Ž n. Matematicki Fakultet, Studentski trg 16, Belgrade, p.p , Yugosla ia. Submitted by Paul S. Muhly. Received December 17, 1997

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1 JOURNAL OF MATEMATICAL ANALYSIS AND APPLICATIONS 6, ARTICLE NO AY O the Iclusio U Ž B ad the Isoerimetric Ieuality Miroslav Pavlovic ad Miluti R Dostaic Matematicki Fakultet, Studetski trg 16, Belgrade, , Yugoslaia Submitted by Paul S Muhly Received December 17, 1997 Let U ad B be the uit olydisc ad the uit all i, resectively We rove that if f is i the ardy sace U, the f is i Ž B ad that the orm of the iclusio is eual to oe If f deeds o oe variable oly, the the result reduces to a ieuality of isoerimetric tye due to Carlema ad Burbea 1998 Academic Press I Carlema gave a beautiful roof of the isoerimetric ieuality, reducig it to a ieuality for holomorhic fuctios o the uit disc Burbea 1 exteded Carlema s ieuality by rovig the followig TEOREM 1 Let g be i the ardy class o the uit disc U, ad let be a iteger The 1 1 it Ž 1 z gž z dxdy gž e dt Ž 1 ž / U Euality occurs if ad oly if gž z cž 1 z for some costats c ad, 1 Ž I the case ieuality 1 reduces to * avlovic@matfbgacyu domi@matfbgacyu ž / 4 1 it gž z dx dy gž e dt, Ž 4 U X98 $500 Coyright 1998 by Academic Press All rights of reroductio i ay form reserved

2 144 PAVLOVIC AND DOSTANIC which is ust the Carlema ieuality Let G be a simly coected subdomai of the comlex lae such that the boudary G is a rectifiable curve By the R iema maig theorem, there exists a coformal maig of U oto G Takig gž z Ž Ž z 1 i Ž oe obtais A L 4, A ad L are the area of G ad the legth of G, resectively For more details ad other roofs ad geeralizatios of the isoerimetric ieuality we refer to 1, 3, ad 4 I this aer we exteded ieuality Ž 1 to several variables We start from the observatio that the left side of Ž 1 is eual to f B d, fž z 1, z,, z gž z, g Ž U, Ž 3 B is the uit Euclidea ball i ad d is the ormalized surface measure o the shere B see 6, 144Ž 1 Therefore, Ž 1 becomes ž / B T fž z d f dm Ž 4 ere T is the uit circle i the comlex lae, T times ad dm is the aar measure o T, ie, 1 it 1 it dmž e,,e dt 1,,dt Ž T T TŽ ŽWe follow the otatio from Rudi s books 5, 6 Our mai result states that Ž 4 holds for a arbitrary fuctio f belog- ig to the ardy class o the olydisc U ot oly for f of the form Ž 3 Before statig the result i a more recise form we recall the basic facts from the theory of saces Žcf 5 ad 6 Let 0 A fuctio f holomorhic i the olydisc U belogs to the ardy class ŽU if su fž r dm Ž 0r1 T It turs out that if f U, the there exists the fiite limit fž lim fž r Ž ae o T r1 Ž ad the boudary fuctio belogs to L T, m Moreover T fž dm su fž r dm Ž r1 T

3 TE ISOPERIMETRIC INEQUALITY 145 The aalogous facts hold for the ardy class o the uit ball; we have oly to relace T, U, ad dm by B, B, ad d, resectively Note a simle fact: if f is holomorhic i the olydisc, the the itegral f d is defied Ž fiite or ot B TEOREM If f U, the f Ž B ad there holds Ž 4 Euality occurs if ad oly if there exist k, 1 k ad costats c, Ž 1 such that 1 fž z,, z cž 1 z 1 k COROLLARY If g U, the g L Ž B ad there holds i- euality ž / B T 1 g d g dm Ž 5 ere d stads for the ormalized Lebesgue measure o B Note that if 1, the Ž 5 coicides with Ž 1 Proof Let g U Defie the fuctio f o U by fž z,, z, z gž z,, z By the formula 145Ž 1 of 6 we have B 1 f d g d 1 Now the result follows from Theorem B For the roof of Theorem we eed two lemmas LEMMA 1 If,, are oegatie itegers, the 1 1 Ł Ž 1 1 1! 1! 1! Ž 6 Euality occurs if ad oly if there exists k, 1 k such that 0 for k Proof Let Ž x log Ž x, is the Euler gamma fuctio, ad let max,, 4 1 Sice the fuctio is strictly covex, ie, 0 we have Ž x Ž y ŽŽ x x y Ž 7

4 146 PAVLOVIC AND DOSTANIC with euality oly if x y ece 1 Ž a 1 1, 1 Ž a 1 1, Ž 1 Ž a Ž 1 1 ece, by summatio, Sice the derivative is icreasig, we have for 1 1 It follows that Ž 8 ad this is euivalet with Ž 6 If euality occurs i Ž 8, the so does i every euality followig Ž 7, which imlies This comletes the roof of the lemma Let k be a ositive iteger The LEMMA The umber of multidices Ž,, 1 satisfyig the k 1 coditio k is eual to Ž 1 k By the term multidex we mea a -tule of oegatie itegers ad Proof This follows from the relatios 1 ž / t 1 t ž / 1 Ž 1 t Ž 1 t 0 1 ž / t 0

5 TE ISOPERIMETRIC INEQUALITY 147 Proof of the Theorem Let f U, fž z cž z, z z 1 z z 1, Ž 1,, The ž / fž z cž cž z, 1 1 Ž 1,,, ad k k1 k k Ž By Fatou s lemma ad Parseval s formula alied to f r,0 r 1 we have B Ž r1 B f d lim if f r d 1 B c c z d z 1 By Lemma, the umber of summads i the ier sum is eual to Ł k1 ad therefore, by the ieuality k 1 ž k / N 1 A N A 1

6 148 PAVLOVIC AND DOSTANIC we get 1 O the other had cž cž Ł k 1 ž k / 1 k1 z Ž 1 1 c c 1! 1!! d 1! B 1 Ž see 6, Proositio 149 It follows that Ž 1 B 1 f d M c c, Ł k1 k 1! M 1 1! 1! Sice M 1, by Lemma 1, we get 1 f d M cž 1 cž B 1 Ž 1 1 ž / T c c f dm The above roof shows that if there holds euality, the M 1 or cž cž 1 0 wheever 1 This imlies, via Lemma 1 that the Taylor exasio of ŽŽ f z is a sum of olyomials of the form cz s Ž 1 k k, ad this is ossible oly whe f deeds o oe variable oly, ie, whe is of the form Ž 3 Now we ca aeal to Theorem 1 to fiish the roof

7 TE ISOPERIMETRIC INEQUALITY 149 REFERENCES 1 J Burbea, Shar ieualities for holomorhic fuctios, Illiois J Math 31 Ž 1987, 4864 T Carlema, Zur theorie der miimalflache, Math Z 9 Ž 191, M Matelevic ad M Pavlovic, New roofs of the isoerimetric ieuality ad some geeralizatios, J Math Aal Al 98 Ž 1984, M Matelevic ad M Pavlovic, Some ieualities of isoerimetric tye cocerig aalytic ad subharmoic fuctios, Publ Ist Math Ž Belgrade, 50Ž 64Ž 1991, W Rudi, Fuctio Theory i Polydiscs, Beami, New York, W Rudi, Fuctio theory i the uit ball of, Sriger-Verlag, Berlieidelberg New York, 1980