Applications of geometric function theory related to mechanical systems

Transcription

1 Rev Téc Ig Uiv Zulia Vol 9 Nº Applicatios of geometric fuctio theory related to mechaical systems CRamachadra ad RAmbrosePrabhu Departmet of Mathematics Uiversity College of Egieerig Villupuram Villupuram Tamil Nadu Idia ID: crjsp004@yahoocom Departmet of Mathematics College of Egieerig Guidy Aa Uiversity Cheai Tamil Nadu Idia ID: acyamb@yahoocom Abstract:- The purpose of the preset paper is to show some applicatio of fractioal calculus to the solutio of time-depedet viscous-diffusio fluid mechaics problems are preseted The classical trasiet viscous-diffusio equatio i a semi-ifiite space is show to yield explicit aalytical (fractioal) solutios for the shear stress ad fluid speed aywhere i the domai Comparig the fractioal results for boudary shear-stress ad fluid speed to the existig for a aalytical results The fractioal methodology is validated ad show to be much simpler ad more powerful tha existig techiques Also we try to derive the Feete-segö iequality for the class R ( ) of ormalied aalytic fuctios f() defied o the ope uit dis for which the class of fuctio ( ) to the real axis Some coefficiet estimatios are also obtaied 00 AMS Subject Classificatio: Primary 0C45 0C50 0C55 R lies i a regio star lie with respect to ad symmetric with respect Key words ad Phrases: Feete-segö iequality Aalytic fuctios Uivalet fuctios Subordiatio Liear operator INTRODUCTION Let A deote the class of all aalytic fuctios f () of the form () f () a which are aalytic i the ope uit dis U={ C: <} ad satisfy the coditio f(0) = 0 & f'(0) = We also deote by S the subclass of A cosistig of all fuctios which are uivalet i a ope uit dis U For fuctios f() ad g() aalytic i U we say that the fuctios f() is said to subordiate to g() if there exist a Schwar fuctio ω() aalytic i U with such that We deote this subordiatio by (0) 0 ad ( ) U f()=g(ω()) f g or f () g() U I particular if the fuctio g() is uivalet i U the above subordiatio is equivalet to f (0) = g(0) ad f(u ) g(u) Let φ() be a aalytic fuctio i U with φ(0)= φ'(0)>0 ad Re{φ()}>0 U which maps the ope uit dis U oto a regio starlie with respect to ad is symmetric with respect to the real axis Deote S * ( ) ad C (φ) respectively we deote the subclasses of the ormalied aalytic fuctio class A which satisfy the followig subordiatio relatios: f '( ) () f() ad f ''( ) ( ) U f '( ) These fuctio classes were itroduced ad studied by (Ma ad Mida 994) I their particular case whe ( ) ( ) U 0 these fuctio classes reduce respectively to the well-ow classes of order α i U ad C(α) (0 α< of covex fuctios of order α i U I the wor by (Ma ad Mida 994) the Feete-Segö iequality for fuctios i the class C(φ) was derived ad i view of the Alexader result U S * ( ) 0 of starlie fuctios 77

2 Rev Téc Ig Uiv Zulia Vol 9 Nº relatig the fuctio classes S * ( ) ad C(φ) the Feete-Segö iequality for fuctios i the class S * ( ) was also deduced For a brief history of the Feete-Segö problems for the starlie covex ad other various subclasses of the ormalied aalytic fuctio class A we refer the reader to the wor doe by (Srivatsava et al 00) ad (Ramachadra et al 009)] of course the mai result shall refer bac to (Feete ad seg o 9) i the year 9 After 0 years or so (Keoghad Meres969)solved the problem for certai subclasses of uivalet fuctios (Koepf 987) gave excellet results for the class of close-to-covex fuctios Recetly (Shamugam et al006) have studied the Feete-Segö problem for subclasses of starlie fuctios with respect to symmetric poits Motivated essetially by the aforemetioed wors we prove the Feete-Segö type coefficiet iequality i Theorem below for a more geeral class of ormalied aalytic fuctios For λ δ N N0 the authors (Darus 008) itroduced the operator defied by f c a ( ) [ ( ) ] ( ) () The basic mathematical ideas of fractioal calculus (itegral ad differetial operatios of oiteger order) were developed log ago by the mathematicias Leibi (695) Liouville (84) Riema (89) ad others brought to the attetio of the egieerig world by Oliver Heaviside i the 890s it was ot util 974 that the first boo o the topic was published by Oldham ad Spaier Recet moographs ad symposia proceedigs have highlighted the applicatio of fractioal calculus i physics cotiuum mechaics sigal processig ad electromagetic As a first part we obtai the Feete-segö iequality for the fuctio f A i the class ( ) follows: Defiitio : Let : Let A A is a liear operator ad f C a ( ) [ ( ) ] ( ) is aalytic i f(u) R defied as Where ( ) C( ) ( ) ( ) Whe λ= δ=0 we get the Sălăgea differetial operator Whe =0 or λ=0 gives Ruscheweyh operator δ=0 gives Al-oboudi differetial operator of order 0 f ( ) f ( ) f ( ) f '( ) 0 0 Defiitio : Let φ() be a uivalet starlie fuctio with respect to which maps the uit dis U oto a regio i the right half plae which is symmetric with respect to the real axis φ(0)= ad φ'(0)> A fuctio f A is i the class R ( ) ( if f ( ))' ( ) N N0 f() I order to prove our mai results we eed the followig lemma: Lemma (Ma ad Mida 994) If p c c is a aalytic fuctio with positive real part ( ) i U the 4v ; v 0 c vc ; 0 v 4v; v whe v<0 or v> the equality holds true if ad oly if p () or oe of its rotatios If 0<v< the the equality holds true if ad oly if p () or oe of its rotatio If v=0 the the equality holds true if ad oly if p ( ) 0 or oe of its rotatios If v= the the equality holds if ad oly if p is the reciprocal of oe of the fuctios such that the equality holds i the case of v=0 Also the above upper boud is sharp it ca be improved as follows whe 0<v<: 78

3 Rev Téc Ig Uiv Zulia Vol 9 Nº c vc v c 0 v ad c vc ( v) c v We also eed the followig result i our ivestigatio Lemma (Ravichadra et al 005) If p ( ) c c is a fuctio with positive real part i U the c vc max v The result is sharp for the fuctios () ad p give by p () p () FEKETE-SZEGÖ PROBLEM FOR THE FUNCTION CLASS ( ) By maig use of Lemma we prove the Feete-segö type for the class R ( ) Theorem Let R ( ) B B If f() give by () belogs to the class R ( ) if a a if if the Where ( )( ) ( B B ) B ( )( ) B ( )( ) ( B B ) B ( )( ) B Further if 4( )( )( ) ( )( ) B B ( ) ( )( ) the Ad if the Where B B ( )( ) ( )( ) a a {( B B ) B } a 4 ( )( ) B ( )( ) a a {( B B ) B } a 4 ( )( ) B ( )( ) ( )( ) ( )( ) The result is sharp Proof: If f R ( ) the there exists a Schwar fuctio () (0) 0 ad ( ) U such that ( f ( )) ( w ( )) f() is aalytic i U with 79

4 4 Rev Téc Ig Uiv Zulia Vol 9 Nº Defie a fuctio p () by w ( ) p ( ) w ( ) Sice () is a Schwar fuctio we see that Re{ p ( )} 0 ad p (0) 0 Defie a fuctio p() by ` p = λδ f = φ ω = + b λδ f + b + () From () we obtai b b ( ) ( ) a a ad a ( ) ( ) ( ) ( )( ) () Sice ( p ( )) () p ( p ( )) The p ( ) p ( ) p ( ) ad c c b b c c Equatig the coefficiets of ad =φ c + c c + () we obtai b = B c ad b = B c c + B 4 c (4) From () ad (4) we get Bc a ( ) ( ) B B a c c B ( ) ( )( ) B Therefore we have B B Bc a a c c B ( ) ( )( ) B 4( ) ( ) B { c vc} ( ) ( )( ) where B B B ( )( ) v B ( )( ) If the by Lemma ad Lemma 4 we obtai a a 4( )( )( ) B B ( )( ) this is the first part of Theorem Similarly if the by Lemma ad Lemma we obtai B ( )( ) a a 4( )( )( ) B B ( )( ) B ( )( ) If we see that B B a a { c vc } ( ) ( )( ) ( ) ( )( ) 80

5 5 Rev Téc Ig Uiv Zulia Vol 9 Nº Further If the a a ( ) a B C vc ( ) ( )( ) ( )( ) ( B B ) B B C ( )( ) B 4( ) ( ) B B c vc v c ( ) ( )( ) ( ) ( )( ) Fially we see that the If a a ( ) a ( )( ) ( B B ) B B C ( ) ( )( ) B ( ) ( ) c vc v c ( )( ) B ( ) ( )( ) To show that the bouds are sharp we defie fuctios ( ) by B C vc ( )( ) B 4( ) ( ) ( ( )) ( ) (0) 0 ( (0)) () G (0 by ad the fuctio F ad ) Clearly the fuctios G ( ( )) ( ) G (0) 0 ( G (0)) G ( ) ad G R ( ) K K F We also write If or the the equality i Theorem holds true if ad oly if f is K or oe of its rotatios the the equality holds true if ad oly if f is Whe If the the equality holds true if ad oly if f is F or oe of its rotatios K or oe of its rotatios If the the equality holds true if ad oly if f is G or oe of its rotatios By maig use of Lemma we ca easily obtai the followig theorem Theorem : Let F ( ) B B where the coefficiets R the If f() give by () belogs to ( ) ( ( )) ( ) F (0) 0 F (0) F ( ) B are real with B 0 ad B 0 a μa B + λ δ + δ + max B B B + μb δ + + λ δ + + λ μ C The result is sharp Remar : The coefficiet bouds for a ad a are special cases of those asserted by Theorem Remar : I its special case whe λ= δ=0 ad =0 we arrive at a ow result due to (Ma ad Mida994) APPLICATIONS TO ANALYTIC FUNCTIONS DEFINED BY USING FRACTIONAL CALCULUS OPERATORS AND CONVOLUTION The subject of fractioal calculus (that is calculus of itegrals ad derivatives of ay arbitrary real or complex order) has gaied cosiderable popularity ad importace durig the early decades Two of the most recet wors o this subject of widespread ivestigatios iclude rather comprehesive treatises o the theory ad applicatios of fractioal differetial equatios by (Podluby999) ad (Kilbas et al 006) 8

6 6 Rev Téc Ig Uiv Zulia Vol 9 Nº For the applicatio of the results give i the precedig sectios we first itroduce the class M ( ) which is defied by meas of the Hadamard product (or Covolutio) ad a certai operator of fractioal calculus ow as the Owa-Srivatsava operator for details see (owa978) (owa984) (owa et al987) ad (Srivatsava 00) Defiitio : The fractioal itegral of order δ is defied for a fuctio f() by f ( ) D f ( ) d( 0) () ( ) ( ) o where the fuctio f() is aalytic i a simply-coected domai of the complex -plae cotaiig the origi ad the multiplicity of ( ) is removed by requirig log( ζ) to be real whe ζ>0 Defiitio 4: The fractioal derivative of order δ is defied for a fuctio f() by f ( ) D f ( ) d(0 ) ( ) ( ) () where the fuctio f() is costraied ad the multiplicity of ( ) o is removed by requirig log( ζ) to be real whe whe ζ>0 Defiitio 5: Uder the hypothesis of Defiitio 4 the fractioal derivative of order (+δ) is defied for a fuctio f() by D +δ f = d D d δ f() (0 δ < ; N 0 ) () Usig Defiitios 4 ad 5 of fractioal derivatives ad fractioal itegrals Owa ad Srivatsava itroduced what is popularly referred to i the curret literature as the Owa-Srivatsava operator ( f )( ) ( ) D f ( )( 4 ) (4) I terms of the Owa-Srivatsava operator i the followig way: defied by (6) we ow itroduce the fuctio class M ( ) M ( ) f : f A ad f M ( ) (5) It is easily see that the fuctio class M g ( ) is a special case of the fuctio class M ( ) whe Suppose ow that The sice if ad oly if ( ) ( ) g( ) ( ) g( ) g ( g 0) (6) (7) f ( ) a M ( ) ( f * g)( ) g a M ( ) g we ca obtai the coefficiet estimates for fuctios i the class M ( ) from the correspodig estimates for fuctios i the class M ( ) By applyig Theorem to the followig Hadamard product (or covolutio): ( f * g)( ) g a g a we get Theorem give below after a obvious chage of the parameter μ Theorem Let 0 μ 0 α 0 β ad 0 λ Suppose ( ) B B B where the coefficiets g belogs to the class M ( ) the B are real with B 0 B 0 adb 0( N{}) If f() give by () 8

7 7 Rev Téc Ig Uiv Zulia Vol 9 Nº if 4 g a a if 4 5 g if 5 g where g ( )( ) ( B B ) B 4 g ( )( ) B g ( )( ) ( B B ) B 5 g ( )( ) B where ad η are defied as i Theorem respectively These results are sharp Sice by () ad the equatio (4) ( ) ( ) ( f )( ) a (8) ( ) we readily obtai () ( ) g (9) ( ) ad (4) ( ) 6 g (0) (4 ) ( )( ) For g ad g give by (9) ad (0) respectively Theorem reduces to the followig iterestig result Theorem Let 0 μ 0 α 0 β ad 0 λ Suppose where the coefficiets B are real with B > 0 B > 0 ad B > 0 ( N{}) g If f() give by () belogs to the class M ( ) the where ( )( ) if 6 6 ( )( ) a a if ( )( ) if 7 6 ( ) B B B ( ) ( )( ) ( B B ) B 6 ( ) ( )( ) B ( ) ( )( ) ( B B ) B 7 ( ) ( )( ) B where ad η are defied as i Theorem respectively Remar : I its special case whe λ = 0 β = α = 0 B = 8 B π = 6 π Theorem coicides with the followig result due to (Srivatsava et al000)for which f() is a parabolic Starlie fuctio defied by (Goodma 99) ad (Roig99) 8 6 Remar 4: Whe 0 0 B ad B Theorem would coicide with the result obtaied earlier by (Ma ad Mida 99) 4 CONCLUDING REMARKS The study of operators plays a importat role i the geometric fuctio theory May differetial ad itegral operators ca be writte as i terms of covolutio of certai aalytic fuctios It is observed that this 8

8 8 Rev Téc Ig Uiv Zulia Vol 9 Nº formalism brigs a ease i further mathematical exploratio ad also helps to uderstad the geometric properties of such operators better Refereces DarusM ad Al-shaqsiK Differetial Sadwich theorem with geeralied derivative operator ProcWorld Acad Sci Eg ad Tech 8 (008)-4 Feete M ad Gsegö EieBermerugberugeradeschlichteFutioeJLodMathSoc 8 (9) Goodma AW Uiformly Covex Fuctios A Polo Math 56 (99) 87-9 Keogh FR ad EPMeres A Coefficiet iequality for certai classes of aalytic fuctios ProcAmerMathSoc0 (969) 8- Kilbas AA HMSrivatsava ad JJTrujillo Theory ad Applicatios of Fractioal Differetial EquatiosNorth-Hollad Mathematics Studies North Hollad Mathematics Studies Elsevier (North-Hollad) Sciece Publishers Amsterdam Lodo New Yor 04 (006) Koepf W O the Feete-segö problem for Close-to-covex fuctios II ArchMath49 (987) Koepf W O the Feete-segö problem for Close-to-covex fuctios II ProcAmerMathSoc0 (987) Owa S A Applicatio of the Fractioal Derivative Math Japa9 (984) 8-89 Ma W ad D Mida A Uified treatmet of some special classes of uivalet fuctios i Proceedigs of the Coferece o Complex Aalysis (Z Li F Re L Yag ad S Zhag Editors) Coferece proceedigs ad Lecture otes i aalysis Vol I Iteratioal Press Cambridge Massachusetts Ma W ad D MidaUiformly Covex FuctiosII A Polo Math58(99) Owa S O the Distortio Theorems KyugpooMathJ 8 (978) 5-59 Owa S ad HMSrivatsava Uivalet ad Starlie Geeralied Hypergeometric Fuctios Caad JMath9 (987) Podluby IFractioal Differetial Equatios:A Itroductio to Fractioal Derivatives Fractioal Differetial Equatios to Methods of their solutios ad some of their Applicatios Mathematics i Sciece ad Egieerig Academic Press New Yor Lodo ad Toroto 98 (999) Ramachadra C SSivasubramaia HMSrivatsava ad ASwamiatha Coefficiet iequalities for certai subclasses of aalytic fuctios ad their applicatios ivolvig the Owa-Srivatsava operator of fractioal calculus Iteratioal joural of Mathematical Iequalities ad Applicatios () (009) 5-6 Ravichadra V MetiBolcal YasarPolotoglu ad A Se Certai Subclasses of Starlie ad Covex fuctios of complex order Hacettepe Joural of Mathematics ad Statistics Hacpatte Joural of Mathematics 4 (005) 9-5 Roig F Uiformly Covex Fuctios ad a correspodig class of starlie Fuctios ProcAmerMathSoc8 (99) Shamugam TN CRamachadra ad VRavichadra "Feete-segöProblem for Subclass of Starlie Fuctios With Respect to Symmetric Poits" Bulleti of the Korea Mathematical Society 4() (006) Srivatsava HM Some families of fractioal derivative ad other liear operators associated with aalytic uivalet ad multivalet fuctios i Aalysis ad its applicatios(cheai 000) Allied Publishers Limited New Delhi Mumbai Calcutta ad Cheai (00) 09-4 Srivatsava HM ad AKMishra Applicatios of Fractioal Calculus to Parabolic Starlie ad Uiformly Covex Fuctios Comput Math Appl 9(-4) (000) Srivatsava HM AKMishra ad MKDas the Feete-segö problem for a subclass of Close-to-covex fuctios Complex Variables Theory Appl 44 (00) 45-6 Srivatsava HM ad Owa Uivalet Fuctios Fractioal Calculus ad their Applicatios Halsted Press(Ellis Horwood Limited Chichester Joh Wiley ad Sos New Yor Chichester Brisbae ad Toroto (989) 84