Note on a Harmonic Measure in the UnivalentMapping

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奈良教育大学学術リポジトリNEAR

Note on a Harmonic Measure in the Univalent Mapping

著者 OGAWA Shotaro

journal or

publication title

奈良学芸大学紀要

volume 4

number 2

page range 21‑22

year 1954‑12‑25

URL http://hdl.handle.net/10105/5037

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Note on a Harmonic Measure in the Univalent Mapping

Sh5taro (JGAWA

Department of Mathematics Nara Gakgei University

(Beceived September 1, 1954)

1. Let /(z)be regular and univalent for|z|<I1 and /(())=(), /' (0)=l, and map the unit circle|z!<1 onto the domain D/ on the w-plane, and let T denote the boundary of D/. Furthere let T'm denote the point set W so that W « /" and

|W]^Af, and ^j»/ wtDf and|tt>j=Af, and let 0 denote the mass of the arcs on 2|=1 which corresponds to T'm-

It is conceivable that the greater Mbecomes, the less 0 becomes and lim 9=0.

Really we get nex theorem. Af->oo

Theorem. ^ 4 Sin-i^(M>1)

Proof. Let 1= F(z)=^-j-J- and corresponding to D/, F, F'm, ^M, vve de ine Df, C, Oil/, -Lit/ respectively. We use the notation of the harmonic measure as used by li. NeVANLINNA £1~). From the principle of the harmonic measure (I'),

9 -^<oCO, Lw, Df(=D/q (Iz'=1))}> «CO, Cm, D^]=ff (1)

Flere we concider the mapping function r/=v{z) which mapsjz\<A onto Dp.

Then next inequality holds.

, 1_ *

sin-^^jy' (O) | 2 sin (JM;s t})e ]engt}, of jjAf) (-2)

*

From Df^DF we can see tliat

i F(O) |=-l^J-^|?'(O) |

therefore

Froni(l), (2j, (S) we obtain

«i"^^1V(O)[ a sin-L^|,'(O)|~"a sin^^/MsinA

Mere /iVf cannot exceed 2;r, so we get

ri

V^y M Siu^-

This proves the theorem.

Journal of Kara Gakugei Uniuerujly, Vol. 4 No. 2, Dec. 20 1954

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( 22 ) Shotaro Ogawa

2. Using above tlieorem we can derive an estimation of 1=1 \f{re^s)\dfi.

Jo OdM (mi= Min f(z)\, mi-Ma,x\f(z)j for|21=1)

Tin

Therefore /= \ 0 dM+("0 dM

•E>«,, Ji

(•E"'2 1

Ji 1/M

r 1 V2 f"s rfM

. . 1 of"3 dM

yWo JiVM-l

So we obtain next corollary.

COROLLARY Let m be Max|/(Z) (|2|<1). Then

f

are 1

\ i/(re8)(d0 <4(w sm-i- -+1/m-l)

Jo i/w

Remark. Ey elementary calculation, i'or #2<^-i ^r we get

Jiegrettly this result is of no use or the estimation of )an\, because for example Littlewood had given

[33

\ \f(re*),d6 <T^r.

References

Cl] K : Nkvanlinna, Eiudeutige AiiiilitisAe Funktionen. Berlin (1936) S. 63.

['23 Y. KoiJATUj Sur la variatioii d' uue fonciiou de representation confornie, lursque le duuiain varie.

Prue Imp. Acud. Tokyo 19 (1943;, 599-608.

[3] J. E. Littlkwood, On inequalities in the theory of functions. Proc. London Math. Soc. 23 (1925J,

4S1-519.