A Class of Harmonic Meromorphic Functions of Complex Order

Transcription

1 Borg Irol Jourl o D Mg Vol 2 No 2 Ju A Clss o rmoc Mromorpc Fucos o Complx Ordr R Elrs KG Surm d TV Sudrs Asrc--- T sml work o Clu d Sl-Smll [3] o rmoc mppgs gv rs o suds o suclsss o complx-vlud rmoc uvl ucos I s ppr clss o rmoc mromorpc ucos o orm () () g() > o complx ordr s roducd I s sow ucos s clss r ss prsrvg d uvl ousd u dsk Suc coc codos r od or ucos s clss wc r lso sow o cssry w co-lyc pr g() s gv cocs W lso o proprs suc s dsoro ouds xrm pos covoluo d covx como or s clss Kywords--- rmoc Fucos Mromorpc Fucos Srlk Fucos I INTRODUCTION rmoc uvl mppgs r kow o ply mpor rol sudy o mml surcs d v oud pplcos dr lds suc s Egrg Opros rsrc d ppld mmcs [2] rmoc mppgs dom D C r uvl complx vlud rmoc ucos u v wr o u d v rl rmoc D rmoc uvl mppgs v drw rmdous o o complx lyss oly r mpor work o Clu d Sl-Smll [3] 984 grr d Scor [5] [6] 986 workd owrds dg ppropr orm o Rm mppg orm or rmoc mppgs T works o s uco orss d svrl or rsrcrs (s or xmpl [7] [] [2]) gv rs o svrl prolms cojcurs d my rgug qusos Svrl clsss o complx vlud rmoc uvl ucos v roducd d vsgd ollowg sc work o Clu d Sl-Smll [3] Tr r svrl survy rcls d ooks ([2] [4]) o rmoc mppgs d rld rs grr d Scor [7] mog or gs vsgd mly M o ucos () () g() wc r rmoc mromorpc oro prsrvg d uvl U { : > } wr R Elrs Asss Prossor Dprm o Mmcs SIVET Collg C 6 73 Id E-ml : lrs28@ymlcom KG Surm Prossor Scool o Compur Sccs Uvrs Ss Mlys 8 Pg Mlys E-ml : kgsm948@yoocom TV Sudrs Assoc Prossor Dprm o Mmcs SIVET Collg C 6 73 Id E-ml: vsudrs@rdmlcom () ; g() U Jgr [8] d Jgr d Slvrm [9] v lso vsgd rmoc mromorpc ucos wc r srlk U r w roduc or clss S ( α γ ) o rmoc mromorpc ucos dd s ollows: For β < l S ( α γ ) coss o ucos M so α ( ) () α R γ () (2) wr () () () γ < α rl d complx umr suc Rmrk : T clss cluds vry o wll-kow suclsss or spcc vlus o α d w S ( α γ ) M ( [] 2 w S ( α γ ) G (α β ) [] β 3 w α S ( γ ) Σ * [8] 2 Also l S (α γ ) suclss o S ( α γ ) cossg o ucos orm () g() ; () g wc d g r o W o suc coc codos or rmoc mromorpc ucos g o clss S ( α γ ) W lso sow s coc codo s lso cssry or S (α γ ) W lso o dsoro ouds xrm pos covoluo codo d covx como or ucos (α γ ) S II COEFFICIENT CONDITIONS Frs w prov suc codo or rmoc ucos S ( α γ ) (3) ISSN Borg

2 Borg Irol Jourl o D Mg Vol 2 No 2 Ju Torm 2: L g so d g r o orm () I [2 (2 - (- )] [2 (2 - (- )] ( w γ < α rl d o-ro complx umr suc s uvl ss prsrvg rmoc mppg U { : < } d S ( α γ ) Proo: Cosdr uco g wr d g r gv y () I [9] s provd < s rmoc oro prsrvg d uvl U For γ < w o 2 (2 - (- ) ( 2 (2 - (- ) ( d Tror s rmoc oro prsrvg d uvl U du o (4) To sow S ( α γ ) w A() oc ccordg o (2) w mus v R > γ wr B() α A() [( ) (() g())] ( )[ () g ()] α ( )[( ) (() g())] B() [( ) (() g())] Usg c R (w) γ d oly γw γw or γ < s oug o sow A() ( B() A() ( B() Drg d g d susug ov quly w o A() ( B() A() ( B() [(2 ( ( (2 g() ( ( α [γ α ) () ( α ) () ( rg() ( α )]( ) (2 () )() )g () ( ]( ) γ () α α α α )() )g () ( α α )g() )g() (4) { (2 γ 2 ( 2 ( [2 (2 2] [2 γ 2] [2 γ 2] [2 (2 - (- ] [2 2 (2 ] [2 (2 - (- ] {[2 (2 - (- ] [2 (2 - (- ] } Now y (4) s ls xprsso s vr gv d so S ( α γ ) W ow gv xmpl o uco clss S ( α γ ) Exmpl 2: T rmoc uco g wr ( g() 4[2 (2 - ( ( () 4[2 (2 - ( )] )] wr γ < d sss suc codo o Torm 2 d c logs o clss S ( α γ ) Nx w sow coc codo (4) s lso cssry or ucos (α γ ) S Torm 22: L g so d g r o orm (3) A cssry d suc codo or o S (α γ) s {[2 (2 - (- )] [2 (2 - (- )] } (5) ( Proo: I vw o Torm 2 w d oly sow S (α γ ) coc quly (5) dos o old W o S (α γ ) w mus v ISSN Borg

3 Borg Irol Jourl o D Mg Vol 2 No 2 Ju α ( )(() g() ) R ( ) (() g() γ α α ( [( ) [(γ ) ]] α α [( ) [(γ ) ]] R 2 α α ( [( ) [(γ ) ]] α α [( ) [(γ ) ]] R 2 Ts quly mus old or ll U d or ll rl α d y suc < < Lg r > α d rl d posv so w v 2 ( [2 [2-(- ]] r ( ) [2 [2-(- ]] r R 2 ( ) ( ) r r α ( ) A(r) B(r) I codo (5) dos o old A(r) s gv or r sucly clos o Tus r xss r > or A(r) wc quo s gv Ts cordcs B(r) A(r) d so proo s compl B(r) T dsoro ouds or ucos S (α γ ) r gv y Torm 23 Torm 23: I S (α γ ) r ( r () r ( r r > Proo: W prov rg d quly T rgum or l d quly s smlr d c s omd L S (α γ ) Tkg solu vlu o w o () r r ( ) {[2 (2-(-)] [2 (2-(-)] } r ( r r III EXTREME POINTS W us coc ouds od sco 2 o drm xrm pos or ucos (α γ ) Torm 3: S (α γ ) d oly c S xprssd s g ) wr U () g () d ( g () () 2 (2 - (- ) ) x y Proo: No or w my wr () x g ) g ( () 2(2-(- ) ( ) x 2 (2-(- ) Now y Torm 22 ( ( y() 2(2-(- ) ( x 2 (2-(- ) ) ( r ( 2) ( 2) ( y [2 (2 - (- )] 2 (2 - (- ) ( x [2 (2 - (- )] 2 (2 - (- ) ( () 2 (2 - (- ) Covrsly suppos S (α γ ) ISSN Borg

4 Borg Irol Jourl o D Mg Vol 2 No 2 Ju (2-(- ) ( Sg x y 2 (2 - (- ) ( 2 (2 - (- ) ( y x 2 (2-(- ) ( x ) w o () g ) s rqurd IV CONVOLUTION AND CONVEX COMBINATION I s sco w sow clss S (α γ ) s vr udr covoluo d covx comos o s mmrs For rmoc ucos () F() A W d covoluo o d F s ( * F)() ()* F() B () () A B () Torm 4: For β γ L S (α γ ) d F S ( αβ ) T * F S (α γ) S (αβ ) Proo: Suppos d F r so * F s gv y ov covoluo Sc S (α γ ) d F S ( αβ ) cocs o d F mus ssy codos gv y Torm 22 So r cocs o * F w c wr {[2 (2-(- )] A [2 (2-(- )] B } { [2 (2-(- )] [2 (2-(- )]} T rg d sd o ov quly s oudd y ( cus (α γ ) S Tus * F S S (α γ) S (αβ ) Flly w xm covx comos o (α γ ) Torm 42: T mly S (α γ ) s closd udr covx como Proo: Suppos () () S (α γ ) wr d 2 3 T y Torm 22 [[2 (2 - (- )] [2 (2 - (- )] ] ( For d covx comos o my wr s Tus () () () S (α γ ) [2 (2-(- )] [2(2-(- )] [2 (2-(- )] ( ( sc V CONCLUSION [2 (2-(- )] I s ppr mp s md o roduc d vsg som proprs or w suclss o rmoc mromorpc ucos o complx ordr Bsd o s work urr usul sudy o dr suclsss o rmoc uvl ucos c slsd REFERENCES [] B Adol Sp P Nrmldv TV Sudrs d KG Surm A clss o mromorpc ucos w gv cocs Cmcur J Ms Vol No Pp [2] OP Auj Plr rmoc uvl d rld mppgs J Iqul Pur Appl M Vol 6 No 4 Ar [3] J Clu d T Sl-Smll rmoc uvl ucos A Acd Sc F Sr A I M Vol 9 Pp [4] PL Dur rmoc mppgs pl Cmrdg Uvrsy Prss 24 [5] W grr d G Scor rmoc mppgs w gv dlos J Lodo M Soc Vol 33 No 3 Pp [6] W grr d G Scor O oudry vor o oro-prsrvg rmoc mppgs Complx Vrls Tory Appl Vol 5 No 2-4 Pp [7] W grr d G Scor Uvl rmoc ucos Trs Amr M Soc Vol 299 Pp [8] JM Jgr rmoc mromorpc srlk ucos Bull Kor M Soc Vol 37 Pp ISSN Borg

5 Borg Irol Jourl o D Mg Vol 2 No 2 Ju [9] JM Jgr d Slvrm Mromorpc uvl rmoc ucos w gv cocs Bull Kor M Soc Vol 36 Pp [] T Rosy B Adol Sp KG Surm d JM Jgr A clss o rmoc mromorpc ucos Tmkg J M Vol 33 Pp [] T Sl-Smll Coss or plr rmoc mppgs J Lodo M Soc Vol 42 No 2 Pp [2] TJ Surdg rmoc uvl polyomls Complx Vrls Tory Appl Vol 35 No 2 Pp ISSN Borg