BMOA estimates and radial growth of B φ functions
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1 c Jounal of echnical Univesiy a Plovdiv Fundamenal Sciences and Applicaions, Vol., 995 Seies A-Pue and Applied Mahemaics Bulgaia, ISSN axiv:87.53v [mah.cv] 3 Jul 28 BMOA esimaes and adial gowh of B φ funcions Peyo Soilov, Roumyana Gesheva, Milena Racheva Absac BMO esimaes and he adial gowh of Bloch funcions have been sudied by B. Koenblum [3]. he pesen pape conains some naual genealizaions of hese esuls. Inoducion LeD denoeheunidisk{z C : z < }and -heunicicle{z : z = }. he space BMOA is he space of funcions f H fo which f = sup f f I dm <, whee mi) I f I = fdm, I. mi) I Hee m is nomalized Lebesgue measue on. I is known ha fo an analyic funcions f in D he following condiions ae equivalen see, fo example, [] o [2]): a)f BMOA; b) f 2 BMOA = sup ξ D D f z) 2 z 2 ) ξ 2 ) ξz 2 dm 2 <, 99 Mahemaics Subjec Classificaion: Pimay 3E2, 3D5. Key wods and phases: Bloch space, BMOA esimaes, adial gowh of funcions. 39
2 4 Peyo Soilov, Roumyana Gesheva, Milena Racheva whee m 2 denoes nomalized Lebesgue measue in D. Le φ) be a posiive and coninuous funcion on,). Le B φ denoes he space of all analyic funcions f in D saisfying he condiion z 2 ) f z) f Bφ = sup z D φ z 2 ) <. Fo φ) = α, < α, B φ = Λ α is he usual Lipschiz class. In he case α =, Λ is he Bloch space usually denoed B ). In his pape some esuls of B. Koenblum fo he Bloch space B ae genealized fo B φ. Noe ha he funcion log z) B, howeve log 2 z) / B. 2 BMOA esimaes fo B φ funcions and applicaions Le φ) saisfies he condiion x φ 2 ) d = gx) < fo all < x <. If f B φ we wie f z) def = fz) fo < <. heoem. Le f B φ. hen f BMOA f Bφ g 2 ), < <. ) Poof. Le ξ D. hen D f z 2 ) ξ 2 ) z) 2 ξz 2 dm 2 =
3 BMOA esimaes and adial gowh of B φ funcions 4 = 2 D f z) 2 z 2 ) 2 φ 2 z 2 ) φ2 z 2 ) z 2 ) z 2 ) ξ 2 ) dm 2 2 ξz 2 2 f 2 B φ = 2 2 f 2 B φ D φ 2 z 2 ) ξ 2 z 2 ξz 2 dm 2 = φ 2 2 ρ 2 ) ξ 2 ) 2 ρ 2 ) ξρζ 2 ρdmζ)dρ 2 2 f 2 B φ ξρζ 2 ξρζ 2dmζ) φ 2 2 ρ 2 ) 2 ρ 2 ρdρ = Hee we used he ideniy heefoe, = f 2 B φ 2 φ 2 ) d. z 2 2 dmζ) =. ζz f 2 BMOA f 2 B φ g 2 ). Coollay. If f B hen f BMOA f Bφ log 2 ), < <. B.Koenblum [3] poved an analogous BMO esimae, applying he Gasia nom.
4 42 Peyo Soilov, Roumyana Gesheva, Milena Racheva heoem 2. hee ae posiive numeical consans γ and M such ha fo all f B φ, f) = exp γ f ζ) f Bφ g 2 ) ) dmς) M. 2) Poof. he John-Nienbeg heoem [, 2] says ha hee ae posiive consans c and C such ha mζ I : fζ) f I > λ) mi) Cexp cλ ) fo all f BMOA, λ >, I. If f) = hen f = 2π fς) dζ = and Eλ) = mζ : fζ) > λ) Cexp cλ ). 3) Since Eλ) is he disiluion funcion of f, hen fo all p > [] f p dm = p λ p Eλ)dλ. 4) If < γ < c, using 4) and 3), we obain exp γ fζ) )dmζ) = + n n! γ n ) n fζ) n dmζ) =
5 BMOA esimaes and adial gowh of B φ funcions 43 γ n = + n! f n BMOA ) n n λ n Eλ)dλ = = + γ n γ n n! ) n λ n Eλ)dλ = = + γ Eλ) γ n λ n dλ = n! f n BMOA ) n = + γ Eλ) exp γλ ) dλ + γ C exp c γ)λ )dλ = + γc c γ def = M <. Puing f = f and applying ), we obain 2). heoem 3. heeisaconsan γ, suchhafoevey f B φ, f ) = fo almos all ζ. fζ) lim sup log log ) g 2 ) γ f Bφ 5) Poof. heoem 2 implies ha < < ) )log 2 e ) exp γ fζ) )dmζ) )d M. f Bφ g 2 )
6 44 Peyo Soilov, Roumyana Gesheva, Milena Racheva heefoe, fo almos all ζ )log 2 e ) exp γ fζ) ) d <, f Bφ g 2 ) which implies ha lim +)/2 ρ)log 2 e ρ ) exp γ fρζ) ) dρ =. f Bφ g ρ2 ) Puing µ,ζ) = min{ fρζ) : ρ +)/2} we ge +)/2 ρ)log 2 e ρ ) exp f Bφ γ fρζ) g ρ2 ) ) dρ +)/2 log 2 e ρ ) exp γ fρζ) f Bφ g ρ2 ) ) dρ +)/2 log 2 e ρ ) exp γ fρζ) f Bφ g ρ2 ) ) dρ log 2 2e ) exp We used he inequaliies γ µ,ζ) f Bφ g )3+)/4)) ) >. log 2 e ρ ) e log 2 +)/2 ) = 2e log 2 ),
7 BMOA esimaes and adial gowh of B φ funcions 45 g ρ 2 ) g +/2) 2 ) = g )3+)/4)). hen lim 2e log 2 ρ ) exp γ µ,ζ) ) =, f Bφ g )3+)/4)) which implies Since lim γ γ µ,ζ) 2loglog 2e f Bφ g )3+)/4)) ) =. i can be seen easily ha lim loglog 2e loglog ) =, µ,ζ) < γ f Bφ g )3+)/4)) log log ) 6) γ f Bφ. g 2 )/2)) log log ). fo almos all ζ, sufficienly close o and γ = 2/γ. In addiion, le µ, ζ) = f ζ), +)/2. hen fζ) µ,ζ) f ρζ) dρ f Bφ +)/2 φ ρ 2 ) ρ 2 dρ =
8 46 Peyo Soilov, Roumyana Gesheva, Milena Racheva = f B φ 2 2 )3+)/4 φ) d f B φ )/2 φ) d f B φ )/2 φ 2 ) d ) /2 f B φ 2 2 )/2 φ 2 ) d ) /2 = Applying 6), we obain = f B φ g 2 )/2)). 2 fζ) f Bφ g 2 )/2)) 2 +γ log log ) ) fo almos all ζ and sufficienly close o, which poves 5). Coollay. Koenblum [3]) If f B, f ) = hen fζ) lim sup k f B log ) log log ) fo almos all ζ, whee k is an absolue consan.
9 BMOA esimaes and adial gowh of B φ funcions 47 Refeences [] J. Gane.Bounded analyic funcions. Academic Pess, New Yok, 98. [2] P. Koosis. Inoducion o H p spaces. Cambidge Univ. Pess, Cambidge, 98. [3] B. Koenblum. BMO esimaes and adial gowh of Bloch funcions. Bul. Ame. Mah. Sos., 2,, 985, Depamen of Mahemaics echnical Univesiy 25, sanko Dijsabanov, Plovdiv, Bulgaia peyyyo@mail.bg
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