Let S*(1-b), (b 0 complex) denote the class of functions f(z) = z + a2z2 + ... analytic in D = {z : |z| < 1} which satisfies for z = eiθ D, f(z) / z 0 in D, and 

Re[1 + 1/b(z f\u27(z) / f(z) - 1)] > 0.

The aim of this paper is to give the Koebe domain of the above mentioned class

Arzu Sen

Metin Bolcal

Yasar Polatoglu

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GLASNIK MATEMATIČKIVol. 37(57)(2002), 89 – 92KOEBE DOMAIN OF STARLIKE FUNCTIONS OFCOMPLEX ORDER WITH MONTEL NORMALIZATIONYaşar Polatoğlu, Metin Bolcal and Arzu ŞenUniversity of Kültür, TurkeyAbstract. Let S∗(1−b), (b 6= 0 complex) denote the class of functionsf(z) = z + α2z2 + ... analytic in D = {z| |z| < 1} which satisfies, forz = eiθ ∈ D,f(z)z6= 0 in D, andRe[1 +1b(zf ′(z)f(z)− 1)]> 0.The aim of this paper is to give the Koebe domain of the above mentionedclass.1. IntroductionLet F denote the class of functions f(z) = z+α2z2+ ... which are analyticin D. We shall need the following definitions.Definition 1.1. A function f(z) ∈ F is said to be starlike function ofcomplex order (1− b), (b 6= 0 complex), that is f(z) ∈ S∗(1− b) if and only iff(z)z6= 0 in D, and(1.1) Re[1 +1b(zf ′(z)f(z)− 1)]> 0, z ∈ D.It should be noticed that by giving specific values to b we obtain thefollowing important subclasses, [5].1. b = 1, S∗(1 − b) = S(0) = S∗ is the well known class of starlikefunctions.2000 Mathematics Subject Classification. 30C45.Key words and phrases. Univalent functions, starlike functions, starlike functions ofcomplex order, Montel normalization, Koebe domain.8990 Y. POLATOĞLU, M. BOLCAL AND A. ŞEN2. b = 1 − α, (0 ≤ α < 1), S∗(1 − b) = S(α) is the class of starlikefunctions of order α.3. b = cosλe−iλ, S∗(1− b) = S∗(1− cosλe−iλ) = S∗λ, |λ| < π2 , is the classof spirallike functions.4. b = (1 − α) cosλe−iλ , |λ| < π2 , 0 ≤ α < 1, S∗(1 − b) = S∗(1 − (1 −α) cosλe−iλ) = S∗λ,α is the class of spirallike functions of order α.Definition 1.2. The Koebe domain for the family F is denoted by K(F )and by the definition this is the largest domain contained in f(D) for everyfunction f(z) in F .From this definition the Koebe domain has the following properties.1. K(F ) is the collection of points w such that w is in f(D) for everyfunction f(z) in F ,(1.2) K(F ) =⋂f(z)∈Ff(D).2. Supposing the set F is invariant under the rotation, so eiαf(e−iαz) isin F whenever f(z) in F. Then the Koebe Domain will be either thesingle point w = 0 or an open disc |w| < R. In the second case R isoften easy to find indeed, supposing that we have a sharp lover boundM(r) for f(reiθ) for all functions in F , and F contains only(1.3) R = limr→1−M(r)gives the disc |w| < R as the Koebe Domain for the set F . (see 1.2)Montel type normalization. We can also impose a Montel type normal-ization. This means that for some fixed r0 with 0 < r0 < 1, we consider thefamily of functions f(z) regular and univalent in D with f(0) = 0, f ′(0) = 1,f(r0) = r0.We note that: If the class of starlike functions is normalized by the Monteltype normalization, then the class is denoted byS∗montel(1 − b)2. Koebe domain for the class of starlike functions of complexorder with Montel normalizationIn this section we shall give the Koebe Domain for the class of starlikefunctions of complex order with Montel normalization.Theorem 2.1. Let f(z) ∈ S∗(1 − b), then2|u|·(1−|v|2)2b(1+|b|)|v||1−uv|2b−2[|1−uv|+|u−v|]2≤∣∣∣ f(u)f(v)∣∣∣ ≤ 2|u|(1−|v|2)2b(1+|b|)|v||1−uv|2b−2[|1−uv|−|u−v|]2KOEBE DOMAIN OF STARLIKE FUNCTIONS 91holds for u, v ∈ D, u 6= v. This bound is sharp, because the extremal functionis w = f(z) defined for |z| < 1 byw = f∗(z) =z(1 − z)2bProof. We consider the Möbius transformation(2.1) u =z + v1 + vz⇐⇒ z = u− v1 − uvwhich is analytic and univalent in D, and this Möbius transformation mapsthe unit disc on to the itself. Now we define the function:(2.2)F (z) =αzf(z+α1+αz)f(z)(z + α)(1 + αz)2b−1⇐⇒ F (z) = v(1 − uv)2b−1(u− v)f(u)u(1 − |v|2)2bf(v) ,for α = v. This function is starlike function of complex order in the unit discD. On the other hand, if g(z) is starlike function of complex order, then(2.3)2|z|(1 + |b|)(1 + |z|)2 ≤ F (z) ≤2|z|(1 + |b|)(1 − |z|)2holds (see [5]). Therefore applying the inequality (2.3) to the function F (z)which is defined by (2.2) and using the relation (2.1) we get2 u−v1−uv(1 + |b|)(1 +∣∣∣ u−v1−uv∣∣∣)2≤∣∣∣∣v(1 − uv)2b−1(u− v)f(u)u(1 − |v|2)2bf(v)∣∣∣∣ ≤2 u−v1−uv(1 + |b|)(1 −∣∣∣ u−v1−uv∣∣∣)2(2.4)Simple calculations from (2.4) show that this teorem is true.Corollary 2.2. The Koebe Domain of starlike function of complex orderwith Montel type normalization is:R =(1 − r20)2b2(1 + |b|)(1 − 2r0 cos θ + r20)2b, 0 ≤ θ ≤ 2πIndeed, if we take v = f(r0) = r0, 0 < r0 < 1, u = z = reiθ in Theorem2.1, we obtain2|reiθ|(1 − r20)2b(1 + |b|)|1 − r0reiθ|2b−2 [|1 − r0eiθ| + |reiθ − r0|]2≤ |f(z)| ≤2|reiθ|(1 − r20)2b(1 + |b|)|1 − r0reiθ|2b−2 [|1 − r0eiθ| − |reiθ − r0|]2(2.5)92 Y. POLATOĞLU, M. BOLCAL AND A. ŞENTherefore we haveM(r) =2|reiθ|(1 − r20)2b(1 + |b|) · |1 − reiθ|2b−2 [|1 − reiθ| + |reiθ|]2.If we take limr→1−M(r), we get(2.6) R = K(S∗montel(1 − b)) =(1 − r20)2b2(1 + |b|) · (1 − 2r0 cos θ + r20)b.Giving the specific values to b we obtain the following results.1. For b = 1, R = K(S∗montel) =(1 − r20)24 · (1 − 2r0 cos θ + r20)2. For b = 1 − α, 0 ≤ α < 1,R = K(S∗montel(α)) =(1 − r20)2(1−α)2α(1 − 2r0 cos θ + r20)1−α3. For b = cosλe−iλ , ( |λ| < π2 )R =K(S∗montel(1 − cosλ · e−iλ))==(1 − r20)2 cos λe−iλ2(1 + cosλ)(1 − 2r0 cos θ + r20)cos λe−iλ4. For b = (1 − α) cosλe−iλ, |λ| < π2 , 0 ≤ α < 1,R = K(S∗montel(1 − (1 − α) cosλ · e−iλ))=(1 − r20)(1−α) cos λe−iλ(1 + (1 − α) cosλ)(1 − 2r0 cos θ + r20)(1−α) cos λe−iλReferences[1] P. L. Duren, Univalent functions, Springer Verlag, New York, 1981.[2] K. Fan, Distortion of Univalent functions, J. Math. Anal. Appl. 66 (1979), 626–631[3] A. W. Goodman, Univalent functions, Volume I and Volume II, Mariner Publ. Co.Inc. Tampa, Florida, 1983.[4] J. Krzyz, On a problem of P. Montel, Ann. Polon. Math., 12 (1962), 55–60[5] Y. Polatoğlu, Koebe Domain for convex fuctions of complex order, J. Math. ThePunjab Univ. 31(1998), 13–24[6] O. J. Puneja, S. Ponnusamy, S. Rajasekaran, Two point distortion and rotationtheorems, Glas. Mat 30(50) (1995), 221–242Department of Mathematics,Faculty of Science,University of Kültür, TurkeyE-mail : m.bolcal@iku.edu.trReceived : 18.06.1999.

Koebe domain of starlike functions of complex order with Montel normalization

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