The Bloch Space of Analytic functions

Transcription

1 Inernaional OPEN ACCESS Jornal O Modern Engineering Research (IJMER) The Bloch Space o Analyic ncions S Nagendra, Pro E Keshava Reddy Deparmen o Mahemaics, Governmen Degree College, Pormamilla Deparmen o Mahemaics, JNTUA Absrac: We shall sae and prove a characeriaion or he Bloch space and obain analogos characeriaion or he lile Bloch space o analyic ncions on he ni disk in he complex plane We shall also sae and prove hree conainmen resls relaed o Bloch space and Lile Bloch space Keyords: Bloch Space, Analyic Fncions, Mobis Transormaion We le I INTRODUCTION C / D For D, he Mobis ransormaion is deined by Then () or D ( ) ( ) ( ) ( ) () So, he ncion maps D on o isel and D on o isel I is easy o veriy ha is is on inverse Noing ha ( ), he above ideniy saes: Bloch space B is he space o all analyic ncions on D or hich Sp ( ) D and B becomes a Banach space ih respec o he semi norm ( ) ( ) () Sp ( ) D IJMER ISSN: ijmercom Vol 4 Iss Nov 4 33

2 The Bloch Space o Analyic ncions Using (), e have ( ) ( ) Sp :, D, ( ) ( ) Sp :, D, ( ) ( ),, D, Sp D ( ) D D Sp ( ) ( ) o Sp o Sp ( ) ( ) D o Ths Bloch space is a Mobis invarian space Sp (3) In he nex secion, e shall sae and prove a crierion or conainmen in he Bloch space and lile Bloch space A THEOREM For an analyic ncion on D II CHARACTERIZATION FOR BLOCH AND LITTLE BLOCH SPACE Proo : Sppose or an analyic ncion on D Taking limi as, e ge For he nex par, sppose Sp D, D (4) IJMER ISSN: ijmercom Vol 4 Iss Nov 4 34

3 The Bloch Space o Analyic ncions Then or each D, e have ( ) () ( ) d ( ) () ( ) d ( ) d d d ( (4)) ( ) () d log log log log log x x, x ( ) (), D No or, D replace in he above ineqaliy by o and le = () Using ( ()) = and ideniies () and (3) e have o o o ( ) ( ) B IJMER ISSN: ijmercom Vol 4 Iss Nov 4 35

4 The Bloch Space o Analyic ncions ( ) ( ) We briely discss he lile Bloch space B The se o all analyic ncions on D or hich lim ( ) For an analyic ncion on D and < < he dilae is he ncion deined by () = () I is knon ha or an analyic ncion on D: ( ) ( ) :, Sp D, In analogy o heorem (), e have he olloing resl B THEOREM For an analyic ncion on D Proo: ( ) ( ) lim Sp :, D, Taking limi as in he condiion o he saemen, e ge Sppose, hen - ( ) ( ) (5) lim Sp Applying ineqaliy (5) or, e have i as lim IJMER ISSN: ijmercom Vol 4 Iss Nov 4 36

5 The Bloch Space o Analyic ncions Applying ineqaliy (5) or -, e have ( ) ( ) (6) (7) Ineqaliy (6), (7) and riangle ineqaliy imply ha No irs leing and hen, e ge ( ) ( ) lim Sp :, D, In he nex secion, e shall prove hree resls relaed o conainmen o Bloch and Lile Bloch space III CONTAINMENT RESULTS OF BLOCH AND LITTLE BLOCH SPACE Le be a bonded analyic ncion on D, hen here exiss a consan M schha M, D From Cachy s inegral ormla, e have d i C IJMER ISSN: ijmercom Vol 4 Iss Nov 4 37

6 here C is any closed disc o radis r in neighborhood o and conaining, hen The Bloch Space o Analyic ncions 4M r in he r concenric disc o radis This implies ha is bonded in any neighborhood o conained in D henever so is A THEOREM 3 I B hen k B Proo: I is very easy o see ha here k C k is a consan Thereore sp sp k D D Hence k B henever B B THEOREM 4 I B is bonded and is any bonded and analyic ncion on D hen B B Proo: Noe ha lim Taking limi as, he irs erm on RHS ends o becase o he hypohesis and is bonded and he second erm since and bonded in he neighborhood o as is bonded on D ends o Hence lim B Thereore C THEOREM 5 I, g are bonded ncions o B, hen g B Proo: From he deiniion o B, g lim lim Noe ha g g g Taking limi as, e ge g lim g B IJMER ISSN: ijmercom Vol 4 Iss Nov 4 38

7 The Bloch Space o Analyic ncions IV CONCLUSION I invie ineresed readers o prse geomeric inerpreaion o characeriaion heorems ha proved in his paper and also similar conainmen resls relaed o he Bloch space e REFERENCES [] JM Anderson, J Clnie and Ch Pommerenke, On Bloch ncions and normal ncions, Proc o he American Mahemaical Sociey 85,974, -37 [] Jose L Fernande, J, On coeiens o Bloch ncions, London mah Soc () 9 (984), 94- [3] R Alaskari, N Danikas, and R Zhao, The Algebra Propery o The Inegrals o Some Banach spaces o Analyic ncions by N Danikar, Arisole niv o Thessaloniki [4] Theory o ncion spaces by Kehe Jh [5] Bloch ncions : The Basic heory by JMAnderson edied by SC Poer, series C: Mahemaical and Physical Sciences Vol 53 [6] Mlipliers o Bloch ncions by Jonahan Aray, Repor 54, 98 [7] The Bloch space and Besov spaces o Analyic ncions by Karel Sroeho, Bll Asral Mah Soc Vol 54, 996, -9 IJMER ISSN: ijmercom Vol 4 Iss Nov 4 39