On Cesáro means for Fox-Wright functions

Transcription

1 Joural of Mathematics ad Statistics: 4(3: 56-6, 8 ISSN: Sciece Publicatios O Cesáro meas for Fox-Wright fuctios Maslia Darus ad Rabha W. Ibrahim School of Mathematical Scieces, Faculty of Sciece ad Techology Uiversity Kebagsaa Malaysia, agi 436 Selagor D. Ehsa, Malaysia Abstract: The olyomial aroximats which retai the ero free roerty of a give aalytic fuctio ivolvig fox-wright fuctio i the uit dis U: {: <} is foud. The covolutio methods of a geometric fuctio that the Cesáro meas of order retais the ero free roerty of the derivatives of bouded covex fuctios i the uit dis are used. Other roerties are also established. Key words: Fox-wright fuctio, cesáro sum, covolutio INTRODUCTION I the theory of aroximatio, the imortat roblem is to fid a suitable fiite (olyomial aroximatio for the outer ifiite series f so that the aroximat reduces the ero-free roerty of f Recall that a outer fuctio (ero-free is a fuctio fεh of the form: it iγ /π π e π it f( e e f log ψ(tdt e where, ψ(t,log ψ (t is i L ad Ψ(t is i L [3]. Outer fuctio lays a imortat role i H theory, arises i characteristic euatio which determies the stability of certai oliear systems of differetial euatios []. We observed that for outer fuctios, the stadard Taylor aroximats do ot, i geeral, retai the ero-free roerty of f It was show i [] that the Taylor aroximatig olyomials to outer fuctios ca vaish i the uit dis. y usig covolutio methods, the classical Cesáro meas retai the ero-free roerty of the derivatives of bouded covex fuctios i the uit dis. The classical Cesáro meas lay a imortat role i geometric fuctio theory [5-7]. I this study, we obtai ew Cesáro aroximats for outer fuctios. Ideed, fox-wright fuctio is ivolved ad stated as follows: For comlex arameters: α α α,..., (,,,...;,..., A Ad β β β,..., (,,,...;,...,, the fox-wright geeraliatio ψ [] of the hyergeometric F fuctio by [4,,] : ( α,a,...,( α,a ; ψ ψ ( α,a,;( β,; (,,...,(, ; β β Γα Γα : Γβ Γβ Γ( α A Γ( β! ( A... ( A (... (! where, A > for all,...,,> for all,.., ad shi A for suitable values For secial case, whe A for all ad for all,, we have the followig relatioshi: Where: F ( α,..., α; β,..., β ; ΩΨ[( α,,;( β,,;] ;, N N U {}, U Γ( β... Γ( β Ω : Γ ( α... Γ ( α Let A be the class of Fox-Wright fuctios i the uit dis U: {: } < tae the form: Corresodig Author: Maslia Darus, School of Mathematical Scieces, Faculty of Sciece ad Techology, Uiversiti Kebagsaa Malaysia, agi 436 Selagor D. Ehsa, Malaysia 56

2 J. Math. & Stat., 4(3: 56-6, 8 With: ϕ ( : Ψ [( α,a ;( β, ;],, Γ( α A, U Γ( β! ( <Γα ( A Γβ ( ( RESULTS AND DISCUSSION This class of fuctio is a geeraliatio to the oe studied by [5]. The author observed the followig results: Lemma : Let <α β If β or αβ 3 the the ( α fuctio of the form f(, U is ( β covex. Note that (x is the Pochhammer symbol defied by: Γ (x, (x Γ(x x(x...(x, {,,...} Lemma : Assume that a ad a for such that {a } is a covex decreasig seuece i.e.: The a a a ad a a R > a, U We aly Lemma., to fid the ext result which [Lemma 5; 8]. is a geeraliatio to Lemma 3: Let ( holds. The: Γ γ Γ γ γ γ γ [( ] ( ( ( (... ( for ositive iteger ad o-egative iteger Thus sice we obtai [9] : α α α A Γα ( A Γα ( ( (...( (A A A A A Also, by usig the fact that (x (x (x, we fid: α α α A A Γ( α( (...( (A A A A β β β Γ( β ( (...( ( (3 The we obtai Moreover, we have ad i terms of : Γα ( A... Γα ( A (... ( ( α α α A Γ( α ( (...( (A A A A β β β Γ( Γβ Γβ Γ ad A Γ( β( (...( ( α α α A A ( (...( (A A A A β β β ( (...( ( Γ( (4 ϕ( R > for all U Proof: From the defiitio of the fuctio ϕ( we have: (... ( ϕ( Γα Γα Γβ (... Γβ ( Where: Γα ( ( A... Γα ( ( A : Γβ ( (... Γβ ( ( Γ ( for. From (, we have > for all N Now by usig Gauss s multilicatio theorem for the Gamma fuctio, it follows that: Γα ( ( A... Γα ( ( A ( (... ( ( ( ( Γβ Γβ Γ α α α α ( ( ( ( A A A A β β β β ( ( ( ( α A α A β β ( Γ( A ( ( (A A A ( ( ( Thus from the assumtio it follows that:... (5 57

3 J. Math. & Stat., 4(3: 56-6, 8 α α α A A ( A A A A β β β ( ( Γ(, ù I the same way ad by usig (3 ad (4, we ca show that:, ù (6 Thus we fid the seuece { } is covex decreasig ad i virtue of lemma, we obtai that: (... ( Γα Γα ϕ( R R > Γβ (... Γβ ( The roof is comlete: We defie S*, C, QS* ad QC the subclasses of A cosistig of fuctios which are, resectively, starlie i U, covex i U, close-to-covex ad uasi-covex i U. Thus by defiitio, we have: Ad ( ϕ'( S* : ϕ Α: R >, U, ϕ( ( ϕ"( C: ϕ Α: R >, U, ϕ'( ( ϕ'( QS*: ϕ Α: g S* s.t. R >, U g( ( ϕ'(' QC : ϕ Α: g Cs.t. R >, U g'( It is easily observed from the above defiitios that: where, ( P the class of all aalytic fuctios of the form: ad ( P...,s.t.( Give two fuctios: f,g A,f( a g( b their covolutio or Hadamard roduct f(*g( is defied by: f(*g( a b, U We ca verify the followig result for f A ad taes the form (. Lemma 4 [5] : If ϕ C ad g S* the ϕ*g S* If ϕ C ad g S*, P with ( the ϕ *g ( ϕ *g where (U close covex hull of (U CONCLUSION Cesáro aroximats for outer fuctios: The Cesáro sums of order where ù U { } of series of the form ( ca defied as: (, * ( Γ( β! σ ϕ σ ϕ Γ ( α A Ad ϕ( C ϕ'( S* (7 ϕ( QC ϕ'( QS* (8 a a! where,. b b!(a b! We begi with the followig result: Note that ϕ QS* if ad oly if there exists a fuctio g S* such that: ϕ '( g(( (9 Theorem : Let ϕ A be covex i U The the Cesáro meas σ (, ϕ, U of order, of ϕ '( are ero-free o U for all. 58

4 J. Math. & Stat., 4(3: 56-6, 8 Proof: I view of Lemma, the aalytic fuctio ϕ of the form ( is covex i Uif Γ( β or Γ( α A Γ( β 3 ( Proof: Uder the coditios of the theorem, we have that f is covex (Lemma., the i virtue of Theorem, we obtai that σ (, ϕ' thus σ (, ϕ ' > Now by alyig Lemma.3, o σ (, ϕ ' ad usig the fact that { } R ad sice: where, ( holds. Let such that: The: ϕ ( : ( be defied ( Γ( α A ϕ '( φ(* ϕ (! Γ( β!(! (!(! for ad,,, yield: ( σ (, ϕ ' ϕ'(* σ ( α ϕ'(*σ ϕ(* ϕ*σ ϕ(*( σ ' I view of Lemma 3, the relatio (8 ad the fact that σ is covex yield that there exists a fuctio g S* ad P with ( such that: ( ϕ(*( σ ' ϕ(*g( ϕ(*g( ( We ow that { } R ( > ad that ϕ (*g( if ad oly if Hece, σ ( ϕ' ad the roof is comlete. Corollary : If f(u is bouded covex domai, the the Cesáro meas σ (, U for the outer fuctio ϕ ( are ero-free ou for all. Proof: It comes from the fact that the derivatives of bouded covex fuctios are outer fuctio [3]. The ext result shows the uer ad lower boud for σ (, ' ϕ. Theorem : Let ϕ A Assume that ( ad ( hold. The: ( <σ(, ϕ' <, U, (!' 59 σ (, ϕ' σ (, ϕ' < R, > ad U For the other side, we ose that: σ (, ϕ ' ϕ'(* σ ( Γ ( α A Γ ( α A ( Γ ( α A! Γ( β ( ( <!!' (! Γ( β (! Γ( β ( Whe, Hece the roof. Fially, we give the followig result: Theorem 3: Let ϕ A ad let ( holds. The: lim (,, U ( λ' σ α ϕ λ > Proof: From the assumtio ( ad by ( yield:

5 J. Math. & Stat., 4(3: 56-6, 8 α σ (, ϕ ( λ Γ( α A ( λ (!! Γ β Γ ( α A Γ( β! ( λ λ ( as! ( λ ( λ!! ACKNOWLEDGEMENT The wor here was fully suorted by esciece Fud: 4---SF45, MOSTI, Malaysia.. Cuigham, W., 958. Itroductio to Noliear Aalysis. MeGraw-Hill, New Yor. 3. Dure, P.L., 97. Theory of H Saces. Academic Press. 4. Fox, C., 98. The asymtotic exasio of the geeralied hyergeometric fuctio. J. Lodo Math. Soc., : Ruscheweyh, S., 98. Covolutios i Geometric Fuctio Theory. Sem. Math. Su., Uiversity of Motreal Press. 6. Ruscheweyh, S., 99. Geometric Proerties of Cesáro Meas. Results Math., : Ruscheweyh, S. ad L. Salias, 993. Subordiatio by Cesáro Meas. Comlex Var. Theor. Al., : Slia, L.T., 8. O certai alicatios of the hadamard roduct. A. Math. Com., 99: Slater, L.J., 966. Geeralied Hyergeometric Fuctios. Cambridge Uiversity Press, Lodo.. Wright, E.M., 935. The asymtotic exasio of the geeralied hyergeometric fuctio. J. Lodo Math. Soc., : Wright, E.M., 94. The asymtotic exasio of the geeralied hyergeometric fuctio. J. Lodo Math. Soc., 46: REFERENCES. arard, R.W., J. Cima ad K. Pearce, 998. Cesáro sum aroximatio of outer fuctios. A. Ui. Marie Curie-Slodowsa Sect. A, 5: -7. 6