An Application of Generalized Bessel Functions on Certain Subclasses of Analytic Functions

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1 Turkish Joural of Aalysis ad Nuber Theory, 5, Vol 3, No, -6 Available olie at htt://ubsscieubco/tjat/3// Sciece ad Educatio ublishig DOI:69/tjat-3-- A Alicatio of Geeralized Bessel Fuctios o Certai Subclasses of Aalytic Fuctios G Murugusudaraoorthy, T Jaai School of Advaced Scieces, VIT Uiversity Vellore - 63, Tailadu, Idia Corresodig author: gsoorthy@yahooco Received Noveber, ; Revised Deceber, ; Acceted Jauary 3, 5 Abstract The urose of the reset aer is to ivestigate soe characterizatio for geeralized Bessel fuctios of first kid to be i the ew subclasses of β uiforly starlike ad β uiforly covex fuctios of order α Further we oit out cosequeces of our ai results Keywords: uivalet, starlike, covex, uiforly starlike fuctios, uiforly covex fuctios, Bessel fuctios Cite This Article: G Murugusudaraoorthy, ad T Jaai, A Alicatio of Geeralized Bessel Fuctios o Certai Subclasses of Aalytic Fuctios Turkish Joural of Aalysis ad Nuber Theory, vol 3, o (5): -6 doi: 69/tjat-3-- Itroductio Let be the class of aalytic fuctios of the for f ( z ) = z + az z () As usual, we deote by the subclass of cosistig of fuctios which are oralized by f() = = f() ad also uivalet i Deote by the subclass of cosistig of fuctios whose ozero coefficiets fro secod o, is give by f ( z ) = z az, a () For fuctios f give by () ad g give by gz = z + bz, we defie the Hadaard roduct (or covolutio) of f ad g by ( f g )( z ) = z + abz, z (3) A fuctio f is said to be starlike of order α( α < ), if ad oly if zf( z) R > α ( z ) f( z) This fuctio class is deoted by () = : where ( α) We also write deotes the class of fuctios f such that f ( ) is starlike with resect to the origi A fuctio f is said to be covex of order α( α < ) if ad oly if zf( z) R + > α ( z ) f( z) This class is deoted by ( α) Further, = (), the well-kow stadard class of covex fuctios It is a established fact that Let f ( α) zf ( α) ( α) ad ( α) are the class of starlike ad covex fuctios of order α( α < ), itroduced ad studied by Silvera [] The class β was itroduced by Kaas ad Wi siowska [], where its geoetric defiitio ad coectios with the coic doais were cosidered The class β was defied ure geoetrically as a subclass of uivalet fuctios, that a each circular arc cotaied i the uit disk with a ceter ξ, ξ β( β < ), oto a covex arc The otio of β uiforly covex fuctio is a atural extesio of the classical covexity Observe that, if β = the the ceter ξ is the origi ad the class β reduces to the class of covex uivalet fuctios Moreover for β = corresods to the class of uiforly covex fuctios itroduced by Gooda [,], ad studied extesively by Røig [9,] The class β is related to the class β by eas of the wellkow Alexader equivalece betwee the usual classes of covex ad starlike fuctios Further the aalytic criterio for fuctios i these classes are give as below(also see [6,9,,]) For < α ad β a fuctio f is said to be i the class

2 Turkish Joural of Aalysis ad Nuber Theory (i) β uiforly starlike fuctios of order α is deoted by ( αβ, ) if it satisfies the coditio zf( z) zf( z) R α> β, z () f( z) f( z) ad (ii) β uiforly covex fuctios of order α deoted by ( αβ, ), if it satisfies the coditio zf( z) zf( z) R + α> β, z (5) f( z) f( z) Ideed it follows fro () ad (5) that f ( αβ, ) zf ( αβ, ) (6) Reark It is of iterest to ote that ( α,) = ( α) ad ( α,) = ( α) Motivated by above defiitios we defie the followig subclasses of due to Murugusudaraoorthy ad Magesh [6] For λ <, α < ad β, we let ( λα,, β) be the subclass of cosistig of fuctios of the for () ad satisfyig the aalytic criterio zf ( z) R α ( λ) f ( z) + λzf ( z) zf ( z) > β, z, ( λ) f ( z) + λzf ( z) (7) ad also, let ( λα,, β) be the subclass of cosistig of fuctios of the for () ad satisfyig the aalytic criterio f ( z) + zf( z) f ( z) + zf( z) R α > β, z (8) f ( z) λzf( z) + f ( z) + λzf( z) We further let ( λα,, β) = ( λα,, β) ad ( λα,, β) = ( λα,, β) Suitably secializig the araeters we ote that (, αβ, ) = ( αβ, ) [6] (,, β) = ( β) [] 3 (, α,) = ( α) [6] ( λα,,) = ( λα, ) [] 5 (, α,) = ( α) [] 6 (, αβ, ) = ( αβ, ) [6] 7 (,, β) = ( β) [3] 8 (, α,) = ( α) [6] 9 ( λα,,) = ( λα, ) [] (, α,) = ( α) [] Now we state the followig characterizatio roerties for the classes ( λα,, β), ( λα,, β), ( λα,, β) ad ( λα,, β) due to Murugusudaraoorthy ad Magesh [6] Theore A fuctio f( z ) of the for () is i ( λα,, β) if ( + β) ( α + β)( + λ λ)] a α (9) [ Theore A fuctio f( z ) of the for () is i ( λα,, β) if ad oly if [ ( + β) ( α + β)( + λ λ)] a () α Theore 3 A fuctio f( z ) of the for () is i ( λα,, β) if [ ( + β) ( α + β)( + λ λ)] a () α Theore A fuctio f( z ) of the for () is i ( λα,, β) if ad oly if [ ( + β) ( α + β)( + λ λ)] a () α We recall here a geeralized Bessel fuctio ω( bc,, ) = ω defied i [] ad give by ω( z) = ω( bc,, ) ( ) c z = (3) b +!Γ( + ) which is the articular solutio of the secod order liear hoogeeous differetial equatio z ω( z) + bzω( z) + [ cz + ( b)] ω( z) = () where bc,,, which is atural geeralizatio of Bessel s equatio The differetial equatio () erits the study of Bessel fuctio, odified Bessel fuctio, sherical Bessel fuctio ad odified sherical Bessel fuctios all together Solutios of () are referred to as the geeralized Bessel fuctio of order The articular solutio give by (3) is called the geeralized Bessel fuctio of the first kid of order Although the series defied above is coverget everywhere, the fuctio ω is geerally ot uivalet i It is of iterest to bc,, ote that whe b = c = we re-obtai the Bessel fuctio of the first kid ω,, = J, ad for c =, b = the fuctio ω,, becoes the odified Bessel fuctio Now, we cosider the fuctio ubc,,( z ) defied by the trasforatio b + bc,, = + ω bc,, u ( z) Γ z ( z), = By usig well kow ochhaer sybol (or the shifted factorial) defied, i ters of the failiar Gaa fuctio, by

3 Turkish Joural of Aalysis ad Nuber Theory 3 Γ( a+ ) ( a) : = Γ( a) ( = ), = aa ( + )( a+ ) ( a+ ) ( : = {,, 3, }) we ca exress u,,( z ) as u bc ( / ) z =, (5) b +! + bc,, z b + where +,,, This fuctio is aalytic o ad satisfies the secod-order liear differetial equatio z u( z) + ( + b + ) zu( z) + czu( z) = Now, we cosidered the liear oerator defied by ( c, ): ( c, ) f( z) = zu ( z) f( z) = z+ bc,, ( / ) az ( ) ( )! (6) ( b + ) where = + For coveiece throughout i the sequel, we use the followig otatios b + ubc,, = u, = + ad for if c<, > (,,, ) let, ( / ) z( u ( z)) = z z (7) ( ) ( )! The geeralized Bessel fuctio is a recet toic of study i Geoetric Fuctio Theory (eg see the work of [,3,,5] ad [] Motivated by results o coectios betwee various subclasses of aalytic uivalet fuctios by usig hyergeoetric fuctios (see [8,3,5,,5]) ad by work of Baricz [,3,,5], we obtai ecessary ad sufficiet coditio for the fuctio z( u ( z)) to belog to the classes ( λα,, β) ad ( λα,, β) Mai Results ad Their Cosequeces Lea [5] If b,, c ad,,, the the fuctio u satisfies the recursive relatio u ( z) = cu ( z) + for all z Theore 5 If c < ad > the z( u ( z)) ( λα,, β) if ad oly if [ + β λα ( + β)] u () + ( α)[ u () ] ( α) (8) roof Sice ( ) z u z = z z (9) ( ) ( )! accordig to Theore, we ust show that [ ] + β α + β + λ λ () ( ) ( )! ( α) Now, [ ( + β) ( α + β)( + λ λ)] ( ) ( )! ( )[ + β λα ( + β)] ( α) + ( ) ( )! = [ + β λα ( + β)] ( ) + α ( ) ( )! ( ) ( )! [ β λα ( β)] = ( α ) ( ) ( )! ( ) ( )! + β = ( ) α λα ( β) + = = + ( ) ( )! ( ) ( )! [ ] = + β λα+ β ( + ) ( )! ( ) + α ( ) ( )! = [ + β λα ( + β)] u+ () + ( α)[ u() ] = [ + β λα ( + β)] u () + ( α)[ u () ] But the last exressio is bouded above by α if ad oly if (8) holds Reark I articular whe c = ad b =, the coditio 8 becoes [ + β λα ( + β)] () + Γ( + ) α, () + ( α) () which is ecessary ad sufficiet coditio for / z( ζ ( z )) to be i ( λα,, β) where / / / = + ζ ( z ) Γ( ) z ( z ) Theore 6 If c < ad > the z( u ( z)) is i ( λα,, β) if ad oly if

4 Turkish Joural of Aalysis ad Nuber Theory ( α) u u + β λ α + β u + 3+ β α λ α + β + ( α) roof I view of Theore,we eed to show that ( α) Now [ ( + β) ( α + β)( + λ λ)] ( ) ( )! ( ) α β λ ( ) ( ) ( )! [ ( + β) ( α + β)( + λ λ)] ( ) ( )! ( )( ) + + β ( ) + ( + λ( + ) λ)( α + β) ( ) ( )! = [ + β λα ( + β)] ( + ) ( ) ( )! + () ( ) ( ) ( ) ( )! ( ) ( )! ( ) ( )! ( α + β)( λ) + ( ) + ( ) ( )! ( ) ( )! ( + ) + β ( ) ( )! = λα ( β) ( ) ( )! ( ) ( )! ( α β)( λ) + + ( ) ( )! ( ) ( )! + β ( ) ( )! = λα ( + β) ( ) ( )! ( ) ( )! α + β ( λ) + ( )! = + ( ) ( )! = [ + β λα ( + β)] ( ) ( )! + [3 + β α λα ( + β)] ( ) ( )! + ( α) ( ) ( )! + β = λα ( + β) ( ) + ( + ) ( )! 3+ β α + λα ( β) + ( + ) ( )! + ( α) ( ) ( )! = + β λ α + β u + ( + ) + 3+ β α λ( α + β ) u + ( α)[ u () ] = [ + β λα ( + β)] u () + [3 + β α λα ( + β)] u () + ( α)[ u () ] + By a silificatio, we see that the last exressio is bouded above by ( α) if ad oly if () holds 3 Iclusio roerties A fuctio f is said to be i the class ( AB, ),, ( \ {}, B < A ), if it satisfies the iequality f ( z) < ( z ) ( A B) B[ f ( z) ] The class ( AB, ) was itroduced earlier by Dixit ad al [9] If we ut =, A = β ad B = β ( < β ), we obtai the class of fuctios f satisfyig the iequality

5 Turkish Joural of Aalysis ad Nuber Theory 5 f ( z) < β ( z ; < β ) f ( z) + which was studied by (aog others) adaabha [7] ad Caliger ad Causey [7] Makig use of the followig lea, we will study the actio of the Bessel fuctio o the classes ( λα,, β) Lea [9] If f ( AB, ) is of for (), the a ( A B), \ {} (3) The result is shar Theore 7 Let If c < ad ad if the iequality > If f ( AB, ) [ + β λα ( + β)] u () ( A B) α + ( α)[ u () ] () is satisfied, the ( c, )( f) ( λα,, β) roof Let f be of the for () belog to the class ( AB, ) By virtue of Theore, it suffices to show that L( α, β, λ) ( ) + β = a α = ( α + β)( + λ λ) ( ) ( )! Sice f ( AB, ) the by Lea we have a ( A B) Hece L( αβλ,, ) = [ ( + β) ( α+ β)( + λ λ)] ( A B) ( ) ( )! ( + β ) ( α + β)( + λ λ) ( A B) ( ) ( )! ( )[ + β λα ( + β)] + ( α) ( A B) ( ) ( )! (5) ( ) + β λα+ β ( ) ( )! ( A B) + ( α) ( ) ( )! Further, roceedig as i Theore 5 L( αβλ,, ) [ ] + β λα+ β ( ) ( )! ( A B) ( α) + ( ) ( )! + + [ ] ( A B) β λα β u + () ( α)[ u () ] [ + β λα ( + β)] u () ( A B) + ( α)[ u () ] But this last exressio is bouded above by α if ad oly if (3) holds Theore 8 Let c < ad > the z (, c, z) = ( u ( t)) dt is i ( λα,, β) if ad oly if iequality [ + β λα ( + β)] u () + ( α)[ u () ] α (6) roof Sice (,, ) z cz = z ( ) ( )! the by Theore we eed oly to show that ( ) + β (,, ) αβλ = ( α β)( λ λ) + + ( )! Now [ ] + β α + β + λ λ ( )! [ ( β) ( α β)( λ λ)] ( ) ( )! Further, roceedig as i Theore 5 ( α, β, λ) + β ( ) + α λα ( + β) ( ) ( )! ( ) ( )! = [ + β λα ( + β)] u+ () + ( α)[ u() ] = [ + β λα ( + β)] u () + ( α)[ u () ] which is bouded above by α if ad oly if (6) holds

6 6 Turkish Joural of Aalysis ad Nuber Theory Cocludig Rearks If we ut c=- ad b= i above theores we obtai aalogous results of () Further by takig β= ad secializig the araeter λ we ca state various iterestig results (as roved i above theores) for the various subclasses listed i the itroductio Ackowledgeet The authors thak the referees for their valuable suggestios to irove the aer i reset for Refereces [] OAltitas ad SOwa, O subclasses of uivalet fuctios with egative coefficiets, usa Kyŏga MathJ, (988), -56 [] A Baricz, Geoetric roerties of geeralized Bessel fuctios, ubl Math Debrece, 73 (-) (8), [3] A Baricz, Geoetric roerties of geeralized Bessel fuctios of colex order, Matheatica, 8 (7) () (6), 3-8 [] A Baricz,Geeralized Bessel fuctios of the first kid, hd thesis, Babes-Bolyai Uiversity, Cluj-Naoca, (8) [5] A Baricz,Geeralized Bessel fuctios of the first kid, Lecture Notes i Math, Vol 99, Sriger-Verlag () [6] RBharati, Rarvatha ad ASwaiatha, O subclasses of uiforly covex fuctios ad corresodig class of starlike fuctios, Takag JMath, 6 () (997), 7-3 [7] T R Caliger ad W M Causey, A class of uivalet fuctios, roc Aer Math Soc 39 (973), [8] NE Cho, SYWoo ad S Owa, Uifor covexity roerties for hyergeoetric fuctios, Fract Cal Al Aal, 5 (3) (), [9] K K Dixit ad S K al, O a class of uivalet fuctios related to colex order, Idia J ure Al Math 6 (9) (995) [] AWGooda, O uiforly covex fuctios, AoloMath, 56, (99), 87-9 [] AWGooda, O uiforly starlike fuctios, JMathAalAl, 55, (99), [] SKaas ad A Wi siowska, Coic regios ad k-uifor covexity, J Cout Al Math, 5 (999), [3] E Merkes ad BT Scott, Starlike hyergeoetric fuctios, roc Aer Math Soc, (96), [] SR Modal ad A Swaiatha, Geoetric roerties of Geeralized Bessel fuctios, Bull Malays Math Sci Soc, 35 () (), 79-9 [5] AOMostafa, A study o starlike ad covex roerties for hyergeoetric fuctios, Joural of Iequalities i ure ad Alied Matheatics, (3), Art87 (9), -8 [6] GMurugusudaraoorthy ad NMagesh, O certai subclasses of aalytic fuctios associated with hyergeoetric fuctios, Al Math Letters, (), 9-5 [7] K S adaabha, O a certai class of fuctios whose derivatives have a ositive real art i the uit disc, A olo Math 3 (97), 73-8 [8] S ousay ad F Røig, Duality for Hadaard roducts alied to certai itegral trasfors, Colex Variables Theory Al 3 (997), [9] FRøig, Uiforly covex fuctios ad a corresodig class of starlike fuctios, rocaermathsoc, 8, (993), [] FRøig, Itegral reresetatios for bouded starlike fuctios, AaloloMath, 6, (995), [] HSilvera, Uivalet fuctios with egative coefficiets, rocaermathsoc, 5 (975), 9-6 [] HSilvera, Starlike ad covexity roerties for hyergeoetric fuctios, JMathAalAl, 7 (993), [3] KGSubraaia, GMurugusudaraoorthy, Balasubrahaya ad HSilvera, Subclasses of uiforly covex ad uiforly starlike fuctios Math Jaoica, (3), (995), 57-5 [] KGSubraaia, TVSudharsa, Balasubrahaya ad HSilvera, Classes of uiforly starlike fuctios, ubl Math Debrece, 53 (3-), (998), [5] ASwaiatha, Certai suffiect coditios o Gaussia hyergeoetric fuctios, Joural of Iequalities i ure ad Alied Matheatics, 5 (), Art 83 (), -