New Power Series Inequalities and Applications

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1 Iteratoa Joura of Mathematca Aayss Vo., 207, o. 20, HIKARI Ltd, htts://do.org/0.2988/jma New Power Seres Ieuates ad Acatos Loredaa Curdaru Deartmet of Mathematcs, Potehca Uversty of Tmsoara P-ta. Vctore, No.2, Tmsoara, Romaa Coyrght c 207 Loredaa Curdaru. Ths artce s dstrbuted uder the Creatve Commos Attrbuto Lcese, whch ermts urestrcted use, dstrbuto, ad reroducto ay medum, rovded the orga wor s roery cted. Abstract The am of ths aer s to reset severa ew euates for ower seres wth rea coeffcets by usg a Youg-tye euaty for seueces of comex umbers ad a trace euaty for ostve oerators. I addto, some acatos for seca fuctos such as oyogarthm ad for eemetary fuctos are reseted ad aso a acato formato theory, amey a refemet of Sgh s euaty, s gve. Mathematcs Subject Cassfcato: 26D20 Keywords: Youg-tye euaty, Power seres, Trace euates. Itroducto The cassca Höder s euaty, whch s very mortat rea ad comex aayss, ca be easy roved by usg Youg s euaty, x + y xy, whch taes ace for ay ostve umbers x, y, > ad + =. Ths euaty s aso ow as the weghted arthmetc mea-geometrc mea euaty for two umbers whe we wrte t as νa + νb > a ν b ν, where a ad b are dstct ostve rea umbers ad 0 ν. Ths euaty has may acatos varous feds ad there are a ot of terestg geerazatos of ths we-ow euaty ad ts reverse, see for exame 2, 3, 0, 7, 8, 9,, ad refereces there.

2 974 Loredaa Curdaru More recety, are gve ew resuts whch exted may geerazatos of Youg s euaty gve before. The beow resut, from, s a geerazato of the eft-had sde of a refemet of the euaty of Youg roved 200 ad 20 by Kttaeh ad Maasrah 2 ad 3, beg a very mortat too the demostrato of our ext theorems. Theorem Let λ, ν ad τ be rea umbers wth λ ad 0 ν τ. The ν λ A ν a, b λ G ν a, b λ λ ν τ A τ a, b λ G τ a, b, λ τ for a ostve ad dstct rea umbers a ad b. Moreover, both bouds are shar. The techue to fd other euates of fuctos usg ower seres was gve by Ibrahm ad Dragomr 4, Mortc 4 ad Ibrahm, Dragomr ad Darus 5 ad by ths method we ca exted some of the ow euates, whch have acatos may feds. We cosder as 5, a aaytc fucto defed by the ower seres fz = =0 a z wth rea coeffcets ad coverget o the ut ds D0, R, R > 0. Let f A z s a ew ower seres defed by =0 a z where a = a sga where sgx s the rea sgum fucto as 5. The ower seres f A z has the same radus of covergece as the orga ower seres fz. It s ecessary to reca the foowg euates whch have bee obtaed by Ibrahm, Dragomr ad Darus 5, Theorem, Theorem 2 ad Theorem 3 for ower seres see Theorems A, B ad C. Theorem A Let fz = =0 z ad gz = =0 z be two ower seres wth rea coeffcets ad coverget o the oe ds D0, R, R > 0. If x, y C, x, y 0, >, + = so that xy, x, y, x, y D0, R the g A x f A y + f A x g A y fxygxy ad g A x f A y + f A x g A y fx y gx y. Theorem B Let fz ad gz be as Theorem A. The oe has the euates g A x f A y + f A x g A y f x y gxy ad f A x g A y 2 + g A x 2 f A y fxyg x 2 y 2.

3 Power seres euates 975 Theorem C Let fz ad gz be as Theorem A. The oe has the euates g A x 2 f A y + f A x g A y 2 f x y g x 2 2 y ad g A x 2 f A y + f A x 2 g A y f x 2 yg x 2 y. As 6, et H be a Hbert sace ad B H the trace cass oerators BH. We defe the trace of a trace cass oerator A B H to be tra = I Ae, e >, where {e } I s a orthoorma bass of H. If H s fte-dmesoa the we ca see that ths cocdes wth the usua defto of the trace. It s ow, see 6 ad refereces there, that revous seres coverges absoutey ad t s deedet of the choce of {e } I. A trace euaty va Kttaeh-Maasrah resut was rove by Dragomr 6, Theorem see Theorem D. Theorem D Let A, B be two ostve oerators ad P, Q B H wth P, Q > 0. The for ay ν 0, we have r trp A trp 2trP A trp 2 trqb 2 trq + trqb trq trp A ν trp + ν trqb trq trp A ν trqb ν trp trq trp A R trp 2trP A 2 trqb 2 + trqb trp trq trq where r = m{ ν, ν} ad R = max{ ν, ν}. The am of ths aer s to reset ew euates for fuctos defed by for ower seres wth rea coeffcets. Ths thg was doe Secto 2 Theorems 2 ad 3. The acatos for some fudameta comex fuctos such as exoeta ad hyerboc fuctos ad aso for oyogarthm fucto are gve. Power seres ad seca fuctos have mortat acatos egeeerg sceces ad aed mathematcs. May euates vovg the oyogarthm, hyergeometrc, Besse ad modfed Besse fuctos ca be foud the terature, see 2, 22, 23, 24, 25, 26 ad refereces there. Secto 3 s devoted to trace euates, see Theorem 4, ad ast secto, Secto 4 s reserved to acatos Iformato Theory. I Iformato Theory aears may euates ad cocets such as Sgh s euaty 8 ad Shao etroy whch ca be obtaed from geerazatos of Höder s euaty as Remar 3.2 ad Remar 3.4 from 6, 7. Tag to accout artcuar sutabe fuctos the Youg-tye euaty, ew euates w be deduced for some of these cocets, see Theorem 5.

4 976 Loredaa Curdaru 2. The Youg-tye euates for ower seres If we tae Theorem, λ = N, ν =, τ =, a stead of a ad b stead of b the we obta the foowg euaty: a b a b a b a b a b a b, where + = ad + =. Next euates, gve Theorem 2 ad Theorem 3, from beow are ew varats ad refemets of Theorem, Theorem 2 from 5 whe N. I artcuar, these euates ca be obtaed aso for the moduus of the roduct of the fuctos f ad g, see for exame Coroary 2b. The some acatos to seca fuctos are gve Coroary 3 ad Coroary 4. Theorem 2 Let fz = =0 z ad gz = =0 z be two ower seres wth rea coeffcets ad coverget o the oe ds D0, R, R > 0. If a, b C, a, b 0 so that a, a, b, b D0, R ad f,,, are e revous euaty ad addto the we have: f A a b g A a b f A a b g A a b f A a b g A a b f A a b g A a b f A a b g A a b f A a b g A a b. Proof. We start the roof tag to accout that the hyothess a, a, b, b D0, R mes the foowg cusos a b, a b, a b, a b D0, R, = 0, by cacuus. Thus, for exame, a b R + = R so a b D0, R.

5 Power seres euates 977 We use the same method as 5. Thus we choose a = a j b, b = a b j, j, {0,, 2,..., } revous euaty ad we have: a j b a b j a j b a b j a j b b j a a j b a b j a j b a b j a j b a b j. We muty ast euaty by ostve uattes j ad the summg over j ad from 0 to m we obta: m j a j b j b a j=0 j=0 j a j b j j=0 b a m j a j b j b a j=0 m j a j b j a b j=0 m j a j b j b a j=0 j a j b j b a. Tag above the mt whe m we get the desred euaty, because a the seres whose arta sums are voved are covergeto the ds D0, R. Theorem 3 Let fz = =0 z ad gz = =0 z be two ower seres wth rea coeffcets ad coverget o the oe ds D0, R, R > 0. If a, b C, a, b 0 so that a, a, b, b D0, R. If,,, be e revous euaty ad addto. The the foowg euaty taes ace: f A a b g A a b f A a b g A a b

6 978 Loredaa Curdaru f A a b g A a b f A a b g A a b f A a b g A a b f A a b g A a b. Proof. We use the hyothess a, a, b, b D0, R whch mes the foowg cusos a b, a b, a b, a b, a b, a b D0, R, = 0, by cacuus. Now we choose a = a j ad b = a ad use the same method e b j b Theorem 2. Coroary Let fz, gz, a, b ad,,, be as Theorem 2, ad addto, we tae = the same Theorem 2. The we obta: f A b g A a + f A a g A b f A a b g A a b f A b g A a + f A a g A b f A ab g A ab f A b g A a + f A a g A b f A a b g A a b. Coroary 2 a If we tae fz = gz revous euaty, Coroary, the we obta: f A b f A a + f A a f A b f A a b f A a b f A b f A a + f A a f A b fa ab 2 f A b f A a + f A a f A b f A a b f A a b. b We ca aso state the foowg form of the eft sde of the euaty from Theorem 2, a form where aears the fuctos fz ad gz. fa b ga b f A a b g A a b +

7 Power seres euates f A a b g A a b. The same d of euates e before ca be stated for the eft sde of the euates from Theorem 3. c The euaty from Theorem 2 ca be aso rewrtte e beow: f A a b g A a b f A a b g A a b f A a b g A a b. Tag to accout that the fuctos exz, z C, z, z D0,, z, z D0,, shz, z C are ower seres wth rea coeffcets ad coverget o the oe ds D0, we ca th to rewrte the euates gve before uder codtos from our theorems. We gve beow some acatos to artcuar fuctos of terest. 2+! z2+, z Coroary 3 a If we cosder the fucto fz = sz = =0 C the f A z = shz, z C ad uder codto from Coroary 2 a, euaty becomes: sh b sh a + sh a sh b sh a b sh a b sh b sh a + sh a sh b sh 2 ab sh b sh a + sh a sh b sh a b sh a b. b If fz = exz = =0! z = f A z, z C the uder codtos from Coroary 2 a the euaty w be the foowg: ex b + a + ex a + b ex a b + a b ex b + a + ex a + b ex 2 ab ex b + a + ex a + b ex a b + a b. c If fz = = f z Az, z D0, ad a, b are comex umbers as Theorem 2, the we have: b a + b a a b b a + b a ab a 2 b

8 980 Loredaa Curdaru b a + b a a b d Usg codtos from a, by cacuus, we get: a b s 2 ab sh b sh a + sh a b sh a b. We otce that smar resuts ca be obtaed for coshx as we. There exst some euates for seca fuctos such as oyogarthm, hyergeometrc, Besse ad modfed Besse fuctos for the frst d. It s ow that L z, 2 F a, b; c; z, J a z ad I a z are ower seres wth rea coeffcets ad coverget o the oe ds D0,. Le 5, we ca th to rewrte the euates gve before uder codtos of our theorems. As acato to seca fuctos we gve beow the foowg euaty for oyogarthm fucto, but we ca aso use hyergeometrc or Besse fuctos. I order to do that t s ecessary to reca the defto of ths fucto. The oyogarthm fucto, L z s defed by the ower seres L z = whch coverges absoutey for a comex vaues of the order ad z whe z. Coroary 4 If L z s the oyogarthm fucto the we have L b L a + L a L b L a b L a b = z. L b L a + L a L b L 2 ab L b L a + L a L b L a b L a b for ay a, b C, a, b 0 uder codtos of Coroary 2 a whe D0, R s D0,. 3. Trace euates The foowg resut s a geerazato of Theorem from 6 whe N. Theorem 4 Let A, B be two ostve oerators ad P, Q B H wth P, Q > 0. The for ay ν 0, we have: ν τ τ τ trqb trp A trqb τ trp A τ

9 ν τ Power seres euates 98 ν ν trqb trp A trqb ν trp A ν τ τ trqb trp A trqb τ trp A τ. Proof. For roof we roceed as 6, but we use stead the euaty 2. for λ = N from as a startg ot. Thus ν τ τ τ a b a τ b τ ν ν a b a ν b ν ν τ τ τ a b a τ b τ we fx b > 0 ad by usg the fuctoa cacuus for the oerator A we have: ν τ τ τ A x, x > b A τ x, x > b τ ν ν A x, x > b A ν x, x > b ν ν τ τ τ A x, x > b A τ x, x > b τ for ay x H. We fx x H\{0}. Usg aga the fuctoa cacuus, but ths tme for the oerator B we get: ν τ τ τ A x, x > B y, y > A τ x, x > B τ y, y > ν τ ν ν A x, x > B y, y > A ν x, x > B ν y, y > τ τ A x, x > B y, y > A τ x, x > B τ y, y > for ay x, y H ad ν 0,. We ut x = P 2 e, y = Q 2 f where e, f H ad we obta the foowg: ν τ τ τ A P 2 e, P 2 e > B Q 2 f, Q 2 f >

10 982 Loredaa Curdaru A τ P 2 e, P 2 e > B τ Q 2 f, Q 2 f > ν ν A P 2 e, P 2 e > B Q 2 f, Q 2 f > ν τ A ν P 2 e, P 2 e > B ν Q 2 f, Q 2 f > τ τ A P 2 e, P 2 e > B Q 2 f, Q 2 f > A τ P 2 e, P 2 e > B τ Q 2 f, Q 2 f > for ay e, f H. Let {e } I ad {f j } j J be two orthoorma bases of H ad we tae revous euates e = e, I ad f = f j, j J. Summg ow over I ad j J we get: ν τ τ τ P 2 A P 2 e, e > Q 2 B Q 2 fj, f j > I j J I P 2 A τ P 2 e, e > j J Q 2 B τ Q 2 fj, f j > ν ν I P 2 A P 2 e, e > j J Q 2 B Q 2 fj, f j > I P 2 A ν P 2 e, e > j J Q 2 B ν Q 2 fj, f j > ν τ τ τ I P 2 A P 2 e, e > j J Q 2 B Q 2 fj, f j > I P 2 A τ P 2 e, e > j J Q 2 B τ Q 2 fj, f j >. The desred euaty hods by usg the roertes of the trace. Remar Uder codtos of Theorem 4 we have: ν τ τtrp A + τtrqb trqbτ trp A τ νtrp A + νtrqb trqb ν trp A ν ν τ τtrp A + τtrqb trqbτ trp A τ.

11 Power seres euates Acatos formato theory Usg the same method as 6, 7 we reset a acato of euaty from Theorem, whe λ =, formato theory. For that we cosder the euaty A + B A B A + B AB A + B A B, where we ut, as the roof of the cassca Höder s euaty, A = ad B = = b over from to we get or = a b = a where a, b > 0, {, 2,..., } here > ad summg = a b = a = b = b = a { If a = h r t r, b = t r h t = = b = a b = a b = a = b = a = b {, = r, s + r { r = h r = a b = a = b = a b } a a b = = a = b = the by cacuus we obta: = h t + s = h t = ts h t { r = r = t s s = h t + s = h t = ts } r } r, }. where 0 r, s 0 ad = r. Let Λ be the utty formato scheme, as Remar 3.2, see 6, 7 where X = x, x 2,..., x s the ahabet; P β = β, β 2,..., β s the ower robabty dstrbuto; U = u, u 2,..., u s the utty dstrbuto u > 0

12 984 Loredaa Curdaru for a =, 2, 3,..., ; β, β > 0, = β decherabe code, Sgh et a. 8 obtaed og D β u D = = β u =. The, for every uuey og 2 = β u = β u og 2 D where > 0,, D 2, tegers, 0, =, 2,.., ad = D. Accordg to 8 the usefu formato of order for ower dstrbuto P β s defed as og β u = β u = ad the exoeta usefu mea egths of codewords weghted wth the fucto of ower robabtes ad uttes s defed as β u = β u D. = The ast euaty s a geerazato of Shao euaty. Theorem 5 Let > 0, β > 0,, β, 0, =, 2, 3,..., ad = β =, et DD 2 s the sze of the code ahabet. If, =, 2, 3,..., are the egths of the codewords satsfyg = D the for every uuey decherabe code, the usefu - average egth of codewords satsfes og β D u D +og D D + og D M, u ; > = where = = β u > og 2 = M, u ; = r whe >. Proof. Usg the substtuto r = h = β β u = β u og 2 D β = D u = D > 0, s = 0, u = u D β = β u,

13 Power seres euates 985 ad t = β u = u β ast euaty ad after sutabe cacuus we obta the desred euaty whe >. Refereces H. Azer, C. M. Foseca, A. Kovacec, Youg-tye euates ad ther matrx aaogues, Lear ad Mutear Agebra, , o. 3, htts://do.org/0.080/ L. Curdaru, N. Mcuete, Ieuates for ower seres, A. Math. If. Sc., 9 205, o. 4, S. S. Dragomr, Cebysev s tye euates for fuctos o sefadjot oerators Hbert saces, Lear ad Mutear Agebra, , htts://do.org/0.080/ A. Ibrahm, S. S. Dragomr, Power seres euates va Buzao s resut ad acatos, Itegra Trasforms ad Seca Fuctos, 22 20, o. 2, htts://do.org/0.080/ A. Ibrahm, S. S. Dragomr, M. Darus, Power seres euates va Youg s euaty wth acatos, Joura of Ieuates ad Acatos, , 34. htts://do.org/0.86/ x S. S. Dragomr, Trace euates for ostve oerators va recet refemets ad reverses of Youg s euaty, Research Grou Math. Ie. ad A., Res. Re. Co., 8 205, Art S. S. Dragomr, A ote o ew refemets ad reverses of Youg s euaty, Research Grou Math. Ie. ad A., Res. Re. Co., 8 205, Art S. S. Dragomr, New refemets ad reverses of Hermte-Hadamard euaty ad acatos to Youg s oerator euaty, Research Grou Math. Ie. ad A., Res. Re. Co.,, 9 206, Art S. S. Dragomr, Some asymmetrc reverses of Youg s scaar ad oerator euates wth acatos, Research Grou Math. Ie. ad A., Res. Re. Co., 9 206, Art Y. Feg, Refemets of Youg s euates for matrces, Far East J. Math. Sc., FJMS, , S. Furuch, N. Mcuete, Ateratve reverse euates for Youg s euaty, Joura of Mathematca Ieuates, 5 20, o. 4, htts://do.org/0.753/jm F. Kttaeh, Y. Maasrah, Reverse Youg ad Hez euates for matrces, Lear ad Mutear Agebra, 59 20, o. 9, htts://do.org/0.080/ F. Kttaeh, Y. Maasrah, Imroved Youg ad Hez euates for matrces, J. Math. Aa. A., , htts://do.org/0.06/j.jmaa C. Mortc, A Power Seres Aroach to Some Ieuates, The Amerca Mathematca Mothy, 9 202, o. 2, htts://do.org/0.469/amer.math.mothy W. Rud, Rea ad Comex Aayss, 3rd ed., McGraw-H, New-Yor, J. Ta, Reversed verso of a geerazed shar Höder s euaty ad ts acatos, Iformato Sceces, , htts://do.org/0.06/j.s

14 986 Loredaa Curdaru 7 J. Ta, New roerty of a geerazed Höder s euaty ad ts acatos, Iformato Sceces, , htts://do.org/0.06/j.s R. P. Sgh, R. Kumar, R. K. Tuteja, Acatos of Höder s euaty formato theory, Iformato Sceces, , htts://do.org/0.06/s B. Smo, Trace Ideas ad Ther Acatos, Cambrdge Uversty Press, Cambrdge, W. H. Youg, O casses of summabe fuctos ad ther Fourer seres, Proc. of the Roya Soc. A: Mathematca, Physca ad Egeerg Sceces, 87 92, o. 594, htts://do.org/0.098/rsa A. Barcz, Fuctoa euates vovg Besse ad modfed Besse fuctos of the frst d, Exo. Math., , htts://do.org/0.06/j.exmath R. W. Barard, K. C. Keda, O euates for hyergeometrc aaogues of the arthmetc-geometrc mea, J. Ieua. Pure A. Math., , o. 3, B. He, B. Yag, O a Hbert-tye euaty wth a hyergeometrc fucto, Commu. Math. Aa., 9 200, o., M. M. Jema, A ma euaty for severa seca fuctos, Comut. Math. A., , htts://do.org/0.06/j.camwa S. R. Yadava, B. Sgh, Certa euates vovg seca fuctos, Proc. Math. Acad. Sc. Ida, Sect. A Phys. Sc., , o. 3, L. Zhu, Jorda tye euates vovg the Besse ad modfed Besse fuctos, Comut. Math. A., , htts://do.org/0.06/j.camwa R. W. Hammg, Codg ad Iformato Theory, New Jersey, 980. Receved: October 5, 207; Pubshed: November 6, 207