Subclasses of Starlike Functions with a Fixed Point Involving q-hypergeometric Function

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1 J. Aa. Num. Theor. 4, No. 1, (2016) 41 Joural of Aalysis & Number Theory A Iteratioal Joural Subclasses of Starlike Fuctios with a Fixed Poit Ivolvig q-hypergeometric Fuctio G. Murugusudaramoorthy, K. Vijaya ad T. Jaai School of Advaced Scieces, VIT Uiversity, Vellore, , Idia Received: 7 Jul. 2015, Revised: 9 Aug. 2015, Accepted: 21 Aug Published olie: 1 Ja Abstract: Recetly, Kaas ad Roig itroduced the classes of fuctios starlike ad covex, which are ormalized with f(w) = f (w) 1 = 0, w is a fixed poit i the ope disc = {z C : z < 1}. The aim of this paper is to cotiue this ivestigatio ad itroduce a ew class S T m l (α,β,w), of fuctios which are aalytic i. We obtai various results icludig coefficiets estimates, distortio ad coverig theorems, radii of close-to-covexity, starlikeess ad covexity for fuctios belogig to this class. Keywords: Aalytic fuctio, Starlike fuctio, Covex fuctio, Uiformly covex fuctio, Covolutio product, q hypergeometric fuctio. 1 Itroductio ad Motivatios Let w( w = d) be a fixed poit i the uit disc := {z C : z < 1}. Deote by H ( ) the class of fuctios which are regular ad A(w) = { f H( ) : f(w) = f (w) 1 = 0}. Also deote by S w ={ f A(w) : f is uivalet i }, the subclass of A(w) cosistig of the fuctios of the form f(z)=(z w)+ a (z w), (1) which are aalytic i. Deote by T w the subclass of S w cosistig of the fuctios of the form a (z w) (a 0). (2) Note that S 0 = S ad T 0 = T be subclasses of A = A(0) cosistig of uivalet fuctios i. By Sw (β) ad K w(β), respectively, we mea the classes of aalytic ( fuctios that satisfy the aalytic coditios (z w) f ) (z) ( ) R > β, R 1+ (z w) f (z) f(z) f (z) > β ad (z w) for 0 β < 1 itroduced ad studied by Kaas ad Roig 10. The class Sw (0) is defied by geometric property that the image of ay circular arc cetered at w is starlike with respect to f(w) ad the correspodig class K w (0) is defied by the property that the image of ay circular arc cetered at w is covex. We observe that the defiitios are somewhat similar to the oes itroduced by Goodma i 8 ad 9 for uiformly starlike ad covex fuctios, except that i this case the poit w is fixed. I particular, K = K 0 (0) ad S0 = S (0) respectively, are the well-kow stadard classes of covex ad starlike fuctios (see 11,15). For complex parameters a 1,...,a l ad b 1,...,b m (b j 0, 1,...; j = 1,2,...,m) the q-hypergeometric fuctio l Ψ m (z) is defied by := lψ m (a 1,...a l ;b 1,...,b m ;q,z) =0 ( with 2) (a 1,q)...(a l,q) (q,q) (b 1,q)...(b m,q) ( 1) q ( 2) 1+m l z (3) = () 2 q 0 whe l > m+1(l,m N 0 =N {0},z. The q shifted factorial is defied for a,q C as a product of factors by { } 1 = 0 (a;q) = (1 a)(1 aq)...(1 aq ) N Correspodig author gmsmoorthy@yahoo.com

2 42 G. M. S. Moorthy et al. : Subclasses of starlike fuctios with... ad i terms of basic aalogue of the gamma fuctio (q a ;q) = Γ q(a+)(1 q),>0. (4) Γ q (a) Now for z,0 < q < 1 ad l = m+1, the basic hypergeometric fuctio defied i (3) takes the form lψ m (a 1 ;...a l ;b 1,...,b m ;q,z) (a 1,q)...(a l,q) = z, (q,q) (b 1,q)...(b m,q) =0 which coverges absolutely i the ope uit disk. It is (q a ;q) of iterest to ote that lim q 1 (1 q) =(a) = a(a+1)...(a+), the familiar Pochhammer symbol ad lψ m (a 1,...a l ;b 1,...,b m ;z)= =0 (a 1 )...(a l ) z (b 1 )...(b m )!. (5) For a i = q α i,b j = q β j,α i,β j C ad β j 0, 1, 2,...,(i=1,...,l, j = 1,..,m) ad q 1, we obtai the well-kow Dziok-Srivastava liear operator 6,7 (for l = m+1). For l = 1,m=0,a 1 = q, may (well kow ad ew) itegral ad differetial operators ca be obtaied by specializig the parameters, for example the operators itroduced i 4, 5, 11, 12, 14. Motivated by the recet work of Mohammed ad Darus 13, we defie a liear operator give by I l ma l,q f(z) : A(w) A(w) satisfyig the coditio ( (z w)(i l R m f(z)) ) Im l f(z) β > α (z w)(im l f(z)) Im l f(z) 1,(z w). (9) I this preset paper, we obtai a characterizatio, coefficiets estimates, distortio theorem ad coverig theorem, extreme poits ad radii of close-to-covexity, starlikeess ad covexity for fuctios belogig to the class S T m l (α,β,w). 2 Characterizatio ad Coefficiet estimates Theorem 1. Let f T w. The f S T m l (α,β,w), 0 β < 1 ad α 0, if ad oly if (r+d) a. (10) 0 z w z + w <r+d < 1, I l ma l,q f(z)=i(a l,b m ;q;z w) f(z) =(z w) l Ψ m (a 1 ;...a l ;b 1,...,b m ;q,(z w)) f(z), (6) Proof. We employ the techique adopted by 1,16. We have f S T m l (α,β,w), if ad oly if the coditio (9) is satisfied, which is equivalet to I l m f(z)=i l ma l,q f(z) =(z w)+ ϒ l,m a 1,qa (z w) (7) = ϒ l,m (a 1 ;q)...(a l ;q) a 1,q=. (q;q) (b 1 ;q)...(b m,q) (8) Makig use of the operator Im l f(z) ad motivated by the results discussed i 1,2,3,16 we itroduce a ew subclass S T m l (α,β,w) of aalytic fuctios with egative coefficiets. For 0 β < 1, α 0, ad for a fixed poit w, let the class S T m l (α,β,w), cosists of fuctios f T w, ( (1 αe iθ )(z w)(im l R f(z)) + αe iθ Im l f(z) ) Im l f(z) > β, π θ < π. (11) Now, lettig G(z) = (1 αe iθ )(z w)(im l f(z)) + αe iθ Im l f(z) ad F(z)=Im l f(z), equatio (11) is equivalet to G(z)+()F(z) > G(z) (1+β)F(z), 0 β < 1. By simple computatio we have

3 J. Aa. Num. Theor. 4, No. 1, (2016) / 43 G(z)+()F(z) (2 β)(z w) (z w) (+)a αeiθ (2 β) z w α (2 β) z w ad similarly, G(z) (1+β)F(z) β z w + ()a (z w) (+)a z w ()a z w +1 a z w 1 a z w. Therefore, G(z)+()F(z) G(z) (1+β)F(z) 2() z w 2 a z w () a z w 0. By puttig z w z + w < r+d with 0<r < 1 ad w = d i above iequality, we obtai(10). O the other had, for all π θ < π, we must have ( (z w)f (z) R (1+αe iθ ) αe )>β. iθ F(z) Now, choosig the values of (z w) o the positive real axis, 0 z w z + w <r+d < 1, ad usig R{ e iθ } e iθ = 1, the above iequality ca be writte as () a (r+ d) R 1 a (r+ d) 0. hece we get the desired result. Corollary 1. If f S T m l (α,β,w), the a (r+d), 2, (12) 0 β < 1 ad α 0. Equality i (12) holds for the fuctio (r+ d) (z w). (13) I the followig sectio we state Distortio ad Growth theorems without proof. 3 Distortio ad Coverig Theorems Theorem 2. Let be defied as i (8). The, for f S T m l (α,β,w), with z w z + w <r+d < 1 i, we have (r+d) B(α, β, µ)(r+d) 2 f(z), (r+d)+b(α, β, µ)(r+d) 2, (14) B(α, β, µ) :=. 2(α+ 1) (α+ β) (r+d)ϒ(2) Theorem 3. If f S T m l (α,β,w), the for z w z + w <r+d < 1 1 B(α, β, µ)(r+d) f (z) 1+B(α, β, µ)(r+d), (15) B(α, β, µ) as i Theorem 2. Note that i Theorem 2 ad Theorem 3 equality holds for the fuctio (z w) 2. 2(α+ 1) (α+ β) (r+ d)ϒ(2) Theorem 4. If f S T m l (α,β,w), the f S (δ), δ = 1. 2(α+ 1) (α+ β) (r+d)ϒ(2) () This result is sharp with the extremal fuctio beig 2(α+ 1) (α+ β) (r+ d)ϒ(2) (z w) 2. Proof. It is sufficiet to show that (10) implies ( δ)a 1 δ 15, that is, δ (r+d) 1 δ, 2. (16)

4 44 G. M. S. Moorthy et al. : Subclasses of starlike fuctios with... Sice, for 2,(16) is equivalet to ()() δ 1 (r+d) () = Φ() ad Φ() Φ(2), (16) holds true for ay 0 β < 1 ad α 0. This completes the proof of the Theorem 4. 4 Extreme poits of the class S T m l (α,β,w) Theorem 5. Let f 1 (z)=(z w) ad f (z)=(z w) (r+ d) (z w), 2 ad be as defied i(8). The f S T m l (α,β,w) if ad oly if it ca be represeted i the form f(z)= ω f (z), λ 0, =1 λ = 1. (17) =1 Proof. Suppose f(z) ca be writte as i(17). The λ (r+d) (z w). Now, λ (r+d) () () (r+d) = λ = 1 λ 1 1. Thus f S T m l (α,β,w). Coversely, let us have f S T m l (α,β,w). The by usig (12), we may write (r+d) λ = a, 2, ad λ 1 = 1 as i the Theorem. λ. The f(z)= =1 λ f (z), with f (z) is Theorem 6. The class S T m l (α,β,w) is a covex set. Proof. Let the fuctio f j (z)= a, j (z w), a, j 0, j = 1,2, be i the class S T m l (α,β,w). It sufficiet to show that the fuctio g(z) defied by g(z)=λ f 1 (z)+(1 λ) f 2 (z), 0 λ 1, is i the class S T m l (α,β,w). Sice g(z)=(z w) λ a,1 +(1 λ)a,2 (z w), a easy computatio with the aid of Theorem 1 gives, (r+d) λa,1 +(1 λ)a,2 +(1 λ) (r+ d) λ ()+(1 λ)(), which implies that g S T m l (α,β,w). Hece S T m l (α,β,w) is covex. 5 Modified Hadamard products For fuctios of the form ( f 1 f 2 )(z)=(z w) a,1 a,2 (z w). (18) we defie the modified Hadamard product as ( f 1 f 2 )(z)=(z w) a,1 a,2 (z w). (19) Theorem 7. If f j (z) S T m l (α,β,w), j = 1, 2, the ( f 1 f 2 )(z) S T m w(α,ξ,w), ξ = (2 β)( 2(α+ 1) (α+ β) ) (r+d)ϒ(2) 2() 2 (2 β) ( 2(α+ 1) (α+ β) ) (r+d)ϒ(2) () 2, with be defied as i (8). Proof. Sice f j (z) S T m l (α,β,w), j = 1, 2, we have (r+d) a, j. The Cauchy-Schwartz iequality leads to (20) (r+d) a, j a,1 a,2 Note that we eed to fid the largest ξ such that 1. (21) (α+ 1) (α+ ξ) (r+d) a, j a,1 a,2 1 ξ 1. (22)

5 J. Aa. Num. Theor. 4, No. 1, (2016) / 45 Therefore, i view of (21) ad (22), wheever ξ a,1 a,2 β 1 ξ, 2 holds, the (22) is satisfied. We have, from (21), a,1 a,2 Thus, if (r+d), ( ) ξ 1 ξ (r+ d) or, if 2. (23) β, 2, ξ ( β) (r+ d) () 2 ( β) (r+ d) () 2, 2, the (21) is satisfied. Note that the right had side of the above expressio is a icreasig fuctio o. Hece, settig = 2 i the above iequality gives the required result. Fially, by takig the fuctio f(z) = (z w) (z (2 β)2(α+1) (α+β)(r+d)ϒ(2) w)2, we see that the result is sharp. 6 Radii of close-to-covexity, starlikeess ad covexity Theorem 8. Let the fuctio f T w be i the class S T m l (α,β,w). The f(z) is close-to-covex of order ρ, 0 ρ < 1 i z w =< R 1, () (r+ d) 1 R 1 = if, () 2, with be defied as i (8). This result is sharp for the fuctio f(z) give by (13). Proof. It is sufficiet to show that f (z) 1, 0 ρ < 1, for z w <r 1 (α, β,l,ρ), or equivaletly ( ) a z w 1. (24) By Theorem 1, (24) will be true if ( ) (r+ d) z w or, if z w =R 1 () (α+ 1) (α + β) (r+ d) 1, 2. () The theorem follows easily from(25). (25) Theorem 9. Let the fuctio f(z) T w be i the class S T m l (α,β,w). The f(z) is starlike of order ρ, 0 ρ < 1 i z w =R 2 R 2 = if () (r+ d) ( ρ)() 1, 2, with be defied as i (8). This result is sharp for the fuctio f(z) give by(13). Proof. It is sufficiet to show that (z w) f (z) 1 f(z), (26) or equivaletly ( ) ρ a z w 1, for 0 ρ < 1 ad z w =R 2. Proceedig as i Theorem 8, with the use of Theorem 1, we get the required result. Hece, by Theorem 1, (26) will be true if ( ρ or, if ) z w (r+d) z w =R 2 (r+d) 1, 2. ( ρ)() (27) The theorem follows easily from(27). Theorem 10. Let the fuctio f(z) T w be i the class S T l m (α, β,w). The f(z) is covex of order ρ, 0 ρ< 1 i z w =R 3, R 3 = if () (r+d) ( ρ)() 2, with be defied as i (8). This result is sharp for the fuctio f(z) give by(13). Proof. It is sufficiet to show that equivaletly (z w) f (z) f (z) ( ) ( ρ) a z w 1, or for 0 ρ < 1 ad z < r 3 (α, β,l,ρ). Proceedig as i Theorem 8, we get the required result. 1,

6 46 G. M. S. Moorthy et al. : Subclasses of starlike fuctios with... 7 Itegral trasform of the class S T l m (α, β,w) For f S T m l (α,β,w) we defie the itegral trasform 1 V µ ( f)(z)= µ(t) f(tz) dt, 0 t µ is real valued, o-egative weight fuctio ormalized so that 1 0 µ(t)dt = 1. Sice special cases of µ(t) are particularly iterestig such as µ(t) = (1+c)t c, c > 1, for which V µ is kow as the Berardi operator ad ( µ(t)= (c+1)δ µ(δ) tc log 1 ) δ 1, c> 1, δ 0 t which gives the Komatu operator. First we show that the class S T m l (α,β,w) is closed uder V µ ( f). Theorem 11. Let f S T m l (α,β,w). The V µ ( f) S T m l (α,β,w). Proof. By defiitio, we have V µ ( f)= (c+1)δ µ(δ) 1 ( 1) δ 1 t c (logt) δ 1 ((z w) a (z w) t )dt 0 = ( 1)δ 1 (c+1) δ µ(δ) 1 lim t c (logt) δ 1 ((z w) a (z w) t )dt, r 0 + r ad a simple calculatio gives V µ ( f)(z)=(z w) We eed to prove that (r+d) ( ) c+1 δ a (z w). c+ ( ) c+1 δ a < 1. c+ (28) O the other had by Theorem 1, f S T l m(α, β,w) if ad oly if (r+d) a < 1. Hece c+1 c+ < 1. Therefore (28) holds ad the proof is complete. Next we provide a starlike coditio for fuctios i S T l m(α, β,w) ad V µ ( f) o lies similar to Theorem 8. Theorem 12. Let f S T m l (α,β,w). The (i) V µ ( f) is starlike of order 0 γ < 1 i z w < R 1 R 1 = ( c+ if c+1 ) δ (1 γ)(α+ 1) (α + β)(r+ d) ()( γ) 1 (ii) V µ ( f) is covex of order 0 γ < 1 i z w < R 2 R 2 = ( c+ if c+1 ) δ (1 γ)(α+ 1) (α + β)(r+ d) ()( γ) This result is sharp for the fuctio 1. (α+ 1) (α + β)(r+ d) (z w), 2. (29) Cocludig Remarks: For suitable choices α,β,l ad m the family S T m l (α,β,w), evetually lead us further to ew class of fuctios defied either by extesio or by geeralizatio. Refereces 1 E. Aqla, J.M. Jahagiri ad S.R. Kulkari, Classes of k uiformly covex ad starlike fuctios, Tamkag J. Math., 35 (2004), o.3, M.K.Aouf ad G.Murugusudaramoorthy, A subclass of uiformly covex fuctios defied by the Dziok-Srivastava operator, Aust.J.Math. Aal. Appl. (AJMAA), 5 ( 2008), o.1, Article 3, M. K. Aouf, A. Shamdy, A. O. Mostafa ad S. Madia, A subclass of m-w-starlike fuctios, Acta Uiv. Apulesis (2010), o. 21, S. D. Berardi, Covex ad starlike uivalet fuctios, Tras. Amer. Math. Soc., 135 (1969), B. C. Carlso ad S. B. Shaffer, Starlike ad prestarlike hypergeometric fuctios, SIAM J. Math. Aal., 15 (1984), J. Dziok ad H. M. Srivastava, Classes of aalytic fuctios associated with the geeralized hypergeometric fuctio, Appl. Math. Comput. 103 (1999), J. Dziok ad H. M. Srivastava, Certai subclasses of aalytic fuctios associated with the geeralized hypergeometric fuctio, Itergral Trasforms Spec. Fuct. 14 (2003), A.W. Goodma, O uiformly starlike fuctios, J. Math. Aal. Appl. 155 (1991), A.W. Goodma, O uiformly covex fuctios, A. Polo. Math. 56 (1991), o. 1, S. Kaas ad F. Roig, Uiformaly starlike ad covex fuctio ad other related classes of uivalet fuctios, A. Uiv. Marie Curie-Sklodowaska, Sect. A 53 (1991),

7 J. Aa. Num. Theor. 4, No. 1, (2016) / R. J. Libera, Some classes of regular uivalet fuctios, Proc. Amer. Math. Soc., 16 (1965), A. E. Livigsto, O the radius of uivalece of certai aalytic fuctios, Proc. Amer. Math. Soc., 17 (1966), A. Mohammed ad M. Darus, A geeralized operator ivolvig the q-hypergeometric fuctio, Mathematici Vesik, Available olie pages. 14 S. Ruscheweyh, New criteria for Uivalet Fuctios, Proc. Amer. Math. Soc., 49 (1975), H. Silverma, Uivalet fuctios with egative coefficiets, Proc. Amer. Math. Soc., 51(1975) H. Silverma,G.Murugusudaramoorthy ad K.Vijaya,O a certai class of aalytic fuctios associated with a covolutio structure, Kyugpook.J. Math., 49(2009), K. Vijaya works as a Professor of mathematics at the School of Advaced Scieces, Vellore Istitute of Techology, VIT Uiversity, Vellore , Tamiladu, Idia. She received her Ph.D. degree i Complex Aalysis (Geometric Fuctio Theory) from VIT Uiversity, Vellore, i She is havig twety five years of teachig ad thirtee years of research experiece. Her research areas icludes special fuctios, harmoic fuctios ad published good umber of papers i reputed refereed jourals. G. Murugusudaramoorthy works as a Seior Professor of mathematics at the School of Advaced Scieces, Vellore Istitute of Techology, VIT Uiversity, Vellore , Tamiladu, Idia. He received his Ph.D. degree i complex aalysis (Geometric Fuctio Theory) from the Departmet of Mathematics, Madras Christia College, Uiversity of Madras, Cheai, Idia, i He is havig twety years of teachig ad research experiece. His research areas icludes special classes of uivalet fuctios, special fuctios ad harmoic fuctios ad published good umber of papers i reputed refereed idexed jourals. T. Jaai is a Research Associate, pursuig Ph.D i Mathematics, VIT Uiversity, Vellore She got Master of Sciece degree from Idia Istitute of Techology Madras, Cheai ad has idustrial experiece at IBM Pvt. Ltd, Cheai for three years. She has few paper i reputed refereed jourals for her credit i Bi-uivalet, Bessel ad Struve fuctios.