Certain Aspects of Univalent Function with Negative Coefficients Defined by Bessel Function

Transcription

1 Egieerig, Tehology ad Tehiques Vol59: e66044, Jauary-Deember 06 ISSN Olie Editio BRAZILIAN ARCHIVES OF BIOLOGY AND TECHNOLOGY A N I N T E R N A T I O N A L J O U R N A L Certai Aspets of Uivalet Futio with Negative Coeffiiets Defied by Bessel Futio Chellakutti Ramahadra *, Kaliyaperumal Dhaalakshmi ad Lakshmiarayaa Vaitha Departmet of Mathematis; Uiversity College of Egieerig; Villupuram; Aa Uiversity; Tamiladu; Idia ABSTRACT Key words: I reet years, appliatios of Bessel futios have bee effetively used i the modellig of hemial egieerig proesses ad theory of uivalet futiosi this paper, we study a ew lass of aalyti ad uivalet futios with egative oeffiiets i the ope uit disk defied by Modified Hadamard produt with Bessel futio We obtai oeffiiet bouds ad exterior poits for this ew lass 00 AMS Mathematis Subjet Classifiatio: Primary 30C45, Seodary 30C50, 30C80 Keywords:Aalyti futio, Uivalet futio, Extreme poit, Bessel futio, Subordiatio ad Hadamard Produt * Author for orrespodee: rjsp004@yahooom Bra Arh Biol Tehol v59: e6044 Ja/De 06 Spe Iss

2 Ramahadram, C et al INTRODUCTION May importat futios i applied siees are defied via improper itegrals or series (or ifïite produts The geeral ame of these importat futios are alled speial futios Bessel futios are importat speial futios whih are playig the importat role i studyig solutios of differetial equatios Espeially, the liear PDE desribig various hemial trasfer proesses, allow the exat solutio expressed i terms of oe speial kid of Bessel s futios ad they are assoiated with a wide rage of problems i importat areas of mathematial physis, modellig of trasfer proesses i hemial egieerig as well as i the related fields like hydrodyamis, heat trasfer, diffusio, bioproesses ad so o By usig the method of seperatio of variables, exat solutio i terms of Bessel futio a be used to alulate several importat parameters whih are eeded i desig ad ostrutio of hemial egieerig apparatuses ad equipmet like heat exhagers ad their ompoets Typial example for the effiiey alulatio is applied i Brailia powdered milk plat [] I aother ase whe the Bessel futios arises is heat trasfer modellig whih osidered i [6] Here the problem of ross-flow streamig of heated objet with large value of legth to diameter ratio (like thermoaemometer is solved for small Pe umbers usig the theory of aalyti futios Reetly Dei [9] has studied the followig: The geeralied Bessel futio of the first kid of order u, is defied as futio,,(, has the familiar represetatio as follows u b ( u, b, ( =, b C ( =0! ( u Where stads for the Euler gamma futio The series ( permits the study of Bessel futio i a uified maer It is alled the partiular solutio of the followig seod-order liear homogeeous differetial equatio (see for details[5]: ( b( [ u ( b u] ( = 0, u, b, C ( Also i Dei et al [7] ad Dei [8] (see also[, 3, 4, 0] studied the futio,,( defïed, i u terms of the geeralied Bessel futio,,( by the trasformatio u b u u b u, b, ( = u u, b, ( (3 By usig the well kow Pohhammer symbol (or the shifted fatorial ( is defied i terms of the Euler futio, by (, = 0 ( = = ( ( (, N it beig uderstood ovetioally that ( 0 = We obtai the followig series represetatio for the futio,,( give by (3: u b ( C (4 u, b, ( =, = 4 (! b where = u Z 0, N ={,, } ad Z 0 = {0,,, } For oveiee, we write ( = ( Next, we itrodue a operator : S S, whih is defied by the Hadamard, u, b, produt B u b Bra Arh Biol Tehol v59: e6044 Ja/De 06 Spe Iss

3 Uivalet Futio defied by Bessel Futio 3 where ( a B f ( =, ( f ( = = 4 (! ( (,, = 4 ( (! D ( a = = D(,, a = 4 ( (! = It is easy to verify that from (5 [ B f ( ] = B f ( ( B f ( b where = p Z 0 I fat the futio B give by (5 is a elemetary trasformatio of the geeralied hypergeometri futio Hee, it is easy to see that B f ( = 0F( ; * f ( ad 4 also, ( = 0 F ( ; Let A be the lass of all aalyti futios 4 f ( = a (6 = i the uit disk U ={ : < } Let S be the sublass of A osistig of uivalet futios Suppose * for 0 <, S ( ad C ( deote the sublasses of A osistig of futios whih satisfy the f ( f ( followig iequalities: >, ad >, f ( f ( are, respetively, starlike ad ovex of order i U I partiular, we set S (0 = S ad C(0 = C Let T deote the sublasses of S osistig of futios f( give by = (5 f ( = a, a 0 (7 with egative oeffiiets Silverma[3] itrodued ad ivestigated the followig sublasses of the futio lass T : T ( = S ( T ad C ( = C( T, 0 < (8 For f A give by (6 ad g A is give by ovolutio of f( ad g ( is give by = g( = b ( * ( = = ( * (, = Reetly Shamugam et al[] ad [5] have studied the followig:, the Hadamard produt (or f g a b g f U Defiitio Let >, 0 <, k 0, 0 < ad U, a futio f T is said to be i the lass UB (,, k, if it satisfies the followig iequality: F( F( > k F ( F( where ( = ( ( F B f ( ( B f ( Lemma [] Let w = u iv The Re( w > if ad oly if w( w( (9 Bra Arh Biol Tehol v59: e66044 Ja/De 06 Spe Iss

4 4 Ramahadram, C et al Lemma 3 [] Let w = u iv ad, are real umber The Re( w > w i i Re w( e e > Coeffiiet Bouds ad Extreme Poits if ad oly if We obtai the eessary ad suffiiet oditio ad extreme poits for the futio f( i the lass UB (,, k, Theorem Let >, 0 <, k 0, 0 < ad U The futio f( defied by equatio (6 is i the lass UB (,, k, if ad oly if k ( ( k ( k( D(,, ( = Proof From the defiitio, we have ( B f ( ( ( B f ( ( B f ( ( B f ( ( B f ( ( B f ( ( ( B f ( ( B f ( k ( B f ( ( B f ( From Lemma 3, we have ( B f ( ( ( B f ( ( B f ( i i Re ( ke ke ( B f ( ( B f (, or equivaletly i i [( B f ( ( ( B f ( ( B f ( ]( ke ke [( B f ( ( B f ( ] ( B f ( ( B f ( ( B f ( ( B f ( ( Let i i F( = [( B f ( ( ( B f ( ( B f ( ]( ke ke [( B f ( ( B f ( ] ad ( = ( E B f ( ( B f ( By Lemma, equatio ( is equivalet to F( ( E( F( ( E(, for 0 <But (,, D(,, a b D a b i ( ( ( (,, F E ke D a b D(,, a b k D(,, a b Bra Arh Biol Tehol v59: e66044 Ja/De 06 Spe Iss

5 Uivalet Futio defied by Bessel Futio 5 (,, D a b i ( ( ( (,, F E ke D a b D(,, a b D(,, a b k D(,, a b Hee, F( ( E( F( ( E( (( a b ( ( D(,, a b = k ( ( D(,, a b ] 0 = or( ( D(,, a k ( ( D(,, ab = = whih is equivalet to = [( [ ( k( ( ] k] D(,, a b oversely, suppose that the equatio ( holds good, the we have to prove that F( i i ( ke ke F( Now hoosig the values of o the positive real axis where 0 = r<, the above iequality redues to i i i ( ( ke ( ( ke ( D(,, abr ke = 0 ( D(,, abr = i i Sie ( e e =, the above iequality redues to ( [ ( k( ( k( ] D(,, abr k = 0 ( D(,, abr = Lettig r, we get the desired result Corollary Let >, 0 <, k 0, 0 < ad U, if f UB (,, k,, the Bra Arh Biol Tehol v59: e66044 Ja/De 06 Spe Iss

6 6 Ramahadram, C et al a k [ ( ( k ( k( ] D(,, b Theorem 3 Let, N, 0 <, >, 0 <, k 0 ad U If f ( = ad f( =, k [ ( ( k ( k( ] D(,, b The f U (,, k, a, if ad oly if it a be expressed i the form Proof Let But = f ( = f ( 0 ad = or = = = = f ( = f(, where 0 ad = or = The = f ( = k [ ( ( k ( k( ] D(,, f( = = = k [ ( ( k ( k( ] D(,, b k [ ( ( k ( k( ] D(,, b = = Usig Theorem, we have f UB (,, k, Coversely, Let us assume that f( is of the form (6 belogs to UB (,, k,, The a, N, k [ ( ( k ( k( ] D(,, b Settig k [ ( ( k ( k( ] D(,, = a b ad = = we have Hee the proof 3 Growth ad Distortio Theorem f ( = f ( = f ( f ( = = Theorem 3 If f UB (,, k, ad = r < the r r f ( r r k ( ( ( k ( ( ( 4 4 (3 Equality i (3 holds true for the futio f( give by Bra Arh Biol Tehol v59: e66044 Ja/De 06 Spe Iss

7 Uivalet Futio defied by Bessel Futio 7 f ( = (3 k ( ( ( 4 Proof we oly prove the seod part of the iequality i (3, sie the first part a be derived by usig similar argumets If f UB (,, k,, by usig Theorem, we fid that k ( ( ( a = k ( ( ( a 4 4 = = k ( ( k ( k( D(,, a = whih readily yields the followig iequality a (33 = k ( ( ( 4 Moreover it follows that f ( = D(,, a r r a r r = = = k ( ( ( 4 whih proves the seod part of the iequality i (3 Theorem 3 If f UB (,, k,, ad = r < the ( ( r f ( r (34 k ( ( ( k ( ( ( 4 4 Equality i (3 holds true for the futio f( give by (3 Proof Our proof of Theorem 3 is muh aki to that of Theorem 3 Ideed, sie f UB (,, k,, it is easily verified from (7 that ad f ( a r a = = (35 f ( a r a = = (36 The assertio (34 of Theorem 3 would ow follow from (35 ad (36 by meas of a rather simple osequee of (33 give by ( a (37 = k ( ( ( 4 The ompletes the proof of Theorem 3 4 Hadamard Produt Theorem 4 Let = = f ( = a ad g( = b belogs to UB (,, k, The the Hadamard Produt of f( ad g ( give by Bra Arh Biol Tehol v59: e66044 Ja/De 06 Spe Iss

8 8 Ramahadram, C et al Proof Sie f( ad ( ( f * g( = a b belogs to UB (,, k, = g belogs to UB (,, k,, we have k [ ( ( k ( k( ] D(,, b a = ad k [ ( ( k ( k( ] D(,, a b = ad by applyig the Cauhy-Shwar iequality, we have k [ ( ( k ( k( ] D(,, ab ab = ( ( k ( k( D(,, b a = However,we obtai = ( ( k ( k( D(,, a b ( ( k ( k( D(,, ab ab = ( ( k ( k( D(,, Now we have to prove that ab = ( ( k ( k( D(,, ab = ( ( k ( k( D(,, ab = ab = Hee the proof 5 Appliatio of the Fratioal Calulus Various operators of fratioal alulus (ie fratioal derivative ad fratioal itegral have bee rather extesively studied by may researhers(see for example [4] Eah of these theorems would ivolve ertai operator of fratioal alulus whih are defied as follows Defiitio 5 The fratioal itegral operator of order is defied for a futio f( by f( t D ( f ( = dt, > 0 0 ( (5 ( t where f( is aalyti futio i a simply oeted regio of -plae otaiig the origi ad the multipliity of ( t is removed by requirig log( t to be read whe ( t > 0 Bra Arh Biol Tehol v59: e66044 Ja/De 06 Spe Iss

9 Uivalet Futio defied by Bessel Futio 9 Defiitio 5 The fratioal derivative of order is defied for a futio f( by d f ( t D ( f ( = dt, 0 < ( d 0 (5 ( t where f( is aalyti futio i a simply oeted regio of -plae otaiig the origi ad the multipliity of ( t is removed by requirig log( t to be read whe ( t > 0 Defiitio 53 The fratioal derivative of order k is defied by k k d D ( f ( = D ( f (, 0 < (53 k d From defiitio (5 ad (5, after a simple omputatio we obtai ( D f ( = a (54 ( ( = ( D f a (55 ( = ( = ( Now usig equatios (54 ad (55, Let us prove the followig theorems: Theorem 54 Let f UB (,, k, The ( D f ( ( (56 ( [ k ( ( ( b ] 4 ( D f ( ( (57 ( [ k ( ( ( b ] 4 The iequalities (56 ad (57 are attaied for the futio f give by f ( = (58 k ( ( ( b 4 Proof From Theorem, we obtai a (59 = k ( ( ( b 4 Usig equatio (55, we obtai ( D f ( = (, a (50 ( ( suh that (, =, ( where (, is a dereasig futio of ad 0 < (, (, = Usig equatio (59 ad (50, we obtai ( ( D f ( (, a = ( [ k ( ( ( b ] 4 Whih is the equatio (56 Similarly we a get equatio(57 Bra Arh Biol Tehol v59: e66044 Ja/De 06 Spe Iss =

10 0 Ramahadram, C et al Theorem 55 Let f UB (,, k, The ( D f ( ( ( [ k ( ( ( ] b 4 ( ( [ k ( ( ( ] b 4 The iequalities (5 ad (5 are attaied for the futio f give by f ( = k ( ( ( b 4 Proof Usig equatio (55, we obtai ( D f ( = (5 (5 (53 ( D f ( = (, a (54 ( ( suh that l(, =, ( where (, is a dereasig futio of ad 0 < (, (, = Usig equatio (59 ad (54, we obtai ( ( D f ( (, a = ( [ k ( ( ( ] 4 Whih is the equatio (5 Similarly we a get equatio (5 Corollary 56 For every f UB (,, k,, we have ( f ( t dt 0 3( k ( ( ( b 4 ad 3( k ( ( ( b 4 ( ( ( k ( ( ( b k ( ( ( b f ( 4 4 Bra Arh Biol Tehol v59: e66044 Ja/De 06 Spe Iss

11 Uivalet Futio defied by Bessel Futio Proof By Defiitio 5 ad Theorem 54 for =, we have Also by Defiitio 5 ad Theorem 55 for =0, we have Hee the result is true 6 Radii of lose-to-ovexity Starlikeess ad Covexity Theorem Theorem 6 Let the futio f( defied by f ( = a D f ( = f ( t dt 0 d D f ( = f ( t dt = f ( d 0 = f( is lose to ovex of order (0 < < i < r (,, k,, where 0, the result is true be i the lass UB (,, k,, The [ k [ ( ( k ( k( ] D(,, b ]( r ( The result is sharp for the futio f( give by f ( = k [ ( ( k ( k( ] D(,, b = (,, k,, = if = Proof It is suffiiet to show that f(, for < r (,, k, a, Hee ad if f ( = a a = a (6 = Thus by Theorem, (6 will holds true if k [ ( ( k ( k( ] D(,, b = k [ ( ( k ( k( ] D(,, b ( (or if ( The theorem follows easily from previous equatio = Theorem 6 Let the futio f( defied by f ( = a be i the lass UB (,, k,, The = f( is starlike of order (0 < < i < r (,, k,, where Bra Arh Biol Tehol v59: e66044 Ja/De 06 Spe Iss

12 Ramahadram, C et al [ k [ ( ( k ( k( ] D(,, b ]( r ( ( The result is sharp for the futio f( give by f ( = k [ ( ( k ( k( ] D(,, b = (,, k,, = if = Proof If f( is starlike it is suffiiet to show that Sie Sie if f (, for < r (,, k, a,, f( = = = = ( a ( a f ( = f( a a f ( f( a (6 = Thus by Theorem, (6 will holds true if k [ ( ( k ( k( ] D(,, b = k [ ( ( k ( k( ] D(,, b ( = (or if ( ( The theorem follows easily from previous equatio Theorem 63 Let the futio f( defied by f ( = a be i the lass UB (,, k,, The = f( is ovex of order (0 < < i < r3 (,, k,, where [ k [ ( ( k ( k( ] D(,, b ]( r ( ( The result is sharp for the futio f( give by = 3(,, k,, = if Bra Arh Biol Tehol v59: e66044 Ja/De 06 Spe Iss

13 Uivalet Futio defied by Bessel Futio 3 f k [ ( ( k ( k( ] D(,, b ( = = Proof Sie f( is ovex it is eough to show that Thus if f ( f( = f (, for < r3 (,, k,,, Sie f( ( a a = f ( f( ( a (63 = Hee by Theorem, (63 will holds true if k [ ( ( k ( k( ] D(,, b ( = k [ ( ( k ( k( ] D(,, b ( (or if ( ( The theorem follows easily from previous equatio REFERENCES = Adrás S, ad Bari A, Mootoiity property of geeralied ad ormalied Bessel futios of omplex order, Complex Var Ellipti Equ 009;54(7: Aqla E S, Some Problems Coeted with Geometri Futio Theory, PhD Thesis; 004 Pue Uiversity, Pue(Upublished 3 Bari A, Geometri properties of geeralied Bessel futios, Publ Math Debree 008; 73(-: Bari A ad Pousamy S, Starlikeess ad ovexity of geeralied Bessel futios, Itegral Trasforms Spe Fut 00; (9-0: Bari A, Geeralied Bessel futios of the first kid, Leture Notes i Mathematis; 994; Spriger, Berli, 00 6 Bertola V, Cafaro E, Thermal istability of visoelasti fluids i horiotal porous layers as iitial value problem, It J Heat Mass Trasfer 006; 49: Dei E, Orha H ad Srivastava H M, Some suffiiet oditios for uivalee of ertai families of itegral operators ivolvig geeralied Bessel futios; Taiwaese J Math 0; 5(: Dei E, Covexity of itegral operators ivolvig geeralied Bessel futios, Itegral Trasforms Spe Fut 03; 4(3: Dei E, Differetial subordiatio ad superordiatio results for a operator assoiated with the geeralied bessel futios, Arxiv 3 Apr 0: v[Math CV] 0 Prajapat J K, Certai geometri properties of ormalied Bessel futios, Appl Math Lett0; 4(: Bra Arh Biol Tehol v59: e66044 Ja/De 06 Spe Iss

14 4 Ramahadram, C et al Ribeiro Jr C P, Adrade M H C, Aalysis ad simulatio of the dryig-air heatig system of a Brailia powdered milk plat, Bra J Chem Eg (004; (: Shamugam T N, Ramahadra C ad Ambross Prabhu R, Certai aspets of uivalet futios with egative oeffiiets defied by Rafid operator, It J Math Aal (Ruse 03; 7(9-: Silverma H, Uivalet futios with egative oeffiiets, Pro Amer Math So 975; 5: Uivalet futios, fratioal alulus, ad their appliatios, Ellis Horwood Series: Mathematis ad its Appliatios, Horwood, Chihester; Srivastava H M, Shamugam T N, Ramahadra C ad Sivasubramaia S, A ew sublass of k - uiformly ovex futios with egative oeffiiets, J Iequal Pure Appl Math 007; 8(:Artile 43, 4 pp Reeived: February 03, 06; Aepted: July 4, 06 Bra Arh Biol Tehol v59: e66044 Ja/De 06 Spe Iss