kr −Kρ f (z) − w where χ = χ(γρ , w) is the winding number (or index ) of the curve γρ with respect to the point w. By the analytic Green’s theorem ( see, for example [6]), the area Z Z Z 1 1 wdw ¯ = χ(γρ , w)dudv. (1.2) Iρ = 2πi γρ π R2 Let Gρ be the set omitted by f on Eρ = {|z| > ρ}. By (1) w ∈ Gρ if and only if χ(γρ , w) = 1. Also, it follows from (1) that χ(γρ , w) is an integer less than or equal to zero if w 6∈ G¯ρ . This together with (2) gives (1.3)
πIρ ≤ area(Gρ ). References
[1] L. Ahlfors: Conformal invariants, McGraw-Hill Book Company, 1973. [2] W. Fuchs, Some Topics in the Theory of Functions of One Complex Variable, D. Van Nostrand Co. 1967 [3] G. M. Goluzin: Geometric function theory, Nauka, Moskva, 1966. (Russian). [4] W. K. Hayman, Some applications of the transfinite diameter to the theory of functions, Journal d Analyse Math´ ematique, Dec 1951 [5] M. Mateljevi´ c, An extension of the area theorem, Complex Variables, Theory and Application, Volume 15, Issue 3, 155-157, 1990 (now Complex Variables and Elliptic Equations 15(3):155-157, Sep 1990) [6] C. Pommerenke: Univalent functions, Vanderhoeck & Riprecht, 1975. Faculty of mathematics, Univ. of Belgrade, Studentski Trg 16, Belgrade, YU E-mail address:
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