Coefficients of Meromorphic Univalent Functions

Y. KΌBOTA KODAI MATH SEM REP. 28 (1977), 253-261 COEFFICIENTS OF MEROMORPHIC UNIVALENT FUNCTIONS BY YOSHIHISA KUBOTA 1. We denote by Σ' the family of functions regular and univalent in !<|^|<oo. Let g(z) be a function belonging to Σ' and let n=i be the inverse function of g(z). The following results are known : Springer [4] proved that |c a ^1 and conjectured that , (2n-2)l « c 2 » -{< ιl= n!(n-l)! _ι -0 ^-1,2 A . ίn In this paper we shall prove that the conjecture is true for the cases n=3, 4, 5. THEOREM. In these inequalities equality occurs only for the inverse function of z+(l/z) and its rotations. Ozawa [3] made use of Grunsky's inequality together with Golusin's inequality to prove the Bieberbach conjecture for the sixth coefficient. We apply his method to prove our theorem. 2. Let be a function belonging to Σ' and let Fm(w) be the m-th Faber polynomial Received Nov. 25, 1975. 253 254 YOSHIHISA KUBOTA which is defined by Then Grunsky's inequality has the form N N 771,71=1 7i = l I Σ namnxmxn\^Σ,n\xn\2 and Golusin's inequality has the form Σn|Σ^α 771=1 71=1 where N is an arbitrary positive integer and xlt x2, •••, τ^ are arbitrary complex numbers. By a simple calculation we have aιn=bn (fl=l, 2, •••), a21=2b2 , +5^3+15 W+^i 6 ... Let *(*)=*+ Σ Λ=l x be a function belonging to Σ and let COEFFICIENTS OF MEROMORPHIC FUNCTIONS 255 be the inverse function of g(z), then by a simple calculation we have 2 4 +28^ In this paper we shall use the following notations : b2=y+iy' , (1) 3. Firstly we are concerned with the case n=3. By Grunsky's inequality with N=3, xλ= x2= 0, ΛΓ3=1 we have 1 < 1 3' = Hence we have _1_ 3 ' We put (2) Since the polynomial S^A+V+CS/S)^3 is homogeneous, it is sufficient to prove that F^2 for |arg&ι|^(ττ/3). Rewriting (2) with the notations (1) we have And it is evident that 0^/>^1 and x/2^3p2 when |arg b,\ g(τr/3). By Grunsky's inequality with N=2, xί=Q, xz=l we have By taking the real part we have 256 YOSHIHISA KUBOTA Hence we have By the area theorem Thus we obtain (3) Since O^ Therefore (3) implies the desired result : Equality occurs only for x=Q. 4. Next we consider the case n— 4. By Grunsky's inequality with Λf=4, *ι— 8b2> ^2—5^ι, ^3—0, ^4=1 we have Hence we have Further by using Grunsky's inequality with N=2, *ι—0, bs+-lb * we have We put (4) COEFFICIENTS OF MEROMORPHIC FUNCTIONS 257 Now it is sufficient to prove that F^5 for I arg ^1^(^/4). Rewriting (4) with the notations (1) we have (5) And it is evident that O^ί^l and x/2^p2 theorem when |arg/?J g(ττ/4). By the area (6) Putting (6) into (5), we obtain (7) P(x)=l— —-X+X2— j-*3. Since * /2 ^£ 2 ^l, we have It is easy to prove that PO)>0 for O^ί^l. Therefore (7) implies that F^ for |arg^!|^(π/4), with equality holding only for x—0. 5. Finally we are concerned with the case n=5. By Grunsky's inequality 7 with A "— 5, X!=Q, x2=10b2, xs=(35/6)blί x±=Q, x5—l we have + Hence we have \c»\ ^ \bs 258 YOSHIHISA KUBOTA Further by using Grunsky's inequality with N=3, x1=x2=Q, xB=l we have 107 We put (8) Now it is sufficient to prove that the notations (1) we have (9) ^14 for \aτgb1\^(π/5). Rewriting (8) with + +y(-72px'y'-2x'ξ')+η(-S4;px'2-36x'η'}-2x'y'ξ. And it is evident that O^ί^l and */2<0.53£2 when |arg ^1^(^/5). Here we make use of Golusin's inequality. We put N=2, Xι=Q, x2=l in Golusin's inequality. Then we have Rewriting this we have (10) 4 COEFFICIENTS OF MEROMORPHIC FUNCTIONS 259 Putting (10) into (9), we have +y(-72px'y'-2x'ξ')+η(-56px'2-36x'η'}-2x'y'ξ. By the area theorem -42f2-42f'2 Hence we obtain I ^ r 2^3 (11) > 3 r 2+4- T5 +63^2-2jOj>f +2yφ-2 2.5ιjφ +2py'ξ'-2y'φ'+2 25η'φf +2 36px'y'y+2x'ξ'y+2 2Spx'2η+2 18x'y/η+2x'y'ζ. It is evident that P(#)>0 for O^ί^l. In order to prove 0^0, we first observe Further we have 260 YOSHIHISA KUBOTA Indeed we consider the discriminant Δ of this quadratic from. 3 Then 4 J>120.279+176.318£+2677.749ί -974.682£ -(103.491+1736.946£+586.192£2+2303.991ί3);c/2 +(1494.504£-196);t'4. If x'2^QA5p2 and O^j^l, then Δ > 120.279+ 17β.318£+2β77.749£3-947.682£4 -0.45ί2(103.491+1736.946ί+586.192ί2+2303.991^3) +(0.45)2£4(1494.504ί-196) +1896.123£3-1278.159ί4-734.159ί5>0. If 0.45£2g*/2<0.53ί2, then Og£<0.84, whence J>120.279+176.318/)-54.851/)2+1757.1β7^3-1340.421ί4-801.31ί5>0. Thus we have the desired inequality. By using these inequalities we have +2 36px'y'y+2x'ξ r Further by using the inequalities 4f / 36z we have We consider the symmetric matrix associated with the quadratic form /63 0 1 0 39 -p 1 -p 13+14^-18ί2-18.25Λ:/2/ Its principal diagonal minor determinants are COEFFICIENTS OF MEROMORPHIC FUNCTIONS 261 63, 2457, 31902+34398ί-44289ί2-4484().25;c/2ΞΞ Δ . If */2rg0.45£2 and Q^p^l, then If 0.45£2^;t/2<0.53£2 and 0^ί<0.84, then J^31902+34398/>-68055£2>0 . Hence it follows that (13+14jO-18/)2-18.25^2)/4-39f2+6302-2^+2^^0 for larg bl I ^(ττ/5). Similarly it follows that (13+14^-18i2-19^2)3;/2+40f/2+63^/2+ 2py'ξ/-2y'φ''^Q for |arg b,\ ^(ττ/5). Consequently we have <2^0. Thus (11) implies that F^14 for l a r g ^ l ^(ττ/5), with equality holding only for x= 0. REFERENCES [1] GOLUSIN, G. M., On £-valent functions. Mat. Sbornik (N.S.) 8 (1940), 277-284. [" 2 ] GRUNSKY, H., Koeffizientenbedingungen fur schlicht abbildende meromorphe Funktionen. Math. Zeits. 4 (1939), 29-61. [ 3 ] OZAWA, M., On the Bieberbach conjecture for the sixth coefficient. Kδdai Math. Sem. Rep. 21 (1969), 97-128. [4] SPRINGER, G., The coefficient problem for schlicht mappings of the exterior of the unit circle. Trans. Amer. Math. Soc. 70 (1951), 421-450. TOKYO GAKUGEI UNIVERSITY.