EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS. 1. Introduction

EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS MICHAEL BOLT Abtct. Loxodomic c e hown to be the mximize of inveive clength, which i invint unde Möbiu tnfomtion. Peviouly, thee c wee known to be exteml. The fit eult y tht t ny loxodomic c, the inveive clength functionl i concve with epect to non-tivil petubtion tht fixe the cicle element t the endpoint. The econd eult y tht mong cuve with monotone cuvtue tht connect fixed cicle element, the loxodomic c uniquely mximize inveive clength. Thee eult pove conjectue mde by Liebmnn in 193. 1. Intoduction In 193, Heinich Liebmnn [6] intoduced notion of clength tht i invint unde Möbiu tnfomtion of the complex plne. The quntity i clled inveive clength nd depend on thee deivtive of the pmeteiztion. Peviou utho knew the coeponding diffeentil invint, clled inveive cuvtue, which depend on five deivtive of the pmeteiztion. Tken togethe, thee notion of clength nd cuvtue completely detemine the inveive diffeentil geomety of plne cuve. Togethe, they exemplify Klein Elngen pogm fo the goup SL, C), nd they hve been of ongoing inteet duing much of the twentieth centuy. See [, 7, 8, 10], fo intnce. Befoe 193, it w known tht the cuve with contnt inveive cuvtue e the logithmic pil nd thei Möbiu imge, the loxodome. In hi ppe, Liebmnn howed tht thee cuve e lo the exteml of inveive clength. Anothe poof of thi fct w given by Med in [8].) Motivted by nlogou eult fom ffine geomety, Liebmnn futhemoe conjectued the following. Conjectue. Liebmnn, 193) Among cuve connecting fixed cicle element, the loxodomic c mximize the inveive clength. In thi ppe we pove Liebmnn conjectue. In poving the conjectue, we etblih two pincipl intemedite eult tht eem not to hve been peviouly known nd my be of independent inteet. The fit of thee, locl eult, y tht petubing loxodomic c eult in n c with tictly mlle inveive clength. Theoem. At loxodomic c, the inveive clength functionl i concve with epect to ny thee time diffeentible petubtion tht fixe the cicle element t Dte: Mch 4, 006. 000 Mthemtic Subject Clifiction. Pimy: 53A35. Secondy: 53A40, 58E35. 1

MICHAEL BOLT the endpoint. In pticul, loxodomic c e tict locl mximize of inveive clength. Ou econd bic geometic eult i globl one; to povide ntul fomultion we will fit intoduce pi of invint fo mooth c, clled epectively the Kezmn-Stein nd Coxete invint. In pt, the Kezmn-Stein invint detect cuve iotopy cl, viewed inide the extended complex plne. Two cuve e id to gee inveively to econd ode t the endpoint if thei coeponding invint gee. Theoem. Conide thee time diffeentible cuve with monotone cuvtue tht gee inveively to econd ode t the endpoint. Among them thee i exctly one loxodomic c, up to Möbiu tnfomtion, nd thi c uniquely mximize the inveive clength. We mention tht the nlogou eult in Eucliden geomety i the fmili fct tht, with epect to clength, the only exteml pth between two point i tight line egment, nd thi pth minimize the clength. Thee i lo eult in ffine geomety tht y tht fte pecilizing to convex cuve, the pbolic c hve contnt zeo) ffine cuvtue, nd thee cuve uniquely mximize the ffine clength mong cuve tht connect fixed line element. See Blchke [1, p.40], fo intnce. The ppe i tuctued follow. In Section, we eview the bic notion of inveive diffeentil geomety, we explin the neceity of eticting to cuve with monotone cuvtue, nd we intoduce pi of invint fo mooth c. In Section 3, we give the pecie ttement of ou min eult, nd in Section 4 nd Section 5, we give thei poof. In Section 6, we ecod two dditionl fct tht emege fom the poof in the peviou ection. The utho thnk Dvid E. Bett fo mny helpful convetion duing the peption of thi ppe.. Peliminie In thi ection, we povide bief oveview of inveive diffeentil geomety, nd we decibe pi of invint fo mooth c..1. Inveive clength nd cuvtue. In one dimenion, inveive geomety efe to the tudy of geometic tuctue tht behve invintly with epect to the ction of the Möbiu goup SL, C) = { µ = µz) = z + b cz + d } :, b, c, d C, d bc = 1 on the complex plne. The goup lw i given by compoition. Of pticul inteet e the integl nd diffeentil invint of mooth cuve γ C, which cn be decibed explicitly in tem of thei Eucliden countept. We biefly ecll the definition hee nd efe to Cin nd Shpe [] nd Ptteon [9] fo moe extended tetment.

EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS 3 If κ = κ) give the Eucliden cuvtue of γ function of the clength pmete,, then the inveively invint one-fom i dλ = κ ) 1/ d nd the inveive length of γ i Lγ) = dλ. At time it will be helpful to ue pmeteiztion fo cuve with epect to the inveive clength pmete; fo intnce, γ γ = γλ) with dγ/dλ κ ) 1/. Defining inveive clength uully equie tht cuve e thee time diffeentible. Moeove, to void n mbiguity tht occu whee κ chnge ign, it i common to etict to cuve with monotone cuvtue. Cuve with monotone cuvtue hve the popety tht thei oiented oculting cicle e popely neted. Thi men tht the egion they bound dic, hlf-plne, o complement of dic) e neted inide ech othe. Möbiu tnfomtion theefoe peeve cuve with deceing ep., inceing) cuvtue. We emk tht whethe cuve h inceing o deceing cuvtue doe not depend on it oienttion. If γ i five time diffeentible, then it inveive cuvtue i the fifth ode invint I 5 = 4κ κ κ )κ 5κ ) 8κ ) 3. The cuve with contnt I 5 e the loxodome, tht i, the Möbiu imge of the logithmic pil. A logithmic pil i decibed mot imply uing R e α C fo ome α C, Re α) 0 Imα). Such pil inteect cicle centeed t the oigin in contnt ngle. In genel, the vlue of I 5 fo cuve t point coepond with the ngle of the loxodome tht bet ppoximte the cuve t tht point. See Med [7] fo othe geometic intepettion of inveive cuvtue. Logithmic pil hve one finite pole nd one pole t infinity. Loxodome genelly hve two finite pole, nd fo thi eon, they e ometime clled logithmic double pil. Finlly, we mention tht when γ h monotone cuvtue, it cn be ecoveed up to Möbiu tnfomtion fom it intinic eqution, I 5 = I 5 λ). Futhemoe, the inveive cuvtue i infinite t vetex, tht i, point of ttiony cuvtue. Cicle nd line hve eveywhee infinite inveive cuvtue; thei inveive clength i zeo... The Kezmn-Stein nd Coxete invint fo mooth c. We hee decibe pi of fit nd econd ode invint fo twice diffeentible c. They e expeed uing ditnce function on the pce of line element nd cicle element, epectively, though thee ditnce function e not ditnce in the uul ene. We y γ connect cicle element p, φ p, κ p ) nd q, φ q, κ q ) if it endpoint e p nd q, it tngent vecto thee hve ngle φ p nd φ q, nd it cuvtue thee e κ p nd κ q, epectively. In thi nottion, the ngle φ p nd φ q e not unique, the they e detemined only up to multiple of π. Fo line element p, φ p ) nd q, φ q ), the Kezmn-Stein ditnce i the diffeence in ngle between the vecto expiφ p ) t p nd the vecto gotten by eflecting the

4 MICHAEL BOLT vecto expiφ q ) t q co the chod connecting p to q. It i given by ) q p 1) θp, φ p ; q, φ q ) = g q p e iφq+φp). Kezmn nd Stein encounteed thi ngle in thei tudy of the Cuchy kenel; ee [5].) Then, fo n c γ tht connect line element p, φ p ) nd q, φ q ), the fit ode invint θ = θ γ i defined uing the ight hnd ide of 1). We chooe the bnch of the gument function tht mke θ γ p, q ) continuou function of q γ whoe vlue t q = p i zeo. The quntity in penthee on the ight hnd ide of 1) ppoche 1 q p.) In thi wy, the θ invint lo identifie cuve iotopy cl, viewed inide the pce of line element on the extended plne Ĉ = C { }. Figue 1 how loxodomic c tht both connect 0, 0) nd 1, π/4) but with θ invint tht diffe by π. 0. 0. 0. 0.4 1.0 0. 0. 0. 0.4 1.0 Figue 1. Loxodomic c tht connect line element 0, 0) nd 1, π/4) with θ invint tht diffe by π. Fo noninteecting cicle, the Coxete ditnce ee [3]) i the quntity δ = coh 1 d 1 )/ 1 ) whee the cicle hve diu 1 nd, nd the ditnce between thei cente i d. Coepondingly, fo cicle element p, φ p, κ p ) nd q, φ q, κ q ), the ditnce i ieiφp p + ) q + ieiφq ) ) δp, q) = coh 1 κ p κ q 1 κ 1 p κ q 1 1. κ p κ q In pticul, fo twice diffeentible cuve γ connecting thee two cicle element, the econd ode invint δ = δ γ i defined uing the ight hnd ide of ). It will not be necey to implify thi expeion fo genel cuve.

EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS 5 Thee ditnce function e not ditnce in the uul ene ince neithe of them tifie genel tingle inequlity. Moeove, the Coxete ditnce i zeo fo cicle tht e tngent to ech othe, nd the Kezmn-Stein ditnce i zeo fo line element tht e tngent to common cicle. By eticting to cuve with monotone cuvtue, howeve, we eliminte thee degenecie. In fct, by eticting to cuve with deceing cuvtue we my ume tht both invint e poitive. Fo n explntion why the θ invint i poitive, ee ubection 6.1. Fo the δ invint, it i umed tht one ue the poitive vlue of coh 1 in ). 3. Sttement of Min Reult Liebmnn [6] howed tht the exteml of inveive clength e the loxodomic c, ubject to petubtion tht fix the cicle element t the endpoint. Med lo poved thi fct in [8, p.56]. We how the loxodomic c e, in fct, mximize of inveive clength. Ou fit eult i locl veion of thi ttement. Theoem 1. At loxodomic c, the inveive clength functionl i concve with epect to ny thee time diffeentible petubtion tht fixe the cicle element t the endpoint. In pticul, loxodomic c e tict locl mximize of inveive clength. Ou econd eult i globl veion. By conideing only cuve with deceing cuvtue, we my ume tht both of cuve invint e poitive. We mention tht the endpoint cicle element only detemine cuve θ invint up to multiple of π, o by pecifying the θ invint in Theoem, we equie tht ll cuve belong to the me iotopy cl. Theoem. Conide thee time diffeentible cuve with deceing cuvtue tht connect two fixed cicle element nd hve the me θ invint. Among them thee i exctly one loxodomic c, nd thi c uniquely mximize the inveive clength. A Möbiu tnfomtion cn end point on cuve to the point t infinity, o it i poible tht the exteml c will p though the point t infinity. Fo thi eon, we lo peent the eult in moe ntully inveive etting, without pecific efeence to the endpoint. Theoem 3. Conide thee time diffeentible cuve with monotone cuvtue tht gee inveively to econd ode t the endpoint. Among them thee i exctly one loxodomic c, up to Möbiu tnfomtion, nd thi c uniquely mximize the inveive clength. In thi fomultion, two cuve e id to gee inveively to econd ode t the endpoint if they hve the me θ, δ) invint. We lo mention tht the eult fo cuve with deceing cuvtue immeditely extend to cuve with inceing cuvtue. Fo intnce, unde conjugtion z = x+iy z = x iy), cuve with deceing cuvtue become cuve with inceing cuvtue; menwhile, thei

6 MICHAEL BOLT inveive clength i unchnged. Loxodome with inceing cuvtue e lo the Möbiu imge of logithmic pil defined in Popoition 1, fo < 0. Following thee obevtion, Theoem 3 follow diectly fom Theoem nd Lemm 6. In the finl ection we povide evidence tht ugget thee eult e optiml. Fo intnce, when conideing only c with the me θ invint, it i poible to mke the inveive length bitily lge o mll, even within the fmily of loxodomic c. Fo thi eon, it i necey to include both invint when fomulting the poblem. 4. Poof of Theoem 1 We ue the vitionl ppoch to how tht loxodomic c e tict locl mximize of inveive clength. By uing n ppopite Möbiu tnfomtion, we my ume tht the loxodomic c i n c fom logithmic pil. Let Lz;, t) denote the inveive length of n c pmeteized by [, t] z). Popoition 1. Suppoe [, t] z = z) = e i log /1 + i) pmeteize logithmic pil fo ome > 0. Thi i pmeteiztion by clength. Conide thee time diffeentible function p : [, t] p = p) R which tify i) p = p t = 0, ii) p = p t = 0, nd iii) p t /p = /t, nd et z p,ǫ = z + iǫp z fo ǫ R. So z p,0 = z fo ll. Then, fo ech uch p, 3) Lz p,ǫ ;, t) = Lz;, t) + ǫ R p;, t) + oǫ ), whee R p;, t) 0. Thee i equlity if nd only if p 0. Poof of Theoem 1. Theoem 1 follow fom Popoition 1 once we how tht the condition on p e tified fo ny petubtion tht fixe the cicle element t the endpoint. The fit two condition on p y peciely tht the petubtion hould fix the line element t the endpoint. Unde thee condition, the lt condition y it hould lo fix the Coxete invint; we omit the detil of thi lt fct. Fo Theoem 1, howeve, it i eentil to know tht the thid condition depend on nothing beyond the econd ode infomtion t the endpoint. It i imple, then, to veify tht it y the petubtion hould fix the tio of cuvtue t the endpoint. Fo thi, let u tempoily ume the concluion of Lemm 1. Then, to fit ode in ǫ, the tio of cuvtue fo the petubed cuve i /t + ǫp t )// + ǫp ). Thi equl the tio of cuvtue fo the unpetubed cuve peciely when p t /p = /t)//) = /t. Poof of Popoition 1. To implify the nottion we lo wite γ = z p,ǫ γ = dγ d = z + iǫp z + iǫp i z = z [ p 1 ǫ ip. Then, )].

EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS 7 Suppoe tht u = u) i the clength pmete fo γ. Then dγ du 1 dγ d d du = 1 ǫ p ) + ǫp ) d du, o tht 4) nd du d = 1 + 1 d du = ǫ p + ǫ p ) + ǫ p ) 1 8 ǫ p ) + oǫ ) = 1 ǫ p + ǫ p ) + oǫ ), ) 1 du = 1 + ǫ p d + p ) ǫ p ) + oǫ ). Fom now on, we will intepet the equl ign to men equl only up to tem of econd ode in ǫ. Tem of ode oǫ ) will be counted zeo. Lemm 1. Neglecting tem of ode oǫ ) the cuvtue of γ t γ i given by k = ) + ǫ p 1 + p + ǫ p ) + p p p p + 3 p ) 3. Futhemoe, dk d = + ǫ p p 3 + p + p p 3p p ) 3p + ǫ p + p p 3 3 + 3 p p 3 p ) 33 p 4 ). Poof. We fit expe the cuvtue of γ in tem of the coodinte: ) ) ) ) d dγ d dγ d d d dγ d d d du du d d du du d d du d du k = i dγ = i dγ = d i dγ +. i du d du d It follow tht d [ z p ))] [ 1 ǫ d ip 1 + ǫ p + p ǫ k = p ) )] [ i z p )] 1 ǫ ip + 1 [ d 1 + ǫ p i d + p )] ǫ p ) = [ 1 + ǫ p + p ǫ p ) )] ǫ [ p i p ] [ ip 1 + ǫ p ][ p )] 1 + ǫ ip + 1 [ p ǫ i p ) [ + ǫ p p p ) ]] p p.

8 MICHAEL BOLT The tem tht hve the fcto of ǫ 1 e peciely p 1 p i p ) ip + 1 p i p ) nd the tem tht hve the fcto of ǫ e peciely p ) p ) 1 p i p ) ) p ip ip + 1 [ p p i p ) = 3 p 3 p ) p + p p ) ) + p p [ p p +i p ) p p p p p = p + p, ] p p ) ] + p p = 1 p ) + p p p p + 3 p 3, climed by the lemm. The expeion fo dk/d i then ey to check; we kip the few detil. Next, the inveively invint one fom cn be witten dk 1/ du du = dk d d 1 du du d d = dk d du 1 d d. Lemm. Neglecting tem of ode oǫ ), we hve dk d du 1 d = + ǫ A 1) + ǫ A ), whee nd A ) = + A 1 ) = [ p p 3p p + p ) p + p ) p + p 3 1 4 + 3p p Poof. Uing 4) nd Lemm 1, we find tht whee nd fte implifying, B ) = 3p p p p ) ) 1 3 p ) 4 p p p p dk d du d = + ǫ B 1) + ǫ B ), B 1 ) = p p ) + p p p 3 + p, 3p p + p p 3 p p + 3 p p 3 )]. 3 p 4.

EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS 9 Then by witing dk d du d = ) 1 ǫ B 1) ǫ B ), we hve dk d du d 1/ = [1 )] ǫ B 1) + ǫ B ) 4 8 B 1). We hve left then to expnd nd implify A 1 ) = B 1 )/ ) nd A ) = B )/ ) 3 B 1 ) /8 3/ ). We find tht B 1) = p nd B ) 3 B 1 ) = = = 8 3/ [ 3p p + 3 p p 3 [ 3p p + p ) + p 3 [ p p climed by the lemm. 4 + 3p p + p ) ) p 3 + p = p ) + p p 3 p ) 4 3 4 p p + 3p p p p ) 1 + p 3 1 p 4 p + 3p p p + p ) p p 3p p p p + p p 3 p 3 + p p p p p )p 3 ) ) 1 3 p ) p p p p 4 ) ] ] 4 p ) p p )] So f, fte neglecting the tem of ode oǫ ), [ ] Lz p,ǫ ;, t) = + ǫ A 1) + ǫ A ) d. Hee, / d = logt/) = Lz;, t), the inveive length of the unpetubed logithmic pil. Futhemoe, A 1 )d = p = 3/ + p d d p p ) d ) d 1 p d.

10 MICHAEL BOLT The fit integl in the lt expeion i zeo ince p = p t = 0. The econd integl cn be evluted uing integtion by pt t p d = p p d = tp t p ) p t p ). Thi vnihe, too, ince p t /p = /t nd p = p t = 0. It follow tht A 1)d = 0, nd fo thi eon thee e no fit ode tem on the ight hnd ide of 3). Thi lo confim the ledy known fct tht the loxodomic c e exteml. We hve yet then to veify tht R p;, t) 0 with equlity peciely when p 0. Notice tht both p p + p 3 d = 1 ) d p d d = 0 nd d d p ) d = 0 3p p d = 3 ince p = p t = 0 nd p = p t = 0. Fo the me eon, p ) + 3p p p p ) d = d p p ) d + d We cn then wite R p;, t) = [ t A )d = 3 p ) 4 d + To futhe implify, we ue the following. p p d = 4 p p ) d p p p p d. 1 p p + p p )d ]. Lemm 3. nd p p d = 1 p p ) d p p + p p d = p ) d. Poof. Fo the fit integl, we fit integte by pt: p p d = p p t t p p p ) d = p p p ) d. Next, define q = p / o tht q = p / p /. Then lo p = q + p / = q + q. We then hve p p d = q + q )q d = = q ) d q t q ) d 1 = q q d p p ) d.

EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS 11 In the lt tep we ue the fct tht q = q t = 0. Fo the econd integl, gin integte by pt: p p + p p d = p + p )p = 0 p ) t t t p p + p p ) d = )d p ) d. The lemm i then poved. It follow tht R p;, t) = 5) whee [ 4 1 p p ) 1 d 4 + 1 p p = 1 8 3/ [ 4 X + 4X + Y ) Z ], X = Y = Z = 1 p p p ) d 3 p ) d. ) d 3 p ) d ) + 1 p ) )d In 5), the quntity in bcket i qudtic in, nd h diciminnt = 4X+Y ) 4XZ. We clim tht the diciminnt i negtive when p i nonzeo; evidently it i zeo when p 0. Thi uffice to pove Popoition 1 fo the following eon. Fo p fixed, the gph of the quntity in bcket, function of, open downwd. If the diciminnt i negtive, thi gph neve coe the hoizontl xi. So the quntity in bcket i negtive fo ll vlue of. A thi would be tue except when p 0, we will hve etblihed Popoition 1. To pove the clim, we intoduce new ubtitution. Let = e µ nd d = e µ dµ. Alo, let y = yµ) = p e µ )e µ pe µ ). Then y = p e µ )e µ nd y = p e µ )e 3µ + p e µ )e µ, ] nd p p = y e µ, p = y e µ, We next ue the following two lemm. nd p = y y e 3µ.

1 MICHAEL BOLT Lemm 4. X = Y = Z = =t = =t = =t = y e µ dµ y ) e µ dµ y ) e µ dµ Poof. The fit two integl e immedite: 1 X = p p ) =t 1 y ) d = e µ e µ e µ dµ = nd Y = Fo the lt integl, Z = = =t p ) d = 3 p ) d = = =t = =t = y ) e µ dµ 4 It then uffice to how tht =t = Agin, integte by pt: =t = = ) y e µ e µ e µ dµ = y e 3µ y ) e µ dµ =t = e 3µ y y e µ dµ = y y e µ dµ + 4 =t = y y e µ dµ = 1 y ) e µ =t The boundy tem vnih ince o the lemm i poved. = =t = =t = y ) e µ dµ. + =t = =t = y e µ dµ y ) e µ dµ. y ) e µ dµ. y ) e µ dµ. y ) e µ =t = = p ) =t = = tp t ) p ) = 0, Lemm 5. X Y = =t = yy e µ dµ Poof. Stting with the expeion fo Y fom the peviou lemm, we integte by pt. Then, Y = =t = y ) e µ dµ = yy e µ =t = =t = = =t yy e µ du + yy y )e µ dµ =t = = yy e µ dµ,

EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS 13 the boundy tem vnihing ince y = p p = 0 fo =, t. Integting by pt in the econd integl on the ight hnd ide give Y + =t = yy e µ dµ = y e µ =t + = =t = y e µ dµ = X. In the econd tep, the boundy tem vnih fo the me eon befoe, o the lemm i poved. Uing thee lemm, we pply the Cuchy-Schwz inequlity to the function ye µ nd y e µ : X Y ) =t y e µ dµ =t = = y ) e µ dµ = X Z. Fom thi it follow tht = 4 [X Y ) XZ] 0 nd we e nely done. Cuchy-Schwz lo y thee i equlity only if one of the following i tue: i) y 0 ii) y 0 iii) cye µ = y e µ fo ll µ; tht i, y cy 0. Hee, c 0 i contnt. In ech ce, we mut how tht p 0. If y 0, then p p = 0 fo ll, o p /) = 0 nd p = c. But p = p t = 0, o p 0. If y 0, then y i line. But befoe, y = 0 fo both =, t, o then y 0. The gument jut given implie p 0. In the finl ce, if c i poitive, thee e no nontivil olution fo y tht vnih t =, t. If c = λ, thee e mny olution [ ] nπ yµ) = in µ log ), logt/) with λ = nπ/ logt/) nd n N. But thee emin the etiction on y tht y y 1 =t µ) e µ = p =t = = p t t p = 0. = Fo thi to hold, it i necey tht conπ)/t 1/ = 0, which i impoible, t > 0 nd t. Agin, thee e no nontivil olution, o Popoition 1 i poved. 5. Poof of Theoem To pove Theoem, we fit detemine how the cicle element t the endpoint of n c cn be nomlized with epect to the cuve invint θ, δ). Then we how tht ech pi of invint θ, δ) i obtined exctly once, up to Möbiu tnfomtion, within the fmily of loxodomic c. Finlly, we etblih context in which the logithmic pil e globl mximize of inveive clength. With thee fct in hnd, we e then edy to pove Theoem.

14 MICHAEL BOLT Lemm 6. Uing Möbiu tnfomtion, the endpoint cicle element p, φ p, κ p ) nd q, φ q, κ q ) of twice diffeentible c γ with deceing cuvtue cn be nomlized o tht p = 0, φ p = 0, κ p = 1, nd q = 1. Afte the nomliztion, the vlue of φ q to multiple of π) nd κ q e detemined by the invint θ, δ). Poof. To pove the lemm, we exhut the ix degee of feedom tht e vilble in SL, C). We fit ue tnltion tht mke p = 0 nd follow tht with ottion nd diltion tht mke q = 1. Thi ue fou degee of feedom, but we my now ume tht γ i nomlized with p = 0 nd q = 1, nd we hve left the ubgoup of Möbiu tnfomtion tht fix p = 0 nd q = 1. Thee Möbiu tnfomtion hve the fom µz) = d 1 z/cz + d) fo ome 0 d C, with c = 1/d d. We clim we cn chooe 0 d C o tht φ p = 0 nd κ p = 1. Next, µ z) = cz + d) nd the unit tngent vecto of the cuve µ γ t p = 0 i d/d)expiφ p ). Afte eplcing d with d expiφ p /) whee 0 d R, thi tngent vecto i 1. So we hve lo nomlized φ p = 0. We my ume then tht γ i nomlized with p = 0, φ p = 0, nd q = 1, nd we hve left the ubgoup of tnfomtion of the fom µz) = d 1 z/cz + d) fo 0 d R, with c = 1/d d. Chooing d o d eult in the me Möbiu tnfomtion, o without lo of genelity, ume d > 0. We clim we cn chooe 0 < d < o tht κ p = 1. At thi point, we my ume tht κ p > 0 ele γ could neve ech q = 1, the it would pil inide the cicle centeed t iκp 1 with diu κ p 1. Suppoe now tht γ) i pmeteiztion by clength, nd = ) i defined o µ γ) i lo pmeteiztion by clength. Then the cuvtue of µ γ cn be expeed by ) ) ) d µ d dµ d d dµ d cγ + d dγ d d d d d d d cγ + d d i dµ = i dµ = d i dµ =. 1 dγ i d d d d cγ + d) d At p = 0, whee ledy φ p = 0 o γ = 0, dγ/d = 1, nd d γ/d = iκ p ), we find tht the cuvtue of µ γ i d κ p. Chooing d = κ 1/ p mke thi cuvtue equl to 1. Finlly, fte the nomliztion, the vlue of φ q nd κ q cn be ecoveed fom the invint θ, δ) by uing 1) nd ). Solving 1) give φ q = θ. Then olving ) give two poibilitie fo κ q, nmely κ ± q = ± cohδ + coφ q + in φ q ). Of thee poibilitie, only κ q = κ q give n oiented) cicle element 0, φ q, κ q ) tht i popely neted with the cicle element 0, 0, 1). Next we how tht ech pi of invint θ, δ) i obtined exctly once, up to Möbiu tnfomtion, in the fmily of loxodomic c. Lemm 7. Ech pi of invint θ, δ) with θ, δ > 0 i obtined exctly once, up to Möbiu tnfomtion, within the fmily of loxodomic c. In pticul, the mp, v) θ, δ) detemined below uing 6) i both one-to-one nd onto.

EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS 15 Once thi i poved, we my conclude fom Lemm 6 nd Lemm 7 the following intemedite eult. Popoition. Given n c γ with deceing cuvtue, thee i peciely one loxodomic c γ tht connect the me cicle element γ nd h the me θ invint γ. To pove Lemm 7 it uffice to conide c of the logithmic pil zu) = exp [1 + i)u/ ] 1)/1 + i) fo > 0. Then lo z u) = expiu ) nd z u) = i exp u/ ) z u) whee the pimed nottion indicte diffeentition with epect to clength. Fo thi pil, the pmete u i the inveive clength pmete nd cn be elted to the clength pmete by u = log. Ech uch pil h one pmete fmily of ymmetie point on logithmic pil cn be tken to ny othe point on the pil by n ppopite tnltion, ottion, nd diltion. We chooe one endpoint to be z0) = 0. Afte Möbiu tnfomtion, then, we conide only the c of logithmic pil 6) u [0, v] zu) = exp [ 1 + i)u/ ] 1 ) /1 + i), tht connect cicle element 0, 0, ) nd zv), v, exp v/ )), fo, v > 0. The Kezmn-Stein invint fo logithmic pil. Hee, θ i exctly the gument of the vecto tht i gotten by eflecting the tngent vecto z v) co the line egment connecting z0) = 0 to zv). We chooe the bnch of the gument to be the one tht mke θ into continuou function tting with θ = 0 t v = 0. Theefoe, ) ) zv) θ, v) = g 1 i e 1+i)v/ 1 zv) e iv = g 1 + i e 1 i)v/ 1 e iv ) 1 i e 1+i)v/ ) e 1+i)v/ ) = g 1 + i e 1 i)v/ ) e 1 i)v/ ) 7) = g 1 i)inh [ 1 + i)v/ ) ]). Uing the identity inhu + iv) = inhucov + i cohu inv, we hve θ, v) = tn 1 coh v ) in v ) inh v ) co v ) inh v ) co v ) + coh v ) in v ) = tn 1 tnv ) tnh v 8) ). tnh v ) + tnv ) Fo fixed > 0, we chooe the bnch of tn 1 tht mke the ight hnd ide ppoch 0 when v ppoche 0. Then we extend continuouly fo v > 0. The Coxete invint fo logithmic pil The dii of the oculting cicle t the endpoint e ecipocl to the endpoint cuvtue nmely, 1/ nd

16 MICHAEL BOLT expv/ )/. The cente of thee cicle e i/ nd zv) + iz v) expv/ )/, nd the ditnce-qued between them i i e1+i)v/ 1 i e iv ev/ 1 + i i1 + i) e 1+i)v/ 1) i1 + i)e 1+i)v/ = 1 + i) i ie 1+i)v/ = = 1 + ev/ e v/ cov ) 1 + i) 1 +. ) So we get 1 + e v/ e v/ cov ) δ, v) = coh 1 1 + 1 ) ev/ 1 ev/ = coh 1 1 + e v/ e v/ cov )) 1 + )1 + e v/ ) e v/ 1 + ) = coh 1 1 + e v/ ) 1 + ) e cov ) v/ 1 + = coh 1 cohv/ ) + cov ) 1 +. In Figue we how contou plot fo the function θ = θ, v) nd δ = δ, v) 50 50 5 5 0 0 5 50 0 0 5 50 Figue. Contou plot fo θ = θ, v) nd δ = δ, v). which wee dwn fo the egion 0.01, v 50.

EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS 17 To pove the injectivity of the mp, v) θ, δ) it i enough to veify tht the tngent line to the level cuve of θ hve negtive lope, nd the tngent line to the level cuve of δ hve poitive lope. Then ech level cuve of θ inteect ech level cuve of δ t mot once. Fo the lope of the line tngent to level cuve of δ = δ, v), we diffeentite implicitly the quntity cohδ = cohv/ ) + cov ) 1 +. It i mino point tht the quntity on the ight hnd ide i poitive fo, v > 0.) Then, v = cohδ) δ=cnt = v coh δ) coh v ) v inh v ) v inv ) 1 + coh v ) + cov )) 1 + ) ] 1 + 3/ inh v ) inv ) = 43/ coh v )+4 3/ cov )+v1+ ) inh v )+inv )) 1 + ) inh v ) inv. )) Likewie, fo the lope of the line tngent to level cuve of θ = θ, v), we diffeentite implicitly the quntity tn θ = tn[v /] tnh[v/ )] tnh[v/ )] + tn[v /] Then, v = tnθ/)) v θ=cnt tnθ/)) = = 1 + )f g fg ) f + g ) 1 + )f v g fg v ) def = f, v) g, v) g, v) + f, v). 1 + v ) ec v 8 3/ )ech v ) inh v ) + inv )) ec v )ech v )coh v ) cov ))/ 1 + )/4 ) ec v )ech v ) inh v ) inv )) = 43/ coh v ) 4 3/ cov ) v1+ ) inh v )+inv )) 1 + ) inh v ) inv. )) Some tep e omitted fom the next-to-lt computtion they ue the hypebolic identity well the tigonometic nd hypebolic double ngle fomulæ. It o hppen tht thi lope i exctly oppoite the lope tht w gotten fo δ. The utho fit dicoveed thi cuiou fct uing Mthemtic. Fo the injectivity, we fit how tht inhv/ ) inv ) > 0 fo, v > 0. Fo thi, ubtitute y = v/ > 0 nd conide fy) = inhy iny) fo fixed

18 MICHAEL BOLT. Then f0) = 0, nd f y) = cohy coy) 0 with equlity if nd only if y = 0. So then fy) > 0 fo ll y > 0, nd the etion i poved. To pove injectivity, then, it i enough to veify tht 9) 4 3/ cohv/ ) 4 3/ cov ) v1 + ) inhv/ ) + inv )) < 0 fo ll, v > 0. Poof of inequlity 9). Afte ubtituting = z/y nd v = yz o y = v/, z = v ), nd fte multiplying by y 5/ / z nd enging tem, inequlity 9) i equivlent to 4yzcoz cohy) + yy + z )in z + zy + z )inhy > 0 fo ll y, z > 0. To etblih thi inequlity, we expnd the left hnd ide in powe eie in tem of y. Then, LHS = 4yz coz 4yz j= j=0 y j j)! + yy + z )in z +zy + z ) j=0 y j+1 j + 1)! = y4z coz 4z + z inz + z 3 ) + y 3 z + inz + z + z 3 /6) y j 4yz j)! + y j 1 zy j 1)! + y j+1 z3 j + 1)!. j= Clely the lt of the five tem on the ight hnd ide i poitive. We how fit tht the um of the 3d nd 4th tem i poitive 4yz j= y j j)! + zy j= y j 1 j 1)! = yz j= j= ) 4 + j y j > 0. j)! To etblih 9) it i enough then to veify the inequlitie fo z > 0: 10) 11) 4 coz 4 + z in z + z > 0 z + in z + z 3 /6 > 0. To veify 11), let fz) = z + in z + z 3 /6. Then f z) = 1 + coz + z / nd f z) = inz + z. Since f0) = 0 nd f 0) = 0, nd f z) 0 fo z 0 with equlity only fo z = 0, we conclude tht fz) > 0 fo z > 0. Thi etblihe 11). To veify 10), let fz) = 4 coz 4+z in z +z, o f z) = 3 inz +z coz +z. Then both f0) = 0 nd f 0) = 0, o it i enough to how tht f z) > 0 fo z > 0. Since 3 inz + z coz 3 + z, it follow tht f z) z 3 + z) = z 3 > 0 fo z > 3. We mut then check tht f z) > 0 fo 0 < z 3, nd fo thi we ue

EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS 19 powe eie 3 inz + z coz + z = 3 z + Hee, = j=1 j=1 1) j z j+1 j + 1)! k=1 + z 1 + j=1 1) j z j + z j)! ] [ 3 1)j j + 1)! + 1)j z j+1 = 1) j j j)! j + 1)! zj+1 j= [ ] 4k 4k = 4k + 1)! z4k+1 4k + 3)! z4k+3 = k=1 [ 4k )4k + 3)4k + ) 4k z ] z 4k+1 4k + 3)!. 4k )4k + 3)4k + ) 4k z 4k )4k + 3)4k + ) 36k = 64k 3 + 48k 5k 1 > 0 ince 0 < z 3 nd k 1. Thi etblihe 10), nd theefoe 9) well. Wht need to be poved, then, i the ujectivity of, v) θ, δ). Fo thi, we fit how tht if > 0 i fixed, then θ, v) incee with v nd ttin ll poitive vlue. We then etict to level cuve of θ, whee we my ume v = v), nd we etblih both 1) lim δ, v)) = + 0 + 13) lim δ, v)) = 0. + Thi i enough to etblih ujectivity, then Lemm 7 nd Popoition will be poved. Fo the fit ttement, we tt with 7) nd find [ )] θ 1 + i)v = g 1 i)inh = tn 1 + tn 1 [ tnv /) tnhv/ )) With > 0 fixed, we chooe the vlue fo tn 1 tht i between 0 nd π/. Fo tn 1 [tnv /)/ tnhv/ ))] we then ue the vlue tht mke the um equl to zeo when v = 0, nd extend continuouly fo v > 0. We fit how tht v tnv /)/ tnhv/ )) i n inceing function wheeve it i defined. ].

0 MICHAEL BOLT Fo thi we mke the ubtitution x = v / nd compute d dx tnx tnhx/) = ec xtnhx/) 1 tn xech x/) tnh x/) = ec xech x/) tnh x/) = ec xech x/) tnh x/) [ inhx/) cohx/) in x co x] [ inhx/) in x]. In the pgph peceding 9), we howed tht the lt quntity in bcket i poitive, o then both x tn x/ tnhx/) nd v tnv /)/ tnhv/ )) e inceing. Futhemoe, if v i lge, then tnhv/ )) 1, nd θ, v) tn 1 + tn 1 tnv /) ) tn 1 + v. The quntity on the ight hnd ide of thi etimte cn be mde bitily lge by tking v lge. We hve left then to veify the two limit, 1) nd 13). Poof of 1)-13). Agin we ue the ubtitution x = v /. Then on the level cuve of θ, whee x = x), we hve 14) tn θ/ + tn 1 ) = nd tn x tnhx/) 15) cohδ, v)) = cohx/) + cox 1 +. Fo the fit limit, tt with vlue of tht e mll enough o tht if 0, then tnθ/ + tn 1 ) tnθ/). In ce θ i n odd multiple of π, tke tnθ/) =. Then, fte eticting to level cuve of θ, we clim tht x = x) i bounded below fo thee mll vlue of. Once thi i known, then 0 +, cohδ, v)) expx/)/ + cox 1 + expx/)/ +, nd the fit limit i poved. To pove the clim, conide the ce 0 < θ < π. If x = x) i not bounded below, then thee i equence { n } with n 0 o tht x n ) 0. But then, ince tn x/ tnhx/) = tnθ/ + tn 1 ) tnθ/) 0, uing 14), it would follow tht tnhx n )/ n ) 0. Thi would lo men tht tnx n ) tnhx n )/ n )) x n) x n )/ n = n 0 tnθ/), contdiction. So the clim i tue fo 0 < θ < π. Fo the ce θ π, the clim follow fom the peviou ce nd fom the fct, hown bove, tht θ, x) incee with x fo ny fixed vlue of. In pticul, uppoe thee i equence { n } with n 0 nd x θ n ) 0, whee x θ = x θ ) i defined uing the level cuve coeponding to fixed θ π. Fo the me equence, one h x θ n ) < x θ n ) fo ny 0 < θ < π, ince θ = θ, x) incee with x. Thi then men tht x θ n ) 0, contdicting the clim fo the peviou ce.

EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS 1 Fo the econd limit, tt with vlue of tht e lge enough o tht if, then tnθ/ + tn 1 ) tnθ + π)/). In ce θ i n even multiple of π, tke tnθ + π)/) = +. Then, on the level cuve of θ, we clim x = x) i bounded bove fo thee lge vlue of. Once thi i known, then +, cohδ, v)) = cohx/) + cox 1 + 1 1, nd the econd limit i poved. The clim follow fom the following obevtion if, the left hnd ide of 14) incee to tnθ + π)/) while x = x) vie continuouly with epect to. So fo 14) to emin tue, it i necey tht x emin bounded between conecutive odd multiple of π/. If x incee pt n odd multiple of π/, then the ight hnd ide of 14) jump fom + to.) So x) i bounded. Remk: A futhe nlyi evel tht 0 + one h x) = θ/ + ǫ) whee ǫ) 0. Thi i lo ppent up to cling) fom the contou plot of θ = θ, x) given in Figue 3. Futhemoe, we find tht + one h 15 10 5 0 0 5 50 Figue 3. Contou plot fo θ = θ, x) whee x = v /. x) = θ/π) π + ǫ) whee ǫ) 0 nd whee i the ceiling function. Thi, too, i ppent up to cling) fom the plot given in Figue 3. We now etblih context in which the logithmic pil e globl mximize of inveive clength. Popoition 3. Suppoe γ i thee time diffeentible cuve fom p to q tht h deceing cuvtue, κ. Suppoe the mount of winding of the tngent vecto long γ i φ = γ κ d, nd uppoe γ h cuvtue κ p > 0 t p nd κ q > 0 t q. If Lγ) denote the inveive length of γ, then Lγ) φ logκ p /κ q ).

MICHAEL BOLT Thee i equlity if nd only if γ i otted nd tnlted imge of the logithmic pil [, t] e i log /1 + i), whee 16) = φ/ logκ p /κ q ), = /κ p, nd t = /κ q. Remk: Since γ h deceing cuvtue nd κ q > 0, the cuvtue of γ mut lwy be poitive. It follow tht φ > 0. Moeove, ince κ p > κ q, it lo follow tht > 0 nd 0 < < t. Poof. We ue the following veion of the Hölde inequlity. See Hdy, Littlewood, nd Póly [4, p.140].) Let 0 < k < 1 o k < 0, nd let 1/k + 1/k = 1. If f 0 nd g 0, then ) 1/k ) 1/k f k g k fg, with equlity if nd only if f k = c g k fo ome contnt c, o if f o g i identiclly zeo. In ou ppliction, both f nd g will be nonzeo. Fit, define contnt,, t > 0 uing 16), nd let [, l] κ) be the cuvtue function fo γ uing the clength pmete. Aume tht = nd = l coepond to p γ nd q γ, epectively. Then, pplying the Hölde inequlity with k = 1/, k = 1, f) = κ ) = κ ), nd g) =, we find ) 1 l Lγ) 1 l d κ ) d = κ) l l + κ) d We conclude tht 17) Lγ) log l = l κl) + κ) + φ = l t + + log t. l t + + log t ), with equlity if nd only if κ ) 1/ = c/ fo ll nd fo ome contnt c. Next we clim tht log l l t + 1 + log t ) log t ) with equlity if nd only if l = t. Fo thi we ue l <, nd we eplce nd t with u = logl/) 0 nd v = logt/) 0. Then lo e u = l/ nd e v = t/, nd the clim y u e u v + 1 + v ) v, with equlity if nd only if u = v. Now the function u u e u v + 1 + v) h vlue 0 t u = 0, nd ppoche u +. It only citicl point i whee e u v + 1 + v) + u e u v ) = 0 e v 1 + v) = e u 1 + u) u = v. Since the vlue of thi function t u = v i v, the clim i poved.

EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS 3 Combining 17) with the clim, we hve l 18) Lγ) = κ ) d) 1/ log ) t = φ logκ p /κ q ), with equlity if nd only if both i) l = t, nd ii) κ ) 1/ = c/ fo contnt c. But thee condition, long with equlity in 18), equie fit tht c = nd then κ ) = /. Since lo κ) = κ p = /, it follow fte integting tht κ) = / fo ll. Thi i exctly the cuvtue eqution fo the pil defined in the popoition, o we e done. We come finlly to the poof of Theoem, which ue both Popoition nd Popoition 3. The only compliction i to check tht in nomlized etting, the θ invint detemine the mount of winding of the tngent vecto. Thi technicl pect of the poof ue continuity gument. Poof of Theoem. Suppoe tht γ i cuve fom p to q tht h deceing cuvtue nd γ i the loxodomic c tht connect the me cicle element nd h the me θ invint γ. Afte tnltion nd ottion, we my ume tht γ tt t p = 0, nd the tngent vecto thee h ngle φ p = 0. Then fte futhe Möbiu tnfomtion we my lo ume tht γ i the logithmic pil tht h the clength pmeteiztion 19) z : [1, ] z) = e i log 1)/1 + i) fo cetin > 0, > 1. The cuvtue function fo γ nd γ e then not only deceing, but lo poitive, ince γ h poitive cuvtue t z). We will how tht the tngent vecto of γ nd γ hve the me mount of winding. Afte thi, Theoem follow immeditely fom Popoition 3. To do thi, we contuct fmily of loxodomic c nd ue continuity gument. Let d = d q > 0 be continuou function of q γ uch tht d p = κ p/ 1/4 nd d q = 1. We will oon pecify the function d.) Hee, κ p i the deivtive of the cuvtue function fo γ tken with epect to clength nd evluted t p. Then, fo q p, d = d q detemine loxodomic c γq follow: 1) Let θ = θ q > 0 be the θ invint of the ubc γ q γ tht connect p to q. ) Uing the fixed) vlue of > 0 detemined by γ in 19), the vlue θ q detemine vlue v = v q > 0 in the poof of ujectivity in Lemm 7. 3) Define = q > 1 ccoding to v q = log q. The pmete nd q then detemine logithmic pil z = z q in 19). 4) Let µ = µ q be the Möbiu tnfomtion µ = d 1 z/cz+d) whee d = d q, def nd whee c = c q = dq ) 1 dz) 1 fo d = d q, z = z q, = q. 5) Let γq be the loxodomic c γ q = µ q z q. The inveive length of γ q i v q nd it θ invint i θ q.

4 MICHAEL BOLT Uing thi contuction, the c γq connect the me line element the ubc γ q γ, nd it lo h the me θ invint. Moeove, ech pmete θ, v,, nd c vie continuouly with epect to q γ, except tht c = c q my not extend continuouly t q = p. Moeove, q p, one h θ 0, v 0, nd 1. Finlly, the condition d p = κ p / 1/4 enue the etimte c q = o1/ q ) fo q ne p, nd the condition d q = 1 enue tht c q = 0, µ q = z, nd γq = γ. Next, fo pticul choice of d = d q, we clim tht the mount of winding of the tngent vecto fo γq vie continuouly with q p nd ppoche zeo q p. Evidently thi i tue fo the mount of winding fo γ q, nd well, the mount of winding fo γq mut gee with the mount of winding fo γ q except fo poibly multiple of π. Once the clim i poved, then, the two mount mut be equl fo ll q p. In pticul, thi i tue fo q = q, nd the theoem follow. To etblih the clim, we fit expe the cuvtue of γq uing computtion like the one fom Lemm 6. The clength pmete fo γq, cll it u, i elted to the clength pmete fo z q ccoding to du/d = cz) + d fo c = c q, z = z q, nd d = d q. Then the cuvtue of γq t µ q z q ) i given by ) dz d 0) d µ z) du dµ z) i du d cz + d d cz + d 1 = dz i cz + d) d [ ] = Im cz )cz) + d) nd the mount of winding of the tngent vecto fo γ q 1) 1 [ ] Im cz )cz) + d) 1 + cz) + d, i given by + cz) + d ) du d d = Im [ cz ) cz) + d ] + ) d = g[cz) + d] + gd + log = g[z)/q ] + log. The pmete in the two integl, nmely, c, nd d, vy continuouly with epect to q, o the fit pt of the clim i poved once we how tht cz)+d 0 fo 1. Fo thi, we chooe the function d = d q tht mke the cuvtue of γq t q gee with the cuvtue of γ q t q. Thi cuvtue i poitive, nd ince γq h deceing cuvtue, it then follow tht γq mut hve eveywhee poitive cuvtue. Fom 0), it then follow tht cz) + d 0 fo 1. To find d, let κ = κ q denote the cuvtue of γ q t q. Then, fte the ubtitution cz) + d = dq ) 1 z) nd c = dq ) 1 dz) 1 in 0), we find ) d q = z) / Im[z )z)] κ q q Im[q z )z)/z)] = 1 1 + in log) + 1/ ) κ q q Im [ q e iθ].

EXTREMAL PROPERTIES OF LOGARITHMIC SPIRALS 5 The numeto in the econd expeion fo d i lwy negtive. To ee thi, ubtitute y = log nd ue the inequlity inhy + iny) < 0 fo y > 0, etblihed duing the poof of Lemm 7. The denominto i lo negtive. Thi ie fom the fct tht the cicle element t the endpoint of γ q e popely neted. In pticul, the cicle centeed t q + ie iφ q /κ q = q + ie iθ q /q κ q ) with diu 1/κ q mut encloe the oigin. We mention tht ince = q, κ = κ q, nd θ = θ q e continuou, it follow tht d = d q > 0 defined in ) i well-defined nd continuou. Futhemoe, if q = q, then z) = q nd κ = /, ince γ nd γ connect the me cicle element. So we conclude fom the fit expeion fo d tht d q = 1. Fo the behvio q p, we etimte 3) 1/ ) + in log ) 1 + = 3 1)3 + O 1) 4 ) = v3 3 + Ov4 ) = 6θ)3/ 3 + Oθ ) = κ p) 3/ 3 3 + o 3 ). Hee, the thid etimte i gotten by expnding 8) to find v = vθ), nd the lt etimte i gotten by expnding the ight hnd ide of 1) in tem of the Eucliden length = q of the c γ = γ q. A imil etimte give κ q q Im [ q e iθ] = κ q q Im [ q e iφ ] q = κ p 3 + o 3 ), 3 o tht d q = κ p/ 1/ + o1) q p. We hve yet to etblih tht the mount of winding in 1) ppoche zeo q p. Uing the me etimte in 3), we find tht = q nd = q e elted by o c = 1 d 1 = v + Ov ) = ) 1 q d = 1 1 z) d 6θ + Oθ) = κ p 1/ κ p 1 1 1 + o 1 1 = o 1/ ) ) ) = o + o ), ) 1. 1 Then, ince z ) = 1 nd cz) = o 1) 1 ) O 1) = o1) fo 1, we find 1 [ cz ] ) Im + ) d cz) + d cz ) 1 cz) + d d + log ) 1 = 1) o O1) + O 1) = o1) 1 fo 1 +, nd the theoem i poved.

6 MICHAEL BOLT 6. Futhe Obevtion Hee we wite down two obevtion tht e elted to wht h ledy been etblihed. 6.1. The function θ = θ γ p, q ) i n inceing ep., deceing) function of q γ fom p to q povided γ h deceing ep., inceing) cuvtue. Poof. Aume tht γ connect p to q nd h deceing cuvtue. If i the clength pmete fo γ, then dθ/d t q γ depend on nothing beyond the econd ode infomtion of γ t q. Let γ q γ be the ubc tht connect p to q, nd let γq be the loxodomic c tht connect the me cicle element γ q nd h the me θ invint γ q. Along γq, the quntity dθ/d i poitive, whee i the clength pmete fo γq. Since γ q nd γ q hve the me econd ode infomtion t q, it mut then be tue tht dθ/d i poitive fo γ t q. So the clim i poved. 6.. Fo c with monotone cuvtue tht connect the me line element nd hve the me θ invint no etiction on econd ode infomtion), the inveive clength cn be mde bitily lge o mll. Poof. Thi i evident even within the fmily of loxodomic c. In pticul, on level cuve of θ = θ, v) we hve een tht θ, v)/ x θ, v)/π π whee x = v /. See Figue 3. Thi men tht the inveive length of the loxodomic c i compble to θ/. Thi cn be mde bitily lge by chooing mll, nd it cn be mde bitily mll by chooing lge. So the clim i poved. Refeence [1] W. Blchke. Voleungen übe Diffeentilgeometie, II Affine Diffeentilgeometie). Spinge-Velg, Belin, 193. [] G. Cin nd R. W. Shpe. On the inveive diffeentil geomety of plne cuve. Eneign. Mth. ), 361-):175 196, 1990. [3] H. S. M. Coxete. Inveive ditnce. Ann. Mt. Pu Appl. 4), 71:73 83, 1966. [4] G. H. Hdy, J. E. Littlewood, nd G. Póly. Inequlitie. Cmbidge Mthemticl Liby. Cmbidge Univeity Pe, Cmbidge, 1988. [5] N. Kezmn nd E. M. Stein. The Cuchy kenel, the Szegö kenel, nd the Riemnn mpping function. Mth. Ann., 361):85 93, 1978. [6] Heinich Liebmnn. Beitäge zu Inveiongeometie de Kuven. Münch. Be. Akdemie de Wienchften, Munich, Sitzungbeichte), pge 79 94, 193. [7] Juku Med. Geometicl mening of the inveion cuvtue of plne cuve. Jp. J. Mth., 16:177 3, 1940. [8] Juku Med. Diffeentil Möbiu geomety of plne cuve. Jp. J. Mth., 18:67 60, 194. [9] Boyd Ptteon. The diffeentil invint of inveive geomety. Ame. J. Mth., 504):553 568, 198. [10] R. Michel Pote. Diffeentil invint in Möbiu geomety. J. Ntu. Geom., 3):97 13, 1993. Deptment of Mthemtic nd Sttitic, Clvin College, 301 Buton St. SE, Gnd Rpid, Michign 49546 USA E-mil dde: mbolt@clvin.edu