KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS

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1 Jounal of Applied Analysis Vol. 14, No ), pp KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS L. KOCZAN and P. ZAPRAWA Received Mach 12, 2007 and, in evised fom, Septembe 4, 2007 Abstact. In this pape we pesent a new method of detemining Koebe domains. We apply this method by giving a new poof of the well-known theoem of A. W. Goodman concening the Koebe domain fo the class T of typically eal functions. We applied also the method to detemine Koebe sets fo classes of the special type, i.e. fo T M,g = {f T : f ) Mg )}, g T S, M > 1, whee = {z C: z < 1} and T, S stand fo the classes of typically eal functions and univalent functions espectively. In paticula, we find the Koebe domains fo the class T M of all typically eal and bounded functions, and fo the class T M) of all typically eal functions with anges in a given stip. 1. Intoduction A function f analytic in the unit disk = {z C: z < 1} is said to be typically eal if it satisfies the condition Im z Im fz) 0, z. Let T denote the class of typically eal functions f with nomalization f0) = 2000 Mathematics Subject Classification. Pimay 30C25; Seconday 30C80. Key wods and phases. Typically eal functions, Koebe domain, subodination. ISSN c Heldemann Velag.

2 44 L. KOCZAN AND P. ZAPRAWA f 0) 1 = 0 see fo example [5]). Let S be the class of functions analytic and univalent in with the same nomalization as in T. The following classes wee consideed in [3] T M,g = {f T : f Mg}, g T S, M > 1. 1) Recall that a function h is subodinated to a univalent function H, witten h H, if h0) = H0) and h ) H ). Choosing g 1 z) = z and g 2 z) = 1 2 log 1 + z 1 z, z, with the pincipal banch of logaithm, we obtain two impotant subclasses of T. Namely, T M,g 1 = {f T : fz) < M, z } and T M,g 2 = {f T : Im fz) < M π 4, z }, which ae biefly denoted by T M and T M)) espectively. The elation { ) } hz) T M,g = Mg : h T M, 2) M which was established in [3], povides the fomula connecting diffeent classes of type T M,g as follows: { )) } hz) T M,f = Mf g 1 : h T M,g. 3) M Fo this eason, instead of eseaching a class T M,f one can conside a class T M,g. We apply this idea in ode to obtain esults in vaious classes T M,g as a consequence of elated esults in T M). Investigating T M) is possible thanks to the integal fomula fo this class. Moeove, extemal points as well as suppoting points ae known in T M) see, [4]). The main aim of this pape is to detemine the Koebe set usually called the Koebe domain) fo T M,g. Recall that fo a given A T, the Koebe set is defined by f A f ) and is denoted by K A. Fom 3) it follows that Lemma 1. Let f, g T S. A set D is the Koebe domain fo T M,g iff M fg 1 D/M)) is the Koebe domain fo T M,f. Poof. w K T M,f w F T M,f F )

3 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS 45 w )) G ) Mf g 1 M G T M,g )) G ) G T M,g w Mf g 1 M G T M,g Mg f 1 w )) G ) M Mg f 1 w )) K M T M,f. 2. The Koebe domain fo the class T In 1977 Goodman detemined the Koebe domain fo T. Let θ) = π sin θ, θ 0, π). 4) 4θπ θ) Theoem A [1]). The Koebe domain fo the class T is a bounded domain, symmetic with espect to both axes of the complex plane. Its bounday in the uppe half plane is given by the pola equation θ) fo θ 0, π), ϱθ) = 1 4 fo θ = 0 o θ = π. A new poof of Theoem A. Let f T omit two values ϱe iθ and ϱe iθ, whee ϱ > 0, θ 0, π). A function fz) ϱe iθ fz) ϱe iθ is analytic in and omits the points 0 and 1. Hence a function hz) = 1 i fz) ϱeiθ log, 5) fz) ϱe iθ with the banch of logaithm chosen in such a way that h0) = 2θ, is also analytic in. Fom 5) we deive fz) = ϱ eiθ e iθ e ihz) 1 e ihz).

4 46 L. KOCZAN AND P. ZAPRAWA We obtain hz) 2nπ, n Z because 1 e ihz) 0. This and the equality ϱ Im fz) = 1 e ihz) 2 1 e 2 Im hz)) sin θ lead to Im z Im hz) 0, z. Since h0) = 2θ, hz) 2nπ, n Z and h is typically eal, we get 0 < hx) < 2π fo eal x. Let a z Hz) = 2θ + 1 2tz + z 2, whee a = 8π θ) θ π, t = 1 2 θ π. Then H is univalent, H0) = 2θ and H ) = C \ {p R: p 0 p 2π}. Fom the popeties of h and H we conclude that the function h is subodinated to a function H. Theefoe hz) = Hwz)), whee wz) = H 1 hz)). This gives wz) z and 2 sin θ ϱ = h 0) = H 0) w 0) = a w 0) a = 8π θ) θ π. Thus ϱ π sin θ 4θπ θ) with equality only in the case of h = H. In the oiginal poof of this theoem Goodman applied some popeties of the so-called univesal typically eal functions. The existence of thei invese functions, which wee defined on Riemann sufaces, played an essential ole hee. In his method all univesal functions wee geneated by the function Gz) = 1 ) πz π tan 1 + z 2. The main advantage of ou new method is that one can easily obtain extemal functions which coespond to bounday points of the Koebe domain. In fact, the bounday points of the Koebe set fo T ae elated to functions F z) = eiθ e iθ e ihz) π sin θ 1 e ihz) 4θπ θ). One can check that functions F coincide with those found by Goodman in [2], i.e. ) z + c G Gc) 1 + cz 1 c 2 )G with c = π + 2 πθ θ 2. c) π 2θ The method pesented above may be applied to othe classes that consist of functions with eal coefficients.

5 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS The Koebe domain fo the class T M,k At the beginning we discuss the class T M,k with kz) = z/1 z) 2. Afte that, we shall detemine the Koebe domains fo the classes T M,g, whee M > 1 and g is a typically eal and univalent function. Obseve that f T M,k iff f T f ) C \ { p R: p M 4 }. 6) Fom now on we assume the banch of agument ag M + 4e iθ) to be in [0, 2π). Theoem 1. A function f T M,k omits e iθ and e iθ, θ 0, π) if and only if thee exists a function h analytic in such that 1. h0) = 2θ, h 0) = 2 sin θ, 2. 2 ag M + 4e iθ) < hx) < 2π fo 1 < x < 1, 3. Im z Im hz) 0, z, 4. fz) = eiθ e iθ e ihz) 1 e ihz). Poof. ) Let f T M,k and fz) e iθ, fz) e iθ fo θ 0, π). Obseve that fz) e iθ )/fz) e iθ ) is an analytic function in and omits 0 and 1. Thee exists the function log fz) eiθ fz) e iθ which we denote by ihz). The banch of logaithm is chosen to be h0) = 2θ. Hence fz) = eiθ e iθ e ihz) 1 e ihz) and e ihz) 1. Theefoe Moeove, which leads to Im fz) = hz) 2nπ, n Z. 7) 1 e ihz) 2 1 e 2 Im hz)) sin θ 8) Im fz) > 0 iff Im hz) > 0,

6 48 L. KOCZAN AND P. ZAPRAWA and Im fz) < 0 iff Im hz) < 0. Consequently, Im z Im hz) 0, z. Then h is typically eal. Fom 7) and h0) = 2θ we conclude that hx) 0, 2π) fo x 1, 1). Fo x 1, 1) the function hx) is inceasing. It follows fom univalence of typically eal functions in the set {z : 1 + z 2 > 2 z }, see [1]. Fo this eason and fom fx) > M/4 by 6)) it follows that hx) > 1 i log M/4 eiθ M/4 e iθ, o equivalently hx) > 2 ag M + 4e iθ). ) Let h be an analytic function in such that the conditions 1 4 of Theoem 1 ae satisfied. With these assumptions 6) holds. Hence Im z Im fz) 0, z, f is nomalized by f0) = f 0) 1 = 0 and fx) > M/4 fo x 1, 1). It means that f T M,k. By the definition of h we know that f omits e iθ and e iθ. Theoem 2. The Koebe domain fo the class T M,k, whee kz) = z/1 z) 2, M > 1, is a bounded domain, symmetic with espect to the eal axis. Its bounday in the uppe half plane is given by the pola equation w = ϱm, θ)e iθ, θ [0, π], whee M 4M 1), θ = 0 ϱm, θ) = M, θ), θ 0, π) 9) 1 4, θ = π, and = M, θ) is the only solution of ag M + 4e iθ) 4θπ θ) π sin θ = 0 10) 4π θ) sin θ in θ), ), and θ) is given by 4). Poof. Let f T M,k and fz) e iθ, fz) e iθ fo θ 0, π), > 0. We assume θ) because T M,k T. By Theoem 1 thee exists a function h satisfying the conditions 1 4 of this theoem and hence Let fz) = eiθ e iθ e ihz) 1 e ihz). H M,θ z) = 2θ + a z 1 2tz + z 2,

7 whee KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS 49 a = 8π θ) θ ag M + 4e iθ)) π ag M + 4e iθ, ) t = π 2θ + ag M + 4e iθ) π ag M + 4e iθ, ) θ 0, π), M > 1. One can check that t 1, 1). The function H M,θ is univalent, H M,θ 0) = 2θ and H M,θ ) = C\ { p R: p 2 ag M + 4e iθ) p 2π }. Fom popeties of h and H M,θ we obtain h H M,θ. Theefoe Hence o equivalently 2 sin θ 2 sin θ = h 0) H M,θ 0) = a. 8π θ) θ ag M + 4e iθ)) π ag M + 4e iθ ) 11) θ θ) θ ag M + 4e iθ) 0. 12) π θ) Let M and θ be fixed. Let us denote by g) the left hand side of 12). It is easily seen that gθ)) < 0. We shall pove that the equation g) = 0 consideed fo [θ), ) has only one solution. We have g sin θ ) = 4 θ 161 M) 2 + 8M cos θ + sin θ ) π θ))2 M + 4e iθ 2 π θ +M M sin2 θ π θ) 2 )). Let W ) = a 2 +b +c with a = 161 M), b = 8M cos θ + sin θ/π θ)), c = M M sin 2 θ/π θ) 2). Since a < 0 and c > 0, the function W ) is zeo in two points of diffeent signs. Let 0 θ) be a positive zeo of W ). We claim that 0 θ) > θ). Indeed, if 0 θ) θ) wee satisfied then we would obtain g ) < 0 fo > θ). In this case it would be a contadiction. We have actually poved that 0 > gθ)) > lim g) = 0, g ) > 0 fo [θ), 0 θ)) and g ) < 0 fo > 0 θ).

8 50 L. KOCZAN AND P. ZAPRAWA Fom lim g) = 0 and g ) < 0 fo > 0 θ) it follows that g) > 0 fo > 0 θ). This and the inequalities gθ)) < 0 and g ) > 0 fo [θ), 0 θ)) lead to the conclusion that fo θ) the equation g) = 0 has only one solution which we shall denote by M, θ). Hence g) 0 holds fo [M, θ), ). Fo the function F M,θ z) = M, θ) eiθ e iθ e ih M,θ 1 e ih, θ 0, π) M,θ thee is M, θ)e iθ F M,θ ). Combining this with M, θ) we deduce that the points M, θ)e iθ, M, θ)e iθ, θ 0, π) belong to the bounday of K T M,k. Consequently K T M,k is a stalike set. Let ϱm, θ)e iθ, θ [0, 2π) be the pola equation of this bounday. Hence ϱm, θ) = M, θ), θ 0, π). Dividing 12) by θ and taking the limit as θ tends to 0 fom the ight we get since ag M + 4e iθ ) lim θ 0 + θ 1/4 = lim θ 0 + Im = 4 M M + 4 0, 13) 4ieiθ 4M cos θ + 4) = lim M + 4eiθ θ 0 + M + 4e iθ 2 As a esult, 12) gives M/4M 1)). Hence ϱm, 0) = M/4M 1)). Witing 10) in the fom π ag M + 4e iθ) ) sin θ 4π θ)θ ag M + 4e iθ )), 14) and taking the limit as θ tends to π fom the left we obtain 1/4. Indeed, fo f T M,k we have fx) > M/4 while x 1, 1). Theefoe f fo negative x takes values fom, b], whee b > M/4. Thus lim θ π ag M + 4e iθ) = 0. The inequality 1/4 means that ϱm, π) = 1/4.

9 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS 51 We have poved that the bounday of the Koebe domain fo T M,k in the uppe half plane is given in the fom w = ϱm, θ)e iθ, θ [0, π], whee M 4M 1), θ = 0 ϱm, θ) = M, θ), θ 0, π) 1 4, θ = π. Since Koebe domains fo classes of functions with eal coefficients ae symmetic with espect to the eal axis, the poof is complete. Obseve that the extemal functions in the cases of θ = 0 and θ = π ae of the fom z z F 1 z) = ) and F 2 z) = 2 1 2z M 1 + z z) 2. These functions belong to T M,k and satisfy F 1 1) = M 4M 1) By Lemma 1 and Theoem 2, and F 2 1) = 1 4. Coollay 1. The Koebe domain fo the class T M is a bounded domain, symmetic with espect to both axes of the complex plane. Its bounday in the uppe half plane is given by the paametic equation w = 2ϱM, θ)e iθ 2 M ϱm, θ)eiθ whee ϱm, θ) is given by 8). 4 M ϱm, θ)eiθ + 1, θ [0, π], 15) Coollay 2. The Koebe domain fo the class T M) is a bounded domain, symmetic with espect to both axes of the complex plane. Its bounday in the uppe half plane is given by the paametic equation w = M ) 4 4 log M ϱm, θ)eiθ + 1, θ [0, π], 16) whee ϱm, θ) is given by 9).

10 52 L. KOCZAN AND P. ZAPRAWA Obseve that the paametic complex equation 16) can be ewitten, using 10), in the fom x = M 8 log ) 16 ϱm, θ) cos θ + M M 2 ϱ2 M, θ) y = M, θ [0, π]. 17) 4ϱM, θ)θπ θ) π sin θ 4 4ϱM, θ)π θ) sin θ Refeences [1] Golusin, G., On typically-eal functions Russian), Mat. Sb. 2769) 1950), [2] Goodman, A. W., The domain coveed by a typically eal function, Poc. Ame. Math. Soc ), [3] Koczan, L., On classes geneated by bounded functions, Ann. Univ. Maiae Cuie Sklodowska Sect. A 522) 1998), [4] Koczan, L., Szapiel, W., Extemal poblems in some classes of measue IV): typically eal functions, Ann. Univ. Maiae Cuie Sklodowska Sect. A ), [5] Rogosinski, W. W., Übe positive hamonische Entwicklungen und tipisch-eelle Potenzeichen Geman), Math. Z ), Leopold Koczan Pawe l Zapawa Depatment of Applied Depatment of Applied Mathematics Mathematics Lublin Univesity of Technology Lublin Univesity of Technology Nadbystzycka 38 D Nadbystzycka 38 D Lublin, Poland Lublin, Poland l.koczan@pollub.pl p.zapawa@pollub.pl

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