Scece Joural of Appled Matheatcs ad Statstcs 05; 3(): 58-6 Pulshed ole Aprl 05 (http://www.scecepulshggroup.co//sas) do: 0.648/.sas.05030.5 ISSN: 376-949 (Prt); ISSN: 376-953 (Ole) O Covergece a Varato of the Coverse of Fary Gap Theore Naser Aas Molood Gor * Departet of Matheatcs Loresta Uversty Khoraaad Islac Repulc of Ira Eal address: aseraas_persa@yahoo.co (N. Aas) olodgor@yahoo.co (M. Gor) To cte ths artcle: Naser Aas Molood Gor. O Covergece a Varato of the Coverse of Fary Gap Theore. Scece Joural of Appled Matheatcs ad Statstcs. Vol. 3 No. 05 pp. 58-6. do: 0.648/.sas.05030.5 Astract: I ths artcle we gve a varato of the coverse of Fary Gap theore cocerg the locato of sgulartes of Taylor-Drchlet seres o the oudary of covergece. whose crcle of covergece s the ut crcle ad for whch the ut crcle s ot the atural oudary. Keywords: Drchlet Seres Etre Fuctos Fary Gap Theore. Itroducto z The gap theore of Fary states that f f ( z) = ae s a power seres whose crcle of covergece s the ut crcle ad l / = the the ut crcle s the atural oudary of f ( z ). Polya ([7]) proved the followg coverse of ths result: Let e a sequece of tegers for whch lf / < ; the there exsts a power seres z ae whose crcle of covergece s the ut crcle ad for whch the ut crcle s ot the atural oudary. Based o a sequece A as etoed earler we costruct a ultplcty sequece B = {( µ ) = that s a sequece where for oe has ad each appears µ tes. For ths sequece B we prove that the fte product G( z) whch vashes exactly o ± B satsfes µ = ε every ε > 0. [ µ ]!/ G ( ) O( exp{ ) That s we have a sharp estate for the fucto of G( z) evaluated o. µ th dervatve We assue that the reader s falar wth the theory of Etre Fuctos ad the theory of Drchlet seres as used the oos [58 ]. We ote that other results cocerg the locato of sgulartes of Taylor Drchlet seres have ee derved y Blaert Parvatha ad Berlad (see [ 4]).. Auxlary Results ad Notos I ths secto we descre the deftos ad also to express ad prove the lea we eed to prove the theore. Defto.. We deote y L( c D) the class of all sequeces A = { a = wth dstct coplex ters a dvergg to fty a a satsfyg the followg codtos: (see also []) () There s a costat c > 0 so that a a c for all. () l / a = D 0. (3) the sup arg a < π /. Defto.. Choose a sequece A = { a = whch elogs to the class L( c D ). suppose ad β e real postve uers so that β <. We say that a sequece B = { = wth coplex ters 0 wth the ot ecessarly a creasg order ad sup N arg π / elogs to the class A β f for all N we have { z C : z a a ad for all oe of the followg codtos holds: () = β β a a () ax e e {. Oe oserves that allows for the sequece B to have cocdg ters. We ay ow rewrte B the for of a ultplcty sequece{ µ = the followg way; frst
Scece Joural of Appled Matheatcs ad Statstcs 05; 3(): 58-6 59 we splt B to groups of ters havg the sae odulus ad the wth each group we order the y the sze of ther arguet egg fro saller to larger. The arguets are tae the terval 0 arg π. We shall say that { µ s the ( µ ) reorderg of B. Rear. We pot out that the spacg codto () of a sequece A L( c D) plays a very portat role throughout the artcle. Let Γ e as Oe deduces that f Γ = { : =. Γ the Γ = Γ. defe to e the uer of ters of refer to as the pseudo-ultplcty of We also Γ ad we shall. I the lea that follows we get a upper oud for. wth respect to Lea.3. There exst postve costats ψ ad χ so that for ay oe has ψ a χ Proof. Frst ote that the relato a / a holds for all > 0 sce a a. Cosder ow ay Γ ( Γ = { : = ). The a = ( a ) ( ) a It follows that a 4 a. a a The oe also gets a a = ( a ) ( ) ( a ) Fally the spacg codto yelds that for ay a a 5 a a a c Γ oe has c 5 a Sce s the uer of ters of Γ the ax{ : Γ. Fro the aove equato t follows that there exsts a postve ψ so that ψ a. Fally the relato a a postve χ so that χ. Lea.4. For ay oe has µ χ yelds Proof. Let = for soe N. Fro the prevous lea we ow that χ for soe χ > 0. But the pseudo-ultplcty ( ) of s the ultplcty µ of. Thus oe otas the relato µ χ. Aother portat lea whch s portat for ths paper ca e stated as follows: Lea.5.Let A L( c D) e a real postve sequece ad let B A β so that B = { s real postve too wth ( µ ) ts reorderg. The the regos of covergece of the three seres where ad f f f as defed = z f ( z) = p ( z) e () = 0 µ p ( z) = µ c z s a polyoal wth c µ 0 µ () * z ** µ z = = = = f z Ae f z A z e are the sae. For ay pot z sde the ope covex rego the three seres coverge asolutely. Slarly f stead of a real sequece A L( c D) we have a coplex sequece A L( c0). Proof. We have to show that log µ l = 0 l = 0 s satsfed. Frst ote that fro Lea.4 oe deduces that the rght lt of log µ l = 0 l = 0 s vald. Thus t reas to verfy the left lt. We cla that Ths ples that a / log log a log = 0 a a sce / a D ad we are doe. Let us ustfy our cla. It s ovous that a /. Assue that a / for soe. We wll prove that
60 Naser Aas ad Molood Gor: O Covergece a Varato of the Coverse of Fary Gap Theore a / as well. Note that there s at least oe so that =. If < a / the a / = < a /. Sce a / < a ths ples that sce a a for all Ν therefore { =. Ths eas that {( µ ) = { = thus there s soe {( µ ) = wth. It follows that a a a > that s a > a whch s false. Thus a / ad ths copletes the proof. 3. Ma Results Ths secto descres the a theore of ths paper ad t ca e fxed y usg prove ethods Polya. Theore 3.. suppose A L( c D) e a real postve sequece. Let B A β so that B = { s real postve too lf = D < ad let ( µ ) e ts reorderg. The ay Taylor-Drchlet seres f ( z) as () satsfyg l l c la = (3) µ sup lsup whose crcle of covergece s the ut crcle ad for whch the ut crcle s ot the atural oudary. Proof. We follow o the les of the proof of Theore [6]. Let f f f ad A as defed () () ad A = ax{ c : = 0... µ. (4) Fro Lea.5 the regos of covergece of the three seres are the sae. Sce the are real postve uers we cosder the o-trval case that s whe the three seres coverge detcal half-plaes of the for R z > x0 x0 R. Wth o loss of geeralty we assue that the ascssa of covergece (ordary ad asolute) s the le x =. I other words the relato la l sup = (5) holds. Thus all three seres coverge asolutely ad uforly ay half-plae x τ >. Oe also otes that fro (3) we have l c µ l su p. = (6) There clearly exst two sequeces of tegers B such that [( ) ] B ad B = c B B > B ad the uer of ( B B ) s greater tha c( B B ) > c3b. (The c's deote asolute postve costats.) The exstece of these sequeces s edate fro lf / <. Deote the the tervals ( B B ) y. We clearly have lf / <. For costructo of f ( z) we shall use oly the. Put z f ( z) = p ( z) e = µ We shall detere the p µ so that the ut crcle wll e the crcle of covergece ad the potwll e a regular pot of f ( z ). It wll suffce to show that there exsts a uer > l > 0 such that the crcle of covergece of f ( z l z l ) p ( z l ) e e = µ = has radus greater tha l. We shall choose l = ( r ) / r r a suffcetly large teger. We have y the oal expaso We have to show that p µ r r ( z ) r p ( z ) e r µ e pµ z C r r l sup / < r for soe choce of / the wth l sup pµ =. Let ε e a sall ut fxed uer; we dstgush two cases. I case does ot le ay of the tervals (( B / r)( ε)( B / r)( ε )). The we show that for every choce of the p µ wth p µ r = r < r δ / l sup δ = δ ( ε ). Ths eas that f s large eough ad does ot le (( B / r)( ε)( B / r)( ε)) the < ( r δ ). Clearly If we defe We fd = e r C. r (7) ( / ) C r r = H H / H = (( r ) / r)( ) / ( ). (8) By studyg the quotet (8) we see that ax ( / ) r H = H = C r r ad y applyg r r
Scece Joural of Appled Matheatcs ad Statstcs 05; 3(): 58-6 6 Strlg's forula We ote that / Hr! ~ ( ) r / / π c there exsts = ( ε) > 0 such that as. It follows fro (8) that H / / H > η for < r ε (9) H / / H < η for > r ε (0) ad hece a sple calculato shows that there exsts a β = β ( ε) > 0 such that for ot ( r / ( ε) r / ( ε )). Now clearly H < ( r β ) () = where the suato s exteded over the < r / ( ε) ad over the > r / ( ε ). (By assupto does ot le (( B / r)( ε)( B / r)( ε)) ad (7) the are all ( B B ) ; thus f < ( B / r)( ε ) > r / ( ε) ad f > ( B / r)( ε ) < r / ( ε ).) Thus fro (9) (0) ad () (y sug a geoetrc seres) or whch copletes the proof. I case ( ) We wrte where < c r β 4 < r δ / lsup ( B / r)( ε) < < ( B / r)( ε ). = r r = pµ ( z ) C r r r r = pµ ( z ) C r r dcates that the suato s exteded oly over those for whch does ot le ( B B ) ad the suato s exteded over the other. We ad we ca show that r C r l sup '/ < r as efore. Now we show that we ca choose the p µ to e such as to ae all the for ( B / r)( ε) ( B / r)( ε) equal to 0. Thus we ust detere thep µ so that r r pµ ( z ) C 0 = r r These are hoogeeous equatos for the p µ. The uer of these equatos s less tha ( B B ) / r for suffcetly sall ε ad the uer of uows s greater tha c 3 B B whch s greater tha the uer of equatos for large eough r. Thus the syste of equatos always has a soluto ad further we ca suppose that the asolute value of the largest s. Ths wll sure that the p µ crcle of covergece of f ( z) wll e the ut crcle whch copletes the proof. 4. Cocluso I ths study we exae a varato of the coverse of Fary Gap theore. Polya's result shows that soe sese Fary's result s the est possle. Perhaps the eleetary ad drect proof that etoed aove ght e of soe terest. To do ths a sequece wth a seres of ew uld ad reorderg the call usg the covergece of three seres f f f ota upper ad lower ouds. Ad usg the Strlg's forula ad we wll acheve the desred result ths paper. Acowledgeets The author gratefully acowledges the help of Prof. E. Zos to prove the orgal verso of the paper. Refereces [] Bereste C.A. ad Gay Roger Coplex Aalyss ad Specal Topcs Haroc Aalyss (New Yor Ic: Sprger-Verlag) (995). [] Blaert M. ad Parvatha R. Ultracovergece et sgualartes pour ue classe de seres d expoetelles. Uverste de Greole. Aales de l Isttut Fourer 9() (979) 39 6. [3] Blaert M. ad Parvatha R. Sur ue egalte
6 Naser Aas ad Molood Gor: O Covergece a Varato of the Coverse of Fary Gap Theore fodaetale et les sgualartes d ue focto aalytque defe par u eleet LC-drchlete. Uverste de Greole. Aales de l Isttut Fourer 33(4) (983) 35 60. [4] Berlad M. O the covergeve ad sgulartes of aalytc fuctos defed y E-Drchleta eleets. Aales des Sceces Matheatques du Queec ()(998) 5. [5] Boas R.P. Jr Etre Fuctos (New Yor: Acadec Press) (954). [6] Erdos P. Note o the coverse of Fary's Gap theore Tras. Aer. Math. Soc. 57 (945) 0-04. [7] Polya G. O coverse Gap theores Tras. Aer. Math. Soc. 5 (94) 65-7. [8] Lev B. Ya. Dstruto of Zeros of Etre Fuctos (Provdece R.I.: Aer. Math. Soc.) (964). [9] Lev B. Ya. Lectures o Etre Fuctos (Provdece R.I.: Aer. Math. Soc.) (996). [0] Levso N. Gap ad Desty Theores. Aerca Matheatcal Socety Colloquu Pulcatos Vol. 6 (New Yor: Aer. Math. Soc.) (940). [] Madelrot S. Drchlet Seres Prcples ad Methods (Dordrecht: D. Redel Pulshg Co.) (97) pp. x 66. [] Zos E. O a theore of Nora Levso ad a varato of the Fary Gap theore Coplex Varales 50 (4) (005) 9-55.