BY JAMES A. JENKINS. r/( r).

Transcription

1 ON WEIGHTED DISTORTION IN CONFORMAL MAPPING BY JAMES A. JENKINS 1. A few yers go Komtu nd Nishimiy [4] rised the question of obtining bounds for the distortion in the sphericl metric of normlized univlent functions in the unit circle, tht is, for the quntity [if(z)[ (1 + [f(z)[2)-1 for given vlue of z I. The explicit vlues they obtined were for the most prt not shrp. Quite recently Oikw [6] obtined the best possible upper nd lower bounds for ll vlues of [z 1, 0 < zl < 1 using the vritionl method. None of the preceding uthors seems to hve relized tht the lower bound hs essentilly been - known for mny yers. Indeed by the stndrd distortion theorem, if zl r, 0 _-< r < 1, If (z) <= (1 + r)/(1 r), while by result of L6wner [5] in form given by Robinson [7] By dding these, If(z) [f (z) - < r/( r). (1 + If(z)[2)[f (z)1- =< ((1 -t- r) -t- r2)/(1 r), which on inverting gives just the lower bound obtined by Oikw. As is well known, equlity occurs in this for the slit functions nd for them only. Of much more interest is the fct tht the form of solution obtined by Oikw hs very little dependence on the explicit form of the sphericl distortion. In fct n nlogous result pplies to expressions of the form F(lf(z) I, If (z)i) whenever F stisfies certin firly simple restrictions to be discussed in detil below As usul we denote by S the fmily of functions f(z) regulr. nd univlent for z < 1 with f(0) 0, f (0) 1. We begin with discussion of the functions which ply the extreml role in our problems. THEOREM 1. Let 0 < r < 1, r/(1 r) <= q <- r/(1 r) Then there exists unique function f(z, r, q) in S with f(r, r, q) q mpping[z[ < 1 conformlly onto n dmissible domin [2; 3, p. 49] with respect to the qudrtic differentil for q r, q Q(w, (w, q)dw )dw w(w- q) Received August 11, Reserch supported in prt by the Ntionl Science Foundtion, Mthemtics Section, through the University of Notre Dme, nd by the Office of Ordnnce Reserch through the Institute for Advnced Study. 28

2 where is determined by ON WEIGHTED DISTORTION IN CONFORMAL MAPPING 29 (1) log ((- q) /) 1/2-1 (( q)/)1/. + 1 nd stisfies ql1/2 -t log 4 l[ logr >= -, r/(1 + r) <= q < r, 1 <, r < q < r/(1 --r) or with respect lo the qudrtic differentil for q r, Q(w, q)dw dw w2(w- q)2" With f(z, r, q) is ssocited (except for q r/(1 + r), r/(1 r)) oneprmeter fmily of functions f(z, r, q, X) e S where f(z, r, q, X) is obtined from f(z, r, q) by trnsltion of mount X in the Q-melric long the trjectories of Q(w,, q)dw (Q(w, q)dw) combined with rottion of the w-plne through the ngle -X where IX =< min (( % q)1/2 (2) (r-1 r) sin tn- [ 2-- (r -t + r) cos]] nd r) Of course f(z, r, q, 0) =- f(z, r, q). If f(z) e S with f(rei) q, then (3) f (re) <= f (r, r, q)[, 1 r_ cos +r) (r-- equlity oecurrin9 only if f(z) eif(ze-i, r, q, X). In the circle [z < 1 we regrd the positive qudrtic differentil Q(z, r, 9)dz where 0 _-< 9 _-< r. The mpping (z e *)(z e-i*) dz z(z r)(z r f r, (/))1/2 dz, where the rdicl is chosen to be positive on the segment 0 < z < r, crries the upper unit semicircle onto domin in the -plne bounded by rectiliner segments of the following nture. Let A, A., A, A*, A be the imges of 0, r, 1, e -1. Then for suitble choice of the lower limit of integrtion,

3 30 JAMES A. JENKINS A1 A2 is the rel xis with both A1 nd A. t the point t infinity nd f incresing s we go from A1 to A A A. is hlf-infinite horizontl segment, the vlue of f on it being positive with 6f decresing s we go from A to A A3 A* is verticl segment with f decresing s we go from A to A* nd with f positive t A*; A* A4 is verticl segment with f incresing s we go from A* to A4 A4 A1 is hlf-infinite horizontl segment with 6" decresing s we go from A4 to A1. Exceptionlly, two of A, A*, A, my coincide, in which cse the corresponding segment reduces to point. Now let the domin bounded by the rel xis nd the segments A2 A, A3 A, A A1 (i.e., the domin obtined from the preceding by deleting the slit penetrting to A*) be mpped conformlly on the upper hlf w-plne with A A2, A3, A*, A going into wl, w2, w3, w*, w4, where wl is the origin nd w3 or w is the point t infinity ccording s sif is lrger t A3 or A4 (these points coinciding t infinity when A nd A coincide). Reflecting cross the segments -1 < z < 1. nd w wl w of the rel xes in the z- nd w-plnes, we obtin conforml mpping of z] < 1 onto the w-plne minus forked slit. The mpping will be ssumed normlized so tht dw/dz 1 t z 0. Then we denote w. by q (which is positive) nd whichever of w nd w4 is not the point t infinity by (provided these points do not coincide). The corresponding mpping function is denoted provisionlly by g(z, r, ). It is cler tht the function g(z, r, ) vries continuously with sense of convergence. Moreover g(z, r, O) z(1 + z) -, g(z, r, r) z(1 z)-; in the usul thus q tkes every vlue between r(1 + nd r(1 t lest once. If we extend r)- f s (non-single-vlued) function of w to the r)-whole w-plne by reflection in the vrious segments w w, w. w, w w, w Wl, we see t once tht -d/": is qudrtic differentil on the w-sphere with double poles t 0 nd q nd, unless A nd A4 coincide, simple pole t the point t infinity nd simple zero t the point. the form (A A) or (A A) Thus the qudrtic differentil hs either (w- )dw v.(w- q). w(w Considertion of the fct tht g (0, r, ) 1 shows tht q:/,. -q. Denoting this qudrtic differentil genericlly by Q(w)dw, we see tht g(z, r, q) mps zl < 1 onto n dmissible domin with respect to it. Thus ix if f e S nd hs f(re) qe x rel (corresponding to this vlue of ), we re in position to pply the Generl Coefficient Theorem [2; 3, p. 51]. In the nottion of tht result we hve now 6t the w-sphere, A the domin g(e, r, ) where E denotes zl < 1. Let q)(w) be the inverse of g(z, r, ) defined in q)"

4 ON WEIGHTED DISTORTION IN CONFORMAL MAPPING g(e, r,,). Then the function e-i f(ei(w)) is n dmissible function ssocited with A. The qudrtic differentil Q(w)dw hs double poles P1 t the origin nd P2 t the point q. The corresponding coefficients re (1) o --17 (1) e () (q-- )/or--l, ( ) ei(x-)g (r, r, )/f (re) The fundmentl inequlity of the Generl Coefficient Theorem then gives (R{- log e (x-) -- (2) log [ei(x-)g (r, r, )/f (re)]} <-_ O. Since in either cse (:) is rel nd negtive, we hve tht If (re)l < g (r, r, )I. Moreover equlity cn occur only when the bove dmissible function constitutes trnsltion in the Q-metric (i.e., the metric Q(w) I/2[dw I) long the trjectories of Q(w)dw"2.. From the ltter remrk we see tht given vlue of q cn occur for t most one vlue of Thus we my now replce the nottion g(z, r, ) by the nottionf(z, r, q). Moreover n elementry clcultion shows tht the cse where A nd A4 coincide occurs precisely when q r. Thus we hve,. >= r/(1 + r) < q < r, <= ---}, r < q <= r/(1 r) In order to find the explicit reltionship between nd q (for fixed r) determine explicitly the extreml function. We hve r-" f (Z ei )(Z e-i )[z(z r)(z r-)] -1 dz, nd we choose the constnt of integrtion so tht A., A*, At lie on the imginry xis. Then r) -] log [(z r-1)(z r) -1] log z + [((r + r-1) 2 COS )(r " -1 + [((r + r-) 2cos)(r -t r)-]logr. Here determintions of the logrithms re principl on the segment (0, r). We hve for this the expnsion bout z 0 (4) nd bout z log z [((r -t- r-) 2 cos )(rr) -] log r r we -t- positive powers of z, --[((r -{- r-) 2 cos )(r " -1 r) -1] log z) -t- log r (5) -t-[((r + r-) 2 r) -1] log (1 r) eos)(r- positive powers of (r- z).

5 32 JAMES A. JENKINS We now get the similr formuls for w, treting for simplicity first the cse where > 0. Then / q-l/e( w)/e[w(q w)]-i dw, The choice of constnt of integrtion corresponding to tht mde for z gives (-) I [ I 1/2 ( w)/:1 ( w) " log L-i + ( w).,,_q log (- w) / + where the rdicls re to be positive nd the determintions of the (- logurithms principl on the segment (0, q). We hve for this the expnsion bout w 0 (6) logw lg4+(-q) 1/2 log + positive powers of w, nd bout w q log (q- w) + - log4(- q) (7) - q)/ Agreement of the expnsions (5) nd (7) shows first tht (8) (( q)/) / [((r r-1) 2 cos r)-]. )(r- Then the fct tht dw/dz 1 t z 0 gives us log r =( 1/2 lull2_ (_q)il21 + log L + ( 2 + positive powers o (q- ). \h- ql log 4-t-log L + (- J" Explicit determintion of the similr developments in the cse where < 0 shows tht (8) is vlid in ny cse, wheres we hve in generl (( q)/) / log r log 4 h l + log (( q)/) 1/2 --{- 1 It remins only to determine for wht functions equlity cn occur in (3). As we hve seen, this is possible only when the mpping given by e-xf(eo(w)) mourts to trnsltion in the Q-metric long the trjectories of Q(w)dw. Such motion is equivlent to verticl trnsltion in the -plne nd my be mde in either sense to the extent of the length of the slit penetrting to the point A*. This length is found by direct clcultion to be + tn-1 2 (r- + r) cos

6 where ON WEIGHTED DISTORTION IN CONFORMAL MAPPING cos llr-1 THEOREM 2. Lel 0 < r < 1, q > 0, nd let (z, r, q) denote the funclion z[1 -+- (q-1 (r- _+_ r))z + z2]-. Lel f(z) be meromorphic nd univlent for z < 1 with f(o) O, f (O) 1. If If(re ) q, 0 rel, then r equlily occurring only if f z eo ze-, r, q This theorem is redily derived from well known result on slit mppings, but for completeness we give its proof here on the sme lines s tht of Theorem 1. Indeed (z, r, q) mps zl < onto domin A(r, q) dmissible with respect to the qudrtic differentil (w, q)dw w2(w- q)2" Denoting by (w) the inverse of (z, r, q) defined in A(r, q), if f(reio) qeix, x rel, we hve e-f(eo(w)) n dmissible function ssocited with A(r, q). We cn now pply the Generl Coeilicient Theorem with (R the w-sphere, Q(w)dw the qudrtic differentil Q(w, q)dw, A the domin A(r, q), nd the bove dmissible function. The qudrtic differentil hs double poles P1 t the origin nd P2 t the point q. The corresponding coefficients re () -2 (2) e(x-) r, r, q) If (re). log e lq- (x-) + q-2 log [e(x-) (r, r, q)/f (reo)]} <= O, The fundmentl inequlity of the Generl Coefficient Theorem then gives OF If (re)l > (r, r, q)[. The equlity sttement follows from the finl sttement in the Generl Coefficient Theorem [2, 3] together with the normliztion f (0) We will now give the exct region of vlues of the pir (If(z) 1, ]f (z) l) for f(z) e S. THEOnEM 3. The region of vlues of the pir ([ f(z) I, f (z) I) for f(z) e S is (where z r, 0 < r < 1) the closed region L(r) of points (q, s) determined by the inequlilies (9) r/(1 + r) <= q <= r/(1 --r),

7 34 JAMES A. JENKINS 21 r (10) q r2 =<s<i=1 where is determined by eqution (1). I4 l ql (4 ll) -/(-q) 1--r2_1 r 2 qr, q-r The boundry of L(r) consists of two rcs (with common end points) Fl(r) corresponding to the lower bound in (10) nd Fe(r) corresponding to the upper bound in (10). A boundry point (q, s) on I l(r) occurs only for the functions ei(ze-io, r, q); boundry point (q, s) on Fe(r) occurs only for the functions eief(ze-, r, q, X) where X -- stisfies the bound (2). Indeed we verify immeditely tht (z, r, q) is in S for r/(1 r) <= q =< r/(1 r)e, ndtht (r, r, q) qe( 1 re)/re. On the other hnd, compring the expnsions (5) nd (7) nd the corresponding ones for < 0, we find for q r log f (r,r,q) --log (1 r2) (/(-- q)) log4 i -t-log41-- ql, or while directly for q If (r,r,q) r 4 l ql (4 ll) -/(-q) 1 r If (r, r, r) 1/(1 re). Thus from Theorem 1 nd Theorem 2 we obtin the bounds (10) together with the corresponding equlity sttements. It is verified immeditely tht f (r, r, q) is continuous t q r; thus the boundry segments re rcs s stted. Now it follows by stndrd rgument due to GrStzsch [1; 3, p. 94] tht the region of vlues of the pir ([ f(re) l, If (re) I), 0 rel, is the closed region defined by the inequlities (9), (10). The bounds (10) were given by Robinson [7], the lower bound s here, the upper bound in different forml expression which necessitted the use of different formule ccording s q < r or q r. He did not obtin the extreml functions or chrcterize the region of vlues, lthough, of course, s we hve just seen nd s is usul in most similr problems, the ltter is strightforwrd once the inequlities re obtined. 4. The result of the preceding section reduces the determintion of bounds for quntity of the form F(f(z) [, If (z)i) to problem in clculus. For the record we stte this in the form of theorem.

8 ON reighted DISTORTION IN CONFORMAL MAPPING 35 THEOREM 4. For ny function F(q, s) defined on L(r) we hve for lz < 1 min(q,)l(r) F(q, s) <= F(] f(z) ], f (z) [) -<_ mx(q,s)l(r)f(q, s). If the function F hs more restrictive properties we cn mke more precise sttements. COROLLARY 1. If the function F(q, s) is defined on L(r) nd hs no extreme vtue t n interior point of L(r), then miil(q,s),r(r)urer) F(q, s) F(I f(z I, If (z) I) mx(q,,)r,(r)ur2r)f(q, s). Equlity cn occur t most t points of Fl(r) o F(r). Further nlysis of the occurrence of n extremum on F(r) or F:(r) cn be mde on the bsis of shrper properties of F. We will not elborte this here. As n illustrtion we will solve explicitly the problem treted by Komtu, Nishimiy nd Oikw. In it the function F(q, s) is s/(1 q). It is cler tht the minimum occurs on F(r), the mximum on F:(r). For the minimum we hve the minimum of (1 r)q/r(l q) on the intervl [r/(1 r), r/(1 r)]. Since this flmction of q is strictly incresing, the minimum vlue is evidently (1 r)[r (1 r)4]-. To del with the mximum problem we observe tht for q r d log F(q, f (r, r, q)]). dq 1 _ q-- q- log41l 1 -F q" where d/dq is determined from the eqution 2(--q) (12) (d) qq-- log41l q obtined by differentiting (1). Substituting from (12) in (11) we obtin d log F(q, [f (r, r, dq q)i) (13) 1 2 2q q- q 2 - q- q( q) 1+ q q(q- )(1 + q)" This lst term is evidently positive if q < r, nd lso for q > r but sufficiently close to r. From(12) we see tht /q increses with q. Writing (13) in the form logf(q, 1 q 2/q f (r,r,q) ) dq (q- )(1 + q:) we see thut F(q, f (r, r, q)) increnses with q until we possibly meet point where -(q q), then decreses. Thus if this vlue of is possible, the

9 36 JAMES A. JENKINS mximum occurs there, otherwise for q r/(1 r)2. Since s q increses, /q increses nd 1 q2 decreses, such solution exists only if r exceeds the smllest positive root r0 of the eqution tht is, of 1 1( r r ) 4 2 (1 r) (1 r) r 4r5-l- 7r 10r A- 7r 2-4r-f It is seen t once tht this is the only such root in the intervl (0, 1). Summrizing these sttements we obtin Oikwt s result. then COnOLLARY 2. where nd q (14) log r If ro denotes the smllest positive root of the eqution r 4r5-4- 7r4-10r-4-7r 2-4r + 1 0, 1 r re io 1 r < if ( )l < r A- (1 -t- r) 1 -k If(re) 12 r -4- (1 r) 4 <- M(r), 2q q2 1 log M (r) log 1 r q2 -t- 1 q(r) is the function defined by 1/2 /q2 q ) log 2(q q), O<r<ro ro<rl (q--1) log 2(q- q) -f- log (q 1/2 -f- 1)1/2 (qe-{- 1)/- (q-- 1)/2 For the lower bound, equlity occurs only for the functions z(1 + e-ez)-2 t the point re, 0 rel. For the upper bound, 0 r <= to, equlity occurs only for the functions z(1 e-iez) -2 t the point re, 0 rel. For the upper bound, ro r 1, equlity occurs only for the.(unctions eef(ze-, r, q, X) t the point re rel, q determined from (14), nd X stisfying the inequlity (2). 5. The following remrk must certinly be known lthough the uthor cnnot recll its occurrence in the literture. Remrk. In most explicit extrcml problems treted by the method of the extreml metric nd the vritionl method, one is led to solution determined by qudrtic differentil. In the present instnce we see, however, tht the function F my redily bc chosen in Theorem 4 so tht mximizing or minimizing functions f do not correspond to qudrtic differentils. This correspondence occurs only when in some sense the problem hs mesure of linerity. It is just to such problems tht the Generl Coefficient Theorem pplies. BIBLIOGRAPHY ]. H. GROTZSCH, Tber die Verschiebung bet schlichter konformer A bbildung schlichter Bereiche, Berichte ilber die Verhndlungen der SKchsischen Akdemie der

10 ON WEIGHTED DISTORTION IN CONFORMAL MAPP1NG [7 Wissenschften zu Leipzig, Mthemtisch-Physische Klssc, vol. 83 (1931), pp JAMES A. JENKINS, A generl coeicient theorem, Trns. Amer. Mth. Soc., vol. 77 (1954), pp , Univlent functions nd conforml mpping, Berlin-GJttingen-Heidelberg, Springer-Verlg, Y. KOMATU AND H. NISHIMIYA, On distortion in schlicht mppings, KJdi Mth. Sere. Rep., vol. 2 (1950), pp K. LOWNE, ber Extremumsdtze ben der konformen A bbildung des Jusseren des Einheitskreises, Mth. Zeit., vol. 3 (1919), pp K. OIKAWA, A distortion theorem on schlicht functions, KJdi Mth. Sem. Rep., vol (1957), pp R. M. ROBINSON, Bounded univlent functions, Trns. Amer. Mth. Soc., vol. 52 (1942), pp THE INSTITUTE FOR ADVANCED STUDY :PRINCETON, NEW JERSEY UNIVERSITY OF NOTRE DAME NOTRE DAME, INDIANA