2000 Mathematics Subject Classification: 30C25, 30C45.K is the exact Koebe domain for the class of functions considered here

Wirths, Karl-Joachim

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Serdica Math. J. 29 (2003), 355-360THE KOEBE DOMAIN FOR CONCAVE UNIVALENTFUNCTIONSKarl-Joachim WirthsCommunicated by S. L. TroyanskyAbstract. LetD denote the open unit disc and f : D → C be meromorphicand injective in D. We further assume that f has a simple pole at the pointp ∈ (0, 1) and an expansionf(z) = z +∞∑n=2an(f)zn, |z| < p.Especially, we consider f that map D onto a domain whose complementwith respect to C is convex. It is proved that this impliesK :={w :∣∣∣∣w + p(1 + p2)(1 − p2)2∣∣∣∣ > 2p2(1 − p2)2}⊂ f(D)and that for any c ∈ C\K there exists a function f satisfying the conditionsmentioned above such that c does not belong to f(D). This means that Kis the exact Koebe domain for the class of functions considered here.2000 Mathematics Subject Classification: 30C25, 30C45.Key words: Koebe domain, concave univalent functions, starlike meromorphic functions.356 Karl-Joachim WirthsFor a family A of functions f analytic in the open unit disc D the Koebedomain K(A) is defined byK(A) :=⋂f∈Af(D).There are many families of functions that play an important role in GeometricFunction Theory for which the Koebe domains have been determined, for examplethe family of schlicht functions and the family of convex functions. Since thesefamilies are invariant under rotations around the origin the Koebe domains arediscs with center in the origin. In the present paper we consider a family whichis not invariant under rotations and we shall determine its Koebe domain. Weare concerned with the family of concave univalent functions with pole p ∈ (0, 1)denoted by Co(p) here. To be precise, we say that a function f : D → C belongsto the family Co(p) if and only if:(1) f is meromorphic in D and has a simple pole at the point p ∈ (0, 1).(2) f has an expansionf(z) = z +∞∑n=2an(f)zn, |z| < p.(3) f maps D conformally onto a set whose complement with respect to C isconvex.There are results on Co(p) that resemble very much those on convexfunctions, for example it has been proved in [3] that |an(f)| > 1 for f ∈ Co(p).Other results look very different from the analogous results on convex functions.Results of this type are those on the domains of variability of the Taylor coeffi-cients an(f), f ∈ Co(p) (compare [6], [1], [3], [2] and especially [5] where furtherreferences on Co(p) may be found).Concerning covering theorems on Co(p) it may be deduced from the results ofJ. Miller (see [6, Theorem 5]) that{−p1 + p2}=⋂f∈Co(p)(C \ f(D)).Further, it was proved in [1] and in [2] that for f ∈ Co(p), c ∈ C\f(D), the sharpinequalities|c| ≥p(1 + p)2and−p(1− p)2≤ Re(c) ≤ −p(1 + p)2The Koebe domain for concave univalent functions 357are valid.The present paper is devoted to the determination of the Koebe domain of Co(p).We showTheorem 1. Let p ∈ (0, 1). ThenK(Co(p)) ={w :∣∣∣∣w + p(1 + p2)(1− p2)2∣∣∣∣ > 2p2(1− p2)2}.(1)The proof of (1) will be divided into the two natural steps. First, we proveTheorem 2. Let p ∈ (0, 1), f ∈ Co(p) and c ∈ C \ f(D). Then theinequality ∣∣∣∣c + p(1 + p2)(1 − p2)2∣∣∣∣ ≤ 2p2(1− p2)2(2)is valid.P r o o f. Since C \ f(D) is starlike with respect to c and f is normalizedas defined above and has a simple pole at the point p, the functionF (z) :=(1 − zp)(1 − z p)f ′(z)f(z) − c,resp. its holomorphic continuation from D \ {p} onto D has a positive real in D(compare [5, Theorem 6]). This and the univalence of f imply that the functionG defined byG(z) =1F (z), z ∈ D,is holomorphic in D and has a positive real part there. Further, we useG(0) = −c and G(p) =p1 − p2.Let c = x+ iy, x, y ∈ R. From the properties of the function G we conclude thatx < 0 and that there exists a function ϕ holomorphic in D such that ϕ(D) ⊂ D,ϕ(0) = 0 and−G(z) + iyx=1− ϕ(z)1 + ϕ(z), z ∈ D.Hence, there exists a function Φ holomorphic in D such that Φ(D) ⊂ D,−G(z) + iyx=1− zΦ(z)1 + zΦ(z), z ∈ D,and−p1− p2+ iyx=1− pΦ(p)1 + pΦ(p).358 Karl-Joachim WirthsThis equation together with Φ(D) ⊂ D yields that for every c = x+iy ∈ C\f(D)there exists a ζ ∈ D such that−p1− p2+ iyx=1− pζ1 + pζ=: u+ iv,(3)where u+ iv varies in the disc described by(u −1 + p21− p2)2+ v2 ≤(2p1− p2)2.(4)From (3) we gety = −x vandp(1− p2)= −xu.Now, we solve these two equations with respect to u and v and insert this into(4). The resulting inequality is easily shown to be equivalent to (2). Remark. An alternative proof of Theorem 2 may be deduced from thefact that the disc described by (2) is the starlike center region for the class of allstarlike meromorphic functions with pole p and the above normalization in theorigin. The definition and determination of this region may be found in [4].To end the proof of Theorem 1, it remains to show that for any c satisfying(2) there exists a function f ∈ Co(p) such that c ∈ C\f(D). In view of the Koebetheorem for the family of schlicht functions, the natural candidates for such fare the functions f ∈ Co(p) that map the unit disc onto C minus a segment of aline. These functions may be written in the formfθ(z) =z −p(1− eiθ)1− p2eiθz2(1 − zp)(1 − zp), z ∈ D, θ ∈ [0, 2pi) fixed.(5)In [6], [1] and [3] it has been proved that these functions belong to Co(p) andthat they have the above mentioned mapping property. For our aims, we needdetailed information about the slit. These informations are the content of thefollowing theorem.Theorem 3. Let θ ∈ [0, 2pi) and fθ be defined by (5) and τ ∈ [0, 2pi) byeiτ =eiθ − p21− p2eiθ.(6)The Koebe domain for concave univalent functions 359ThenC \ fθ(D) = f(∂D) ={c : c = −p1 + p2(1 + eiτ22p cos(ψ + τ2)1 + p2 − 2p cos ψ), ψ ∈ [0, 2pi)}.(7)Let dk, k = 1, 2, denote the endpoints of the slit (7). Both of them lie on theboundary of the disc described by (2), i. e.∣∣∣∣dk + p(1 + p2)(1 − p2)2∣∣∣∣ = 2p2(1− p2)2 , k = 1, 2.(8)P r o o f. The verification of (7) is accomplished setting z = eiψ, multiply-ing nominator and denominator of fθ(eiψ) by e−iψ and using the abbreviation (6).To prove (8) we have to determine the values ψk, k = 1, 2, such that dk = eiψk .These are the roots of the equationddψ(2p cos(ψ + τ2)1 + p2 − 2p cos ψ)= 0.Hence, we getsin(ψk +τ2)=2p1 + p2sinτ2.For the proof of (8) we derive from thiscos(ψk +τ2)=±√(1 + p2)2 − 4p2 sin2 τ21 + p2andcos ψk =2p sin2 τ2 ± cosτ2√(1 + p2)2 − 4p2 sin2 τ21 + p2,where, as in the following, the upper sign belongs to ψ1 and the lower one to ψ2.Using these formulae and the abbreviation dk = xk + iyk, xk, yk ∈ R, k = 1, 2,and some elementary computation, we getxk +p(1 + p2)(1− p2)2=2p2(1 + p2)(1− p2)2(2p sin2τ2∓ cosτ2√(1 + p2)2 − 4p2 sin2τ2)andyk =2p2(1 + p2)(1 − p2)2(−2p sinτ2cosτ2∓ sinτ2√(1 + p2)2 − 4p2 sin2τ2).360 Karl-Joachim WirthsAdding the squares of the right sides reveals the truth of (8). This ends the proofof Theorem 3. The formulae (7) and (8) show that for any c in the disc defined by (2)there exists a function fθ such that c ∈ C \ fθ(D). This fact together withTheorem 2 proves Theorem 1.Acknowledgement. The author thanks F. G. Avkhadiev and Ch. Pommerenkefor helpfull discussions.REFERENCES[1] F. C. Avkhadiev, K.-J. Wirths. Convex holes produce lower bounds forcoefficients. Complex Variables, Theory Appl. 47, 7 (2002), 553–563.[2] F. C. Avkhadiev, K.-J. Wirths. On a conjecture of Livingston. Mathe-matica (Cluj) 46(69) (2004), to appear.[3] F. G. Avkhadiev, Ch. Pommerenke, K.-J. Wirths. On the coefficientsof concave univalent functions. Math. Nachr., to appear.[4] C. Y. Lin. On the representation formulas for the functions in classΣ∗(p,w0). Proc. Amer. Math. Soc. 103 (1988), 517–520.[5] A. E. Livingston. Convex meromorphic mappings. Ann. Pol. Math. 59, 3(1994), 275–291.[6] J. Miller. Convex and starlike meromorphic functions. Proc. Amer. Math.Soc. 80 (1980), 607–613.Institut fu¨r AnalysisTU Braunschweig38106 BraunschweigGermanye-mail: k-j.wirths@tu-bs.de Received June 26, 2003

The Koebe Domain for Concave Univalent Functions

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The Koebe Domain for Concave Univalent Functions

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