Some Wgh Inequalities for Univalent Harmonic Analytic Functions

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ppled Mathematc 464-469 do:436/am66 Publhed Ole December (http://wwwscrporg/joural/am Some Wgh Ieualte for Uvalet Harmoc alytc Fucto btract Pooam Sharma Departmet of Mathematc ad troomy Uverty of

1 ppled Mathematc do:436/am66 Publhed Ole December ( Some Wgh Ieualte for Uvalet Harmoc alytc Fucto btract Pooam Sharma Departmet of Mathematc ad troomy Uverty of Lucow Lucow Ida E-mal: Receved ugut ; reved September 7 ; accepted October I th paper ome Wgh eualte for uvalet harmoc aalytc fucto defed by Wrght geeraled hypergeometrc (Wgh fucto to be certa clae are oberved ad proved Some coeuet reult are alo dcued Keyword: Harmoc Fucto Harmoc Starle Fucto Wrght Geeraled Hypergeometrc Fucto Itroducto ad Prelmare Let u ad v be real valued harmoc fucto a mply coected doma D the complex plae the a cotuou fucto f u v called a complex valued harmoc map D lue ad Shel- Small [] troduced a cla SH of complex valued harmoc map f whch are uvalet ad ee-preervg the ope ut d : ad aume a ormaled repreetato h g where ( h ( h h g ( g g are aalytc ad uvalet Let SH * ( deote the cla of map f hg SH atfyg the codto f( re f ( arg( f( re Im Re f( re f( for re r ad where ( ( ( Deote by TSH the ubcla of fucto f hg SH uch that f h g lo deote * * h ( h g ( g THS ( SH ( TSH We have followg reult from the wor of Jahagr []: Lemma Let f hg SH h ( ad g( are gve by ( atfe h g h g ( the f ee preervg harmoc uvalet ad f SH * ( Furthermore f TSH * ( f ad oly f ( hold For ome 3 correpodg to h ( ad g( defed ( let f( h( g( SH (3 where for h( h( * h (4 g( g( * g * tad for covoluto Sce (5 m h h h m m h ( ad g ( for ome (3 repreet ere of mg term whch creae wth Ivolvg f ( defed (3 a cla SH ( defed a follow: Defto fucto f hg SH ad to be the cla SH ( f t atfe the codto f ( Re (6 f ( opyrght ScRe

2 P SHRM 465 where for ome f ( defed by (3 Fucto the cla SH ( are called harmoc tarle fucto wth repect to -ymmetrc pot of order Note that SH ( SH * ( SH ( ( SH ad TSH ( SH ( TSH TSH ( SH ( TSH The cla SH ( tuded by huja ad Jahagr [3] (ee alo [4] They alo proved followg reult [3] Lemma Let f hg SH h ( ad g( are gve by ( atfe h g h g (7 the f ee preervg harmoc uvalet ad f SH ( Furthermore f TSH ( f ad oly f (7 hold Sha ad Daru [56] proved that for f f SH ( the f * SH ( ad proved followg reult Lemma 3 Let f hg SH h ( ad g( are gve by ( atfe for ome h g h g (8 the f ee preervg harmoc uvalet ad f SH ( Furthermore f TSH ( f ad oly f (8 hold Obvouly Ieualty (8 a geeraled eualty eurg f to be clae SH * ( ad SH ( for ad repectvely We ee that f eualty (8 hold eualty ( mut hold for ay ad for both are ame Hece eualty (8 for ad eure that f SH * ( ad thu t ued th tudy If g ( we deote SH ( S ( whch ( tuded by Wag et al [7] for ( ( * the repectve cla The cla S ( S troduced by Saaguch [8] whoe member atfy the codto h ( Re h ( where h ( h( h ( oectvty of hypergeometrc fucto wth harmoc fucto ee through ome of the recet paper [9-] Specally volvemet of Wrght geeraled hypergeometrc (Wgh fucto tuded [-3] Some Wgh eualte for tarle ad covex clae have already bee obtaed [3] for certa harmoc fucto The Wrght geeraled hypergeometrc (Wgh fucto [45] for potve real umber a ( p ad b B ( wth p B defed by a a ap p b B b B bp Bp p ; Z p a ; b B! a p Z b B (9 p f B ad f p t abolutely coverget for Referrg to [6] the ere (9 abolutely coverget B B B p p b a ad for p p Ivolvg Wgh fucto a defed (9 we coder a uvalet harmoc fucto W( of the form: where B B W( H( G( SH ( a b B ( b H ( ; ( a ( ad ( d c ( G ( ; d D ( c ( a ( ( b : (3 b ( B ( a c ( ( d : (4 d ( D ( c Deote for ome j ad for ay opyrght ScRe

3 466 P SHRM j a j ( j : ; b jb B j c j ( j : ; d jd D It oted that at correpodg ere of j j j j coverge abolutely to j j repectvely f B D B D B B wth D D or ad ether or B B b a j D D d c j (5 Hece from (3 ad (4 we ca ealy derve followg dette for ome j ad ( b j j (6 ( a j j ( d j j (7 ( c j j provded codto ( or ( of (5 hold The ymbol ( called Pochhammer ymbol for o egatve defed by ( ( ( ( The object of th paper to exame ome Wgh eualte a a eceary ad uffcet codto for uvalet harmoc aalytc fucto aocated wth certa Wgh fucto to be the fucto cla SH ( for ome ad partcular SH * ( ad SH ( Some coeuet reult ad a covoluto property are alo derved Some Wgh Ieualte I order to derve Wgh eualte we ue Lemma 3 Theorem Let W( H( G( SH be gve by ( f for B D or B D ad ether for B B ome B D B D d c D D or wth b a ad for ome Wgh eualty ( b ( a ( d (8 ( c hold the W( ee preervg harmoc uvalet ad W( SH ( H( Furthermore W ( G( TSH ( f ad oly f (8 hold Proof To how W( ee preervg harmoc uvalet ad W( SH ( we eed to how by Lemma 3 that S : ( (9 From the gve hypothe ad wth the ue of dette (6 ad (7 for j ad for ay we oberve that S ( ( ( b ( a ( d ( c f eualty (8 hold Furthermore fw ( ( TSH by Lemma 3 eualty (9 hold ad hece (8 hold Th prove Theorem Tag Theorem we get followg reult orollary Wth the ame hypothe of Theorem for f Wgh eualty opyrght ScRe

4 P SHRM 467 ( b ( a ( d ( ( c ( hold the W( ee preervg harmoc uvalet ad W( SH ( Furthermore * H( * W ( G( TSH ( f ad oly f ( hold Remar Tag B ( ad D ( the eualty of orollary cocde wth Theorem 3 [3] for p Tag Theorem we get followg reult orollary Wth the ame hypothe of Theorem for f Wgh eualty ( b ( a ( d ( ( c hold the W( ee preervg harmoc uvalet ad W( SH ( H( Furthermore W ( G( TSH ( f ad oly f ( hold 3 oeuece of Wgh Ieualte Ivolvg Mttag-Leffler fucto [5]: E ( B b ( ; ( E b B D d( ; for potve d D real umber b B ad d D we coder a uvalet harmoc fucto E ( for of the form: E ( ( b E ( ( d E ( SH ( Deote for ome B b D d j ad j ( j EB b jb ( ; b jb B j ( j ED d jd ( ; d jd D j t correpodg ere of EB b jb ( j ED d jd ( coverge abolutely to E j j B b jb E B b jb ( E ( E j j D d jd D d jd repectvely Followg reult ca be drectly obtaed from Theorem orollary 3 Let E ( be defed by ( f for ome eualty D d D d D d D ( b E E E B b B b B b B ( d E E E (3 hold the E ( ee preervg harmoc uvalet ad E ( SH ( Furthermore E ( ( b E ( ( d E ( TSH ( B b D d f ad oly f (3 hold Reult mlar to the orollare ad for E ( ad E ( ca be obtaed by tag ad repectvely orollary 3 O tag B 3 ad D 3 W( reduce to F( F ([ a ] F ([ c ] SH (4 whch volve the geeraled hypergeometrc fucto: a ( F ([ a] ; b c ( F ([ c] ; d lo f = B = =3 ad = D = =3 for ome j ad we get b ( a j j j = ( j F a ( b where = = j d ( c j j j = ( j F = c = ( d j ( a j ( j F := F ([ a ] j j l l l = = ( b j l l! ( c j ( j j j l l := ([ ] l = = ( d j l l! F F c provded b a j d c j From Theorem we obta followg reult orollary 4 Let F( be defed by (4 f for ome ad b a d c eualty a F F F b = opyrght ScRe

5 468 P SHRM c F F F (5 = d hold the F( ee preervg harmoc uvalet ad F( SH ( Furthermore F ( F ([ a] F ([ c] TSH( f ad oly f (5 hold Reult mlar to the orollare ad for F( ad F ( ca be obtaed by tag ad repectvely orollary 4 Further tag b d orollary 4 we get followg reult for a harmoc uvalet fucto defed by Gau hypergeometrc fucto orollary 5 Let for potve real value of a a b c c d ad for a harmoc uvalet fucto: G( F a a ; b; F c c ; d ; SH If for ome ad b a a d c c > eualty a a > F ([ a] F ([ a] baa cc F ([ c] F ([ c] dcc ( (6 hold the G ( ee preervg harmoc uvalet ad G ( SH( Furthermore G F a a b F c c d TSH ( ; ; ; ; ( f ad oly f (6 hold Reult mlar to the orollare ad for G ( ad G ( ca be obtaed by tag ad repectvely orollary 5 4 ovoluto Property I th ecto we obta a covoluto property for fucto belogg to the cla SH ( Theorem fucto f = hg SH ( for ome f ad oly f h ( g ( = < < Proof From the defto of the fucto cla SH ( f SH ( f ad oly f f ( ( f ( for = < < Hece by mple calculato we get f ( f( ( f( Ug (3 we get h ( g ( h( g( ( ( ( h g whch ealy derve the reult Baed o Theorem we get that harmoc fucto W( H( G( E ( ( b E ( ( d E ( ad B b D d F( F ([ a] F ([ c] defed ( ( ad (4 repectvely belog to the cla SH ( for ome f ad oly f for = < < H( G ( ( b EB b( ( d ED d( ad F ([ a] F ([ c] repectvely hold opyrght ScRe

6 P SHRM Referece [] J lue ad T Shel-Small Harmoc Uvalet Fucto ale cademae Scetarum Fecae Sere I Mathematca Vol pp 3-6 [] J M Jahagr Harmoc Fucto Starle the Ut D Joural of Mathematcal aly ad pplcato Vol 35 No 999 pp [3] O P huja ad J M Jahagr Saaguch-Type Harmoc Uvalet Fucto Scetae Mathematcae Japocae Vol 59 4 pp [4] H Ö Ġuey Saaguch-Type Harmoc Uvalet Fucto wth Negatve oeffcet Iteratoal Joural of otemporary Mathematcal Scece Vol No 7 pp [5] M l Sha ad M Daru O Subcla of Harmoc Starle Fucto wth Repect to K-Symmetrc Pot Iteratoal Mathematcal Forum Vol No 57 7 pp [6] M l Sha ad M Daru O Harmoc Uvalet Fucto wth Repect to K-Symmetrc Pot Iteratoal Joural of otemporary Mathematcal Scece Vol 3 No -4 8 pp -8 [7] Z G Wag Y Gao ad S M Yua O erta Subclae of loe-to-ovex ad Qua-ovex Fucto wth Repect to K-Symmetrc Pot Joural of Mathematcal aly ad pplcato Vol 3 No 6 pp 97-6 [8] K Saaguch O erta Uvalet Mappg Mathmatcal Socety of Japa Vol 959 pp 7-75 [9] O P huja Plaar Harmoc ovoluto Operator Geerated by Hypergeometrc Fucto Itegral Traform ad Specal Fucto Vol 8 No 3 7 pp [] O P huja Harmoc Starle ad ovexty of Itegral Operator Geerated by Hypergeometrc Sere Itegral Traform ad Specal Fucto Vol No 8 9 pp [] O P huja ad H Slverma Ieualte ocatg Hypergeomatrc Fucto wth Plaer Harmoc Mappg Joural of Ieualte Pure ad ppled Mathematc Vol 5 No 4 4 [] M K ouf ad J Do Dtorto ad ovolutoal Theorem for Operator of Geeraled Fractoal alculu Ivolvg Wrght Fucto Joural of ppled aly Vol 4 No 8 pp 83-9 [3] M K ouf ad J Do erta la of alytc Fucto ocated wth the Wrght Geeraled Hypergeometrc Fucto Joural of Mathematc ad pplcato Vol 3 8 pp 3-3 [4] J Do ad R K Raa Famle of alytc Fuc to ocated wth the Wrght Geeraled Hypergeometrc Fucto Demotrato Mathematca Vol 37 No 3 4 pp [5] J Do ad R K Raa Some Reult Baed o Frt Order Dfferetal Subordato wth the Wrght Geeraled Hypergeometrc Fucto ommetar Mathematc Uvertat Sact Paul Vol 58 No 9 pp [6] J Do R K Raa ad H M Srvatava Some lae of alytc Fucto ocated wth Operator o Hlbert Space Ivolvg Wrght Geeraled Hypergeometrc Fucto Proceedg of the Jagjeo Mathematcal Socety Vol 7 4 pp [7] G Muruguudaramoorthy ad R K Raa O a Subcla of Harmoc Fucto ocated wth the Wrght Geeraled Hypergeometrc Fucto Hacettepe Joural of Mathematc ad Stattc Vol 38 No 9 pp 9-36 [8] G Muruguudaramoorthy ad K Vjaya Subcla of Harmoc Fucto ocated wth Wrght Hypergeometrc Fucto ppled Mathematc Vol No pp [9] R K Raa erta Subclae of alytc Fucto wth Fxed rgumet of oeffcet Ivolvg the Wrght Fucto Tamu Oxford Joural of Mathematcal Scece Vol No 6 pp 5-59 [] R K Raa O Geeraled Wrght Hypergeometrc Fucto ad Fractoal alculu Operator Eat a Joural of Mathematc Vol No 5 pp 9-3 [] R K Raa ad P Sharma Harmoc Uvalet Fucto ocated wth Wrght Geeraled Hyper- geometrc Fucto Itegral Traform ad Specal Fucto ommucated for Publcato [] P Sharma la of Multvalet alytc Fucto wth Fxed rgumet of oeffcet Ivolvg Wrght Geeraled Hypergeometrc Fucto Bullet of Mathematcal aly ad pplcato Vol No pp [3] P Sharma Multvalet Harmoc Fucto Defed by M-Tuple Itegral Operator ommetatoe Mathematcae Vol 5 No pp 87- [4] E M Wrght The ymptotc Expao of the Geeraled Hypergeometrc Fucto Proceedg Lodo Mathematcal Socety Vol 46 No 946 pp [5] H M Srvatava ad H L Maocha Treate o Geeratg Fucto Halted Pre Ell Horwood Lmted Hcheter 984 [6] Klba M Sago ad J J Trujllo O the Geeraled Wrght Fucto Fractoal alculu ad ppled aly Vol 5 No 4 pp opyrght ScRe

International Journal of Pure and Applied Sciences and Technology

International Journal of Pure and Applied Sciences and Technology It J Pure Appl Sc Techol, () (00), pp 79-86 Iteratoal Joural of Pure ad Appled Scece ad Techology ISSN 9-607 Avalable ole at wwwjopaaat Reearch Paper Some Stroger Chaotc Feature of the Geeralzed Shft Map