Avll t http:pvu.u Appl. Appl. Mth. ISSN: 9-9466 Vol. 0 Issu Dr 05 pp. 007-08 Appltos Appl Mthts: A Itrtol Jourl AAM Etso oruls of Lurll s utos Appltos of Do s Suto Thor Ah Al Atsh Dprtt of Mthts A Uvrst A Y h--tsh@hotl.o Rv: Jur 0 05 Apt: Otor 6 05 Astrt Th of ths rsrh ppr s to ot two tso foruls for th frst so k of Lurll s futos of thr vrls wth th hlp of grlz Do s suto thor whh ws ot Lvo t l. I to to ths two tso foruls for th so thr k of Appll s futos r ot s osqu of th ov to rsults. urthror so trsforto foruls volvg Eto s oul hprgotr srs r ot s ppltos of our rsults. Kwors: Pohhr sol G futo grlz hprgotr futo Lurll futos Appll futos Eto hprgotr srs Do s thor Etso foruls MSC 00 No. : C0 C65 C70. Itrouto I th usul otto lt p ot grlz hprgotr futo of o vrl q wth p urtor prtrs q otor prtrs f s Srvstv Moh 984 p.4 Srvstv Krlsso 985 p.9 whr p... p... p q.... q 0... q ots th Pohhr s sol f f 0... f.. 007
Ah Al Atsh008 Th srs. s ovrgt for vrgt for ll 0 f p q. urthror f w st f p q for f p q whl t s w q t s kow tht th srs. wth p q s p solutl ovrgt for f R w 0 otoll ovrgt for f R w 0 vrgt for f R w. Th Lurll s futos Moh 984 p.60 A A r f rprst s follows Srvstv 0. 0.4 Clrl w hv { }. A whr r Appll oul hprgotr futos Srvstv Moh 984 p.5.5 0
AAM: Itr. J Vol. 0 Issu Dr 05 009.6 0 { }. I 98 Eto 98 p.7 f th followg oul hprgotr srs : A: : X C : D D : 0 '.7 ' whh s th grlzto ufto of Hor s o-oflut oul hprgotr futo H 4 Srvstv Moh 984 p.57 Hor s oflut oul hprgotr futo H 7 Srvstv Moh 984 p.57. or th sk of ov th sol ots th prout. I th thor of hprgotr grlz hprgotr srs Do s suto thor for th su of pl portt rol. Appltos of th ov to thor r ow wll kow s for pl l 9 Lvo t l. 994 K t l. 00 L K 00 Shkhwt 0 Shkhwt Thkor 0. Th of ths ppr s to ot two tso foruls for th Lurll s futos A wth th hlp of th followg grlz Do s thor Lvo t l. 994: A A ] ] ] ] ].8 ] R 0 0 whr ] ots th grtst tgr lss th or qul to ots th usul solut vlu of. Th offts A r gv rsptvl Lvo t l. 994. Wh 0. rus tl to th lssl Do s suto thor Rvll 960 p.9 s lso l 9 p.
Ah Al Atsh00.9 R. W lso rqur th followg wll-kow tts Srvstv Krlsso 985 p.6-7 0.0....4..5. M Etso oruls I ths sto th followg tso foruls wll stlsh : orul. A ' 0 0 4 ' ] ] ] C 0 0 4 '
0 AAM: Itr. J Vol. 0 Issu Dr 05 ] ] ] D. for 0 0. Th offts D C ot fro th offts A rplg rsptvl. orul. 0 0 4 ] ] ] G 0 0 4 ] ] ] H. for 0 0. Th offts H G ot fro th offts A rplg rsptvl.
Ah Al Atsh0 Proof of orul. : Dotg th lft h s of. S pg srs usg th rsults Srvstv Krlsso 985 p.7 A powr. 0 0 0 A A.4 0 w gt p ' p p S. p p.5 0 0 p0 p p Nt usg th tts Srvstv Krlsso 985 p.7 0 0.6 w gt ' S. 0 0.7 th pplto of grlz Do s thor.8.7 os S ' 0 0 A A.8 whr A ] ] ].9 ] ]. ].0
0 AAM: Itr. J Vol. 0 Issu Dr 05 Th offts A ot fro th offts A rplg rsptvl. Now sprtg.8 to v o powrs w gt 0 0 ' S A C 0 0 '. E A D. S 0 0 whr ots th G futo Srvstv Krlsso 985 p.6. W hv s fro. tht 0. A Thrfor. os 0 0 ' S ] ] ] C
Ah Al Atsh04 0 0 ' ] ]. ] D. ll. f w us th rsults.-.5 th ftr so splfto w rrv t th rght h s of.. Ths oplts th proof of forul. I tl th s r forul lso prov. Rrk. Tkg 0.. w u rsptvl th followg tso foruls for Appll s futos : orul. 0 4 ] ] ] C 0 4 ] ]. ] D. orul 4.
05 AAM: Itr. J Vol. 0 Issu Dr 05 0 4 ] ] ] G 0 4 ] ] ] H..4 Th offts D C ot fro th offts A rplg rsptvl th offts H G ot fro th offts A rplg rsptvl. Rrk. It s trstg to to hr tht th rsults..4 r grlzto of th followg wll-kow trsfortos u to l 95 p.9 : 4.5 4..6. Appltos Tkg 0.. usg th rsults.0-. w gt A '
Ah Al Atsh06 : : X. 0: : 4 rsptvl. urthr tkg 0:4 : X 4 :0 :. w gt A ' : : X. 0: : 4.. Tkg 0.. usg th rsults.0-.5 w gt A ' : : X 0: : 4 : X 0: : : 4.4 0 : 4 : X 4 : 0 : rsptvl. 0:4 : X 4 :0 :.5 urthr tkg.4 w gt A ' : : X 0: : 4
AAM: Itr. J Vol. 0 Issu Dr 05 07 : : X. 0: : 4.6 Slrl othr rsults lso ot. 4. Coluso W olu our prst vstgto rrkg tht th rsults stlsh ths ppr ppl to ot lrg ur of trsforto foruls for th Lurll s futos A of thr vrls Appll s futos trs of Eto s oul hprgotr srs grlz hprgotr futo. urthr th tso foruls f w tk th w ot w tso forul for Lurll s futo A ' futo Appll s. Also spl ss of ths tso foruls ot trs of Eto s oul hprgotr srs grlz hprgotr futo. Akowlgts Th uthor s hghl grtful to th oous rfrs th Etor--Chf Profssor Alkr Motzr Hghgh for usful ots suggstos towrs th provt of ths ppr. REERENCES l W. N. 9. Grlz Hprgotr Srs Crg Uvrst Prss Crg. l W. N. 95. O th su of trtg. Qurt. J. Mth. Ofor 4: 7-40. Eto H. 98. Rul oul hprgotr futos ssot tgrls. A. C. Uv. Porto 6-4 : 7-4. K Y.S. Rth A.k. Cho J. 00. Suto foruls rv fro th Srvstv s trpl hprgotr srs H C. Cou. Kor Mth. So. 5: 85-9. Lvo J. L. Gro. Rth A.K. Aror K. 994. Grlztos of Do's thor o th su of. Mth. Cop. 6 05 : 67-76. L S. W. K Y. S. 00. A tso of th trpl hprgotr srs Eto. Ho Mthtl J. : 6-7. Rvll E.D. 960. Spl utos Th Mll Cop Nw York. Shkhwt N. 0. O two sutos u to Ru thr grlzto. Avs Coputtol Mthts ts Appltos : 7-8 Coprght Worl S Pulshr Ut Stts www.worlpuplshr.org. Shkhwt N. Thkor V. 0. O o of sutos u to Ru ts Grlzto. Avs Coputtol Mthts ts Appltos 4 : 9-4. Coprght Worl S Pulshr Ut Stts www.
Ah Al Atsh08 worlpuplshr.org. Srvstv H.M. Krlsso P.W. 985. Multpl Guss Hprgotr Srs Hlst Prss Nw York. Srvstv H.M. Moh H.L. 984. A trts o Grtg utos Hst Prss Nw York.