Bangi 43600, Selangor Darul Ehsan, Malaysia (Received 12 February 2010, accepted 21 April 2010)

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1 O Cesáro Meas of Order μ for Outer Fuctios ISSN (prit), (olie) Iteratioal Joural of Noliear Sciece Vol9(2010) No4,pp Maslia Darus 1, Rabha W Ibrahim 2 1,2 School of Mathematical Scieces, Faculty of sciece ad Techology, Uiversity Kebagsaa Malaysia, Bagi 43600, Selagor Darul Ehsa, Malaysia (Received 12 February 2010, accepted 21 April 2010) Abstract: The polyomial approximats which retai the ero free property of a give aalytic fuctios i the uit dis U : { : < 1} of the form f() (α) +1, U is foud By usig covolutio method of geometric fuctio, Cesáro meas of order μ, μ 0 retai the ero free property of the derivatives of bouded covex fuctios i the uit dis Other properties are also established Keywords: uivalet fuctio; cesáro sum; covolutio product AMS Mathematics Subject Classificatio (2000): 30C45 1 Itroductio, Defiitios ad Prelimiaries I the theory of approximatio, the importat problem is to fid a suitable fiite (polyomial) approximatio for the outer ifiite series f so that the approximat reduces the ero-free property of f Recall that a outer fuctio (ero-free) is a fuctio f H p of the form f() e iγ e 1/2π π 1+e it π 1 e it logψ(t)dt where γ R, ψ(t) 0, logψ(t) L 1 ad ψ(t) L p for details see [1] Outer fuctios play a importat role i H p theory, arise i characteristic equatio which determies the stability of certai oliear systems of differetial equatios (see [2]) We observed that for outer fuctios, the stadard Taylor approximats do ot, i geeral, retai the ero-free property of f It was show i [3] that the Taylor approximatig polyomials of outer fuctios ca vaish i the uit dis By usig covolutio method, the classical Cesáro meas retai the ero-free property of the derivatives of bouded covex fuctios i the uit dis The classical Cesáro meas play a importat role i geometric fuctio theory (see [4,5,6]) Not may problems were discussed i this area, but for differet results o Cesáro meas ca be foud i [10-13] Let A be the class of aalytic fuctios i the uit dis U : { : < 1} tae the form f() where (x) is the Pochhammer symbol defied by (x) Γ(x + ) Γ(x) (α) +1, α, β > 0, U (1) { 1, 0 x(x + 1)(x + 1), {1, 2, } This class of fuctios was studied by Ruscheweyh [4] He observed the followig result: Correspodig author address: maslia@ummy, rabhaibrahim@yahoocom Copyright c World Academic Press, World Academic Uio IJNS /373

2 456 Iteratioal Joural of NoliearSciece,Vol9(2010),No4,pp Lemma 1 ([4]) Let 0 < α β If β 2 or α + β 3, the the fuctio of the form (1) is covex Splia [7] gave the followig: Lemma 2 Let 0 < α β The for f of the form (1) R{ f() } > 1 2 We deote by S, C, QS ad QC the subclasses of A cosistig of fuctios which are, respectively, starlie i U, covex i U, close-to-covex ad quasi-covex i U Thus by defiitio, we have ( f S ()) : {f A : R( > 0, U}, f() ad It is easily observed from the above defiitios that ad C : {f A : R ( 1 + f ()) > 0, U}, f () ( f QS : {f A : g S ()) st R( > 0, U}, g() ( f ()) ) QC : {f A : g C st R( > 0, U} g () Note that f QS if ad oly if there exists a fuctio g S such that where p() P, the class of all aalytic fuctios of the form f() C f () S (2) f() QC f () QS (3) f () g()p() (4) p() 1 + p p 2 2 +, with p(0) 1 Give two fuctios f, g A, f() (α) +1, ad g() (a) (b) +1, their covolutio or Hadamard product f() g() is defied by (a) (α) f() g() +1 (b) We ca verify the followig result for f A ad taes the form (1) Lemma 3 ([4]) (i) If f C ad g S, the f g S (ii) If f C ad g S, p P with p(0) 1, the f gp (f g)p 1 where p 1 (U) close covex hull of p(u) 2 Cesáro approximats for outer fuctios The Cesáro sums of order μ where μ N {0} of series of the form (1) ca be defied as σ μ (, f) σμ f() (α) +1 + μ ( a where b ) a! b!(a b)! We have the followig result Theorem 4 Let f A be covex i U The the Cesáro meas σ μ (, f), U of order μ 1, of f () are ero-free o U for all IJNS for cotributio: editor@oliearscieceorgu

3 M Darus, R W Ibrahim: O Cesáro Meas of Order μ for Outer Fuctios 457 Proof I view of Lemma 1, the aalytic fuctio f of the form (1) is covex i U if β 2 or α+β 3 where 0 < α β Let φ() : ( + 1)+1 be defied such that The f () φ() f() ( + 1) (α) +1 σ α (, f ) f () σ μ () f () σ μ f() φ() σμ f() ( σ μ I view of Lemma 3, the relatio (4) ad the fact that σ μ is covex yield that there exists a fuctio g S ad p P with p(0) 1 such that ) f() ( σ μ ) f() gp() f() g() p1 () 0 We ow that R{p 1 ()} > 0 ad that f() g() 0 if ad oly if 0 Hece, σ μ (, f ) 0 ad the proof is complete Corollary 5 If f(u) is bouded covex domai, the the Cesáro meas σ μ (), U for the outer fuctio f () are ero-free o U for all Proof It comes from the fact that the derivatives of bouded covex fuctios are outer fuctio (see [3]) The ext result shows the upper ad lower bouds for σ μ (, f ) Theorem 6 Let f A Assume that 0 < α β with β 2 or α + β 3, ad μ 0 The 1 2 < σμ (, f ) ( + 1), 1, U, 0 Proof Uder the coditios of the theorem, we have that f is covex (Lemma 1), the i virtue of Theorem 1, we obtai that σ μ (, f ) 0 thus σ μ (, f ) > 0 Now by applyig Lemma 2, o σ μ (, f ) ad usig the fact that R{} ad sice ( + μ )!()! ( )!( + μ)! 1 (5) for μ 0 ad 0, 1,, yield 1 2 < R{σμ (, f ) } σμ (, f ), > 0 ad U IJNS homepage:

4 458 Iteratioal Joural of NoliearSciece,Vol9(2010),No4,pp For the other side, we pose that σ μ (, f ) f () σ μ () ( + 1) (α) + μ whe Hece the proof ( + 1) ( + μ ( + 1) ( + μ ( + 1) ( + 1) Theorem 7 Let f A, ad assume that (α) 2 for all The ) (α) ) (α) lim σα (, f), U 1 Proof By the assumptio, ad i view of (5), we obtai σ μ (, f) 1 (α) μ [ (α) ] μ as Bouded Turig of σ μ (, f) For 0 ν < 1, let B(ν) deote the class of fuctios f of the form (1) so that R{f } > ν i U The fuctios i B(ν) are called fuctios of bouded turig (cf [8, Vol II]) By the Nashiro-Warschowsi Theorem (see eg [8, Vol I]), the fuctios i B(ν) are uivalet ad also close-to-covex i U I the sequel we eed to the followig results Lemma 8 ([9]) For U we have { j } R > 1, ( U) Lemma 9 ( [8, Vol I ]) Let P be aalytic i U, such that P (0) 1, ad R(P ()) > 1 2 i U For fuctios Q aalytic i U the covolutio fuctio P Q taes values i the covex hull of the image o U uder Q IJNS for cotributio: editor@oliearscieceorgu

5 M Darus, R W Ibrahim: O Cesáro Meas of Order μ for Outer Fuctios 459 Theorem 10 Let g H the class of ormalied fuctio taes the form g() : + 2 a, ( U) Assume that 1 If 1 2 < ν < 1 ad g() B(ν), the (α) σ μ (, f)g() B( 3( + μ)! (μ + 1)!!(1 ν) ) 3( + μ)! Proof Let g() B(ν) that is Implies Now for 1 2 < ν < 1 we have R{g ()} > ν, ( 1 < ν < 1, U) 2 R{1 + a 1 } > ν > It is clear that { R a { R 1 + { 1 ν 1} > R 1 + a 1} Applyig the covolutio properties of power series to [σ μ (, f)g()], we may write [σ μ (, f)g()] 1 + [ (α) 1 ν a 1} > 1 2 (6) ( 1) + μ ( 1) (α) a μ 2 : P () Q() I virtue of Lemma 4 ad for j 1, we obtai Sice ad i view of (8), (α) (1 ν) a 1] [ 1 + { R ( 1) + μ ( 1) (1 ν) 1] + μ 2 { 1 } R (8) 2 2 1} { R Thus whe, a computatio gives ( 1) + μ { } { ( 1) R Q() R 1 + (1 ν) 1} > 2 + μ 2 (7) 1 }, (9) + 1 { R 1} 1 3 (10) 2 3( + μ)! (μ + 1)!!(1 ν) 3( + μ)! IJNS homepage:

6 460 Iteratioal Joural of NoliearSciece,Vol9(2010),No4,pp O the other had, the power series satisfies: P (0) 1 ad Therefore, by Lemma 5, we have P () [ 1 + { } { R P () R 1 + { R [σ μ (, f)g()] } > This completes the proof of Theorem (α) (1 ν) a 1], ( U) (α) (1 ν) a 1} > 1, ( U) 2 3( + μ)! (μ + 1)!!(1 ν), ( U) 3( + μ)! Corollary 11 Let the assumptios of Theorem 4 hold The for ( 1) + μ ( 1) 1, σ μ + μ (, f)g() B( 2 + ν ) 3 Acowledgemet The wor preseted here was supported by UKM-ST-06-FRGS The authors would lie to tha the referee for the commets give to the article Refereces [1] Dure P L Theory of H p spaces, Acad Press 1970 [2] Cuigham W Itroductio to Noliear Aalysis, MeGraw-Hill, New Yor 1958 [3] Barard RW Cima J ad Pearce K Cesáro sum approximatio of outer fuctios A Uiv Maria Curie- Slodowsa Sect, A52(1998)(1):1-7 [4] Ruscheweyh St Covolutios i Geometric Fuctio Theory Les Presses deĺ Uiversité de Motréal, Motréal 1982 [5] Ruscheweyh St Geometric properties of Cesáro meas Results Math, 22(1992): [6] Ruscheweyh St ad LSalias Subordiatio by Cesáro meas Complex Var Theory Appl, 21(1993): [7] Splia LT O certai applicatios of the Hadamard product Appl Math ad Comp, 199(2008): [8] Goodma A W Uivalet Fuctios, Vols I ad II, Polygoal Publishig House, Washigto, New Jersey 1983 [9] Jahagiri J M ad Farahmad K Partial sums of fuctios of bouded turig J Iequal Pure ad Appl Math, 4(4) Art 79(2003):1-3 [10] Mucehoupt B ad Webb D W Two-weight orm iequalities for Cesro meas of Laguerre expasios, Tras Amer Math Soc, 353(2001)(3): [11] Darus M ad Ibrahim R W Coefficiet iequalities for a ew class of uivalet fuctios, Lobachevsii J Math, 29(2008): [12] Darus M ad Ibrahim R W O Ces aro meas for Fox-Wright fuctios, J Math Statist, 4(2008): [13] Castella W Z, Filbir F ad Xub Y Ces aro meas of Jacobi expasios o the parabolic biagle, Joural of Approximatio Theory, 159(2009): IJNS for cotributio: editor@oliearscieceorgu