Journal of Mathematical Inequalities

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1 Journl of Mhemicl Inequliies Wih Comlimens of he Auhor Zgreb, Croi Volume 10, Number 2, June 2016 Mrin J. Bohner nd Smir H. Sker Snek-ou rincile on ime scles JMI

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3 Journl of Mhemicl Inequliies Volume 10, Number 2 (2016), doi: /jmi SNEAK OUT PRINCIPLE ON TIME SCALES MARTIN J. BOHNER AND SAMIR H. SAKER (Communiced by A. Peerson) Absrc. In his er, we show h he so-clled snek-ou rincile for discree inequliies is vlid lso on generl ime scle. In riculr, we rove some new dynmic inequliies on ime scles which s secil cses conin discree inequliies obined by Benne nd Grosse- Erdmnn. The min resuls lso re used o formule he corresonding coninuous inegrl inequliies, nd hese re essenilly new. The echniques emloyed in his er re elemenry nd rely minly on he ime scles inegrion by rs rule, he ime scles chin rule, he ime scles Hölder inequliy, nd he ime scles Minkowski inequliy. 1. Inroducion In [4], Benne nd Grosse-Erdmnn sudied he roblem of deducing he convergence of one series from h of noher during he course of heir invesigions on Hrdy ye inequliies. In riculr, hey hve considered generl yes of series which cnno be reed by ny of he usul convergence ess (rio es, comrison es, Rbe s es, ec.). Afer hey hd severl successes, hey noiced somehing quie remrkble in subjec s old s his: The emergence of new echniques. They clled hese he five series heorem, he snek-ou rincile nd he heds/ils oion (see [3]). All hree re vlid in considerble generliy nd ll hve significn licions beyond Hrdy s inequliy. Now he following quesion rises: Is i ossible o exend hese echniques o ime scles? In oher words, is i ossible o exend hese echniques o conin corresonding inegrl inequliies nd summion inequliies s secil cses? Our im in his er is o give n ffirmive nswer for he second one which is he snek-ou rincile. The obined resuls will suor he dvice of Hrdy, Lilewood nd Póly [12, ge 11] by giving unificion of he coninuous nd he discree inequliies nd showing h wh goes for sums goes for inegrls nd vise vers. The snek-ou rincile h hs been considered by Benne nd Grosse-Erdmnn [3] ws concerned wih he equivlence of he wo series ( n Ak k) n=1 α x (1.1) k=n Mhemics subjec clssificion (2010): 26A15, 26D10, 26D15, 39A13, 34A40, 34N05. Keywords nd hrses: Coson s inequliy, Hrdy s inequliy, ime scles. c D l,zgreb Per JMI

4 394 M. J. BOHNER AND S. H. SAKER nd n=1 n A α n ( ) x k. k=n In oher words, when is i ossible o snek he erm Ak α ou of he inner sum in (1.1). Benne nd Grosse-Erdmnn, using his rincile, roved severl inequliies of he form n=1 n ( k=n A α k x k) K(α, ) n=1 n A α n ( k=nx k ) nd heir inverses for differen vlues of nd α. Our im in his er is concerned wih he equivlence of he wo ime scles inegrls ( () A (σ(s))x(s)δs) α Δ (1.2) nd ( ()A α (σ(s)) x(s)δs) Δ, where he domin of he unknown funcion is so-clled ime scle T (which is n rbirry nonemy closed subse of he rel numbers R). In oher words, we wn o deermine recisely when i is ossible o snek A α (σ(s)) ouside of he inner inegrl in (1.2). More recisely, we re concerned wih new dynmic inequliies of he form ( ( () A (σ(s))x(s)δs) α Δ K ()A α (σ(s)) x(s)δs) Δ (1.3) nd heir converses on ime scles for differen vlues of nd α, which s secil cses wih T = N conin he discree inequliies obined by Benne nd Grosse- Erdmnn [3, Secion 6] nd cn be lied lso wih T = R o formule he corresonding inegrl inequliies. The resuls lso cn be lied o oher ime scles such s T = hz for h > 0ndT = q N 0 wih q > 1. We include he deils of he roofs since hey rovide sregy which cn be used in oher siuions. For reled dynmic inequliies on ime scles, we refer he reder o [1, 2, 5, 8, 13 17]. This er is orgnizeds follows: In Secion 2, we rovide some uxiliry resuls such s he ime scles inegrion by rs rule, he ime scles chin rule, he ime scles Hölder inequliy, nd he ime scles Minkowski inequliy. Secion 3 feures wo new dynmic inequliies of Coson ye h re needed in he roofs of our min resuls. In Secion 4, we resenourmin resuls, which re hreedynmicinequliies of he ye (1.3) for differen vlues of 1ndα. Firs he cse α 1 is reed (see Theorem 4.1 below), hen he cse 0 α 1 (see Theorem 4.4 below), nd finlly he cse 1/ < α 0 (see Theorem 4.6 below). The corresonding cses when 0 < < 1 re sill oen roblems nd will be considered in he fuure.

5 SNEAK-OUT PRINCIPLE Auxiliry resuls For comleeness, we recll he following conces reled o he noion of ime scles. For more deils of ime scle nlysis, we refer he reder o he wo books by Bohner nd Peerson [6, 7]. A ime scle T is n rbirry nonemy closed subse of he rel numbers R. The cses when T = R nd T = Z reresen he clssicl heories of differenil nd difference clculus. In his er, we ssume h sut = nd define he forwrd jum oeror by σ() := inf{s T : s > }. For ny funcion f : T R, we wrie f σ for f σ.for T, wedefine f Δ () o be he number (if i exiss) wih he roery h given ny ε > 0, here is neighborhood U of wih f (σ()) f (s) f Δ ()(σ() s) ε σ() s for ll s U. In his cse, we sy f Δ () is he (del) derivive of f.if f is (del) differenible ny T,hen f Δ : T R is clled he del derivive of f. Nex, if F : T R is n niderivive of f, i.e., F Δ = f, hen he Cuchy del inegrl of f is defined by f (s)δs := F() F(), where T is fixed. I is known [6, Theorem 1.74] h so-clled rd-coninuous funcions lwys ossess niderivives. EXAMPLE 2.1. Noe h if T = R,hen σ()=, f Δ = f, nd for,b R wih < b. IfT = Z, hen b b f ()Δ = f ()d σ()= + 1, f Δ = Δ f, nd b for,b Z wih < b. IfT = hz wih h > 0, hen b 1 f ()Δ = = f () σ()= + h nd b f ()Δ = h b h h k=0 f ( + kh) for,b hz. IfT = q N 0 wih q > 1, hen σ()=q nd b log q (b) 1 f ()Δ =(q 1) k=log q () q k f (q k ) for,b q N 0.

6 396 M. J. BOHNER AND S. H. SAKER Now we collec hose known ime scles resuls h will be used frequenly hroughou his er. The roduc nd quoien rules [6, Theorem 1.20] for he derivive of he roduc fg nd he quoien f /g (wih g() 0forll T) of wo differenible funcions f,g : T R se ( fg) Δ = f Δ g + f σ g Δ = fg Δ + f Δ g σ nd ( ) f Δ = f Δ g fg Δ g gg σ. (2.1) The chin rule [6, Theorem 1.90] for he γ -h ower (γ R) of differenible funcion f : T R sys (see [6, Theorem 1.90]) 1 ( f γ ) Δ = γ f Δ (hf σ +(1 h) f ) γ 1 dh. (2.2) 0 The inegrion by rs formul [6, Theorem 1.77] for wo differenible funcions f,g : T R is given (,b T) by b f ()g Δ ()Δ = f ()g() b b f Δ ()g σ ()Δ. (2.3) Hölder s inequliy [6, Theorem 6.13] ses h wo rd-coninuous funcions f,g : T R sisfy b { b { 1 b q f ()g() Δ f () Δ g() Δ} q, (2.4) where > 1, q = /( 1),nd,b T. Minkowski s inequliy [6, Theorem 6.16] ssers h hree rd-coninuous funcions f,g,h : T R sisfy { b { b { 1 b h() f ()+g() Δ h() f () Δ + h() g() Δ}, (2.5) where > 1nd,b T. Inequliies (2.6)nd(2.7) below re simle consequences of he chin rule (2.2), bu for convenience of furher reference, we now se hese four imorn inequliies (which re subsiues for he ower rule from differenil clculus) in he following lemm, sulemened by wo new inequliies which re, however, merely simle consequences of he roduc rule (2.1). LEMMA 2.2. Suose f : T R is differenible nd osiive. Le γ R. If f Δ is eiher lwys osiive or lwys negive, hen γ f Δ ( f γ 1) σ ( f γ ) Δ γ f Δ f γ 1 if 0 γ 1 (2.6) nd γ f Δ f γ 1 ( f γ ) Δ γ f Δ ( f γ 1) σ if γ 1. (2.7)

7 SNEAK-OUT PRINCIPLE 397 If f Δ is lwys osiive, hen nd ( f γ ) Δ f Δ ( f γ 1) σ ( f γ ) Δ f Δ ( f γ 1) σ if 0 γ 1 (2.8) if γ 1. (2.9) Proof. Inequliies (2.6)nd(2.7) follow direcly from (2.2). Nex, if f is incresing nd if 0 γ 1, hen f γ 1 is decresing nd hus ( f γ 1) Δ < 0soh ( f γ ) Δ = ( ff γ 1) Δ (2.1) = f Δ ( f γ 1) σ + f ( f γ 1 ) Δ. This shows (2.8), nd (2.9) follows similrly. 3. Dynmic inequliies of Coson ye In his secion, we rove wo new dynmic inequliies of Coson ye (see [9, 10] for he originl Coson inequliies). These will be used in he roofs of our min resuls in he nex secion. Throughou, we re using he following ssumions: sut =, T, : T (0, ) is rd-coninuous, A() := (3.1) (s)δs, T. THEOREM 3.1. Assume (3.1). Suose ϕ : T R is such h (s) Φ() := A(σ(s)) ϕ(s)δs, T is well defined. Le k 1. Then ()Φ k ()Δ k k ()ϕ k ()Δ. (3.2) Proof. We use inegrion by rs (2.3), he lef r of he inequliy (2.7) wih f = Φ nd γ = k, ndhölder s inequliy (2.4) wih = k nd q = k/(k 1) (unless k = 1 in which cse (2.4) is no needed) o obin ()Φ k ()Δ = (2.3) A Δ ()Φ k ()Δ = A()Φ k () = (2.7) A(σ()) ( A(σ()) Φ k) Δ ()Δ ( Φ k) Δ ()Δ A(σ())kΦ Δ ()Φ k 1 ()Δ

8 398 M. J. BOHNER AND S. H. SAKER = k ()ϕ()φ k 1 ()Δ = k 0 [ { (2.4) k 1 k ()ϕ() ][ k 1 k ] ()Φ k 1 () Δ [ ] { 1 k k [ k ()ϕ() Δ { { = k ()ϕ k k ()Δ k 1 k 1 ()Φ k k ()Δ. ] k } k 1 ()Φ k 1 () k 1 k Δ Dividing he enire inequliy by he righ-hnd fcor of he ls exression nd hen rising he resuling inequliy o he k-h owerconfirms he vlidiy of (3.2). REMARK 3.2. I is worh o menion here h our echnique of he roof of Theorem 3.1 is differen from he echnique due o Coson o rove he discree form of (3.2). In riculr, he followed he echnique due o Ellio [11] h hs been used o rove he Hrdy inequliy. THEOREM 3.3. Assume (3.1). Suose ϕ : T R is such h Φ() := (s)ϕ(s)δs, T is well defined. Le k 1 nd 0 c < 1. Then ( ) () k k A c (σ()) Φk ()Δ ()A k c (σ())ϕ k ()Δ. (3.3) 1 c Proof. Firs we define n uxiliry funcion à : T R by (s) Ã() := A c Δs, T. (σ(s)) Noe h he lef r of he inequliy (2.6) wih f = A nd γ = 1 c imlies σ() Ã(σ()) = ( (s) σ() A 1 c ) Δ (s) A c (σ(s)) Δs 1 c Δs = A1 c (σ()). (3.4) 1 c Now we use inegrion by rs (2.3), he lef r of he inequliy (2.7) wih f = Φ nd γ = k, (3.4), nd Hölder s inequliy (2.4) wih = k nd q = k/(k 1) (unless

9 SNEAK-OUT PRINCIPLE 399 k = 1 in which cse (2.4) is no needed) o obin () A c (σ()) Φk ()Δ = Ã Δ ()Φ k ()Δ (2.3) = Ã()Φ k () ( Ã(σ()) Φ k) Δ ()Δ ( = Ã(σ()) Φ k) Δ ()Δ (3.4) (2.7) = k 1 c = k 1 c (2.4) A 1 c (σ()) kφ Δ ()Φ k 1 ()Δ 1 c ()A 1 c (σ())ϕ()φ k 1 ()Δ [ ] [ 1 k ()A 1 c k (σ())ϕ() k { 1 c [ 1 k ()A 1 c k (σ())ϕ() ] k Δ k [ k 1 k () A c(k 1) k (σ()) Φ k 1 () k 1 k () A c(k 1) k (σ()) ] k k 1 Δ k 1 k ] Φ k 1 () Δ = k { { ()A k c (σ())ϕ k k } () 1 1 k ()Δ 1 c A c (σ()) Φk ()Δ. Dividing he enire inequliy by he righ-hnd fcor of he ls exression nd hen rising he resuling inequliy o he k-h owerconfirms he vlidiy of (3.3). 4. Dynmic snek-ou inequliies In his secion, we rove he min resuls of his er. We lso ly hese resuls o he secil ime scles T = R nd T = Z. The resuls for T = Z re known (see [3]) while he resuls for T = R re new. Throughou his secion, we le 1. For he cses α 1, 0 α 1, nd 1/ < α 0, we resen hree disinc inequliies. In ddiion o (3.1), for given vlueof α, we requirehe ssumions x : T (0, ) is rd-coninuous, y() := x(s)δs nd Ψ() := A α (σ(s))x(s)δs, T (4.1) re well defined. THEOREM 4.1. Le 1 nd α 1. Assume (3.1) nd (4.1). Then ()Ψ ()Δ (1 + α ) ()A α (σ())y ()Δ. (4.2)

10 400 M. J. BOHNER AND S. H. SAKER Proof. We use inegrion by rs (2.3) nd he righ r of he inequliy (2.7) wih f = A nd γ = α o obin Ψ() = A α (σ(s))y Δ (s)δs { (2.3) = A α (s)y(s) = A α ()y()+ (2.7) A α ()y()+ A α (σ())y()+α (A α ) Δ (s)y(s)δs (A α ) Δ (s)y(s)δs } αa Δ (s)a α 1 (σ(s))y(s)δs (s)a α 1 (σ(s))y(s)δs, where we hve uilized A A σ in he ls inequliy. Now we use Minkowski s inequliy (2.5) ndtheorem3.1 wih k = nd ϕ =(A α ) σ y o find he esime { ()Ψ ()Δ (2.5) = (3.2) { { { { [ () A α (σ())y()+α (s)a (σ(s))y(s)δs] α 1 Δ ()[A α (σ())y()] { Δ + () ()A α (σ())y { [ ()Δ +α () { ()A α (σ())y ()Δ + α { = (1 + α ) ()A α (σ())y ()Δ. [ α (s)a (σ(s))y(s)δs] α 1 Δ ] (s) A(σ(s)) Aα (σ(s))y(s)δs Δ ()A α (σ())y ()Δ Rising his inequliy o he -h ower confirms he vlidiy of (4.2). EXAMPLE 4.2. Le T = Z nd = 1. As secil cse of Theorem 4.1,wehve from (4.2) he inequliy ( ( (n) A α (k + 1)x(k)) (1 + α ) (n)a α (n + 1) x(n)), (4.3) n=1 k=n n=1 k=n where n 1 A(n)= (k), n N. k=1 Noe h (4.3) is he discree inequliy [3, Theorem 8] due o Benne nd Grosse- Erdmnn.

11 SNEAK-OUT PRINCIPLE 401 EXAMPLE 4.3. Le T = R. As secil cse of Theorem 4.1, we hve from (4.2) he inequliy ( ) ( () A α (s)x(s)ds d (1 + α ) ()A α () x(s)ds) d, (4.4) where A()= (s)ds, R. Noe h (4.4) is new inequliy. THEOREM 4.4. Le 1 nd 0 α 1. Assume (3.1) nd (4.1). Then ()Ψ ()Δ (1 + ) ()A α (σ())y ()Δ. (4.5) Proof. In his cse, we cnno use he righ r of he inequliy (2.7) wih f = A nd γ = α s we did in he roof of Theorem 4.1. However, we cn use inequliy (2.8) wih f = A nd γ = α nd follow he exc sme ses s in he roof of Theorem 4.1 o obin (4.5). REMARK 4.5. For T = Z, Theorem 4.4 reduces o he corresonding discree inequliy by Benne nd Grosse-Erdmnn [3, Theorem 9]. For T = R, (4.5) is new. THEOREM 4.6. Le 1 nd 1/ < α 0. Assume (3.1) nd (4.1). Then ( ) 1 + α ()Ψ ()Δ ()A α (σ())y ()Δ. (4.6) 1 + α + Proof. We use inegrion by rs (2.3) nd he inequliy (2.8) wih f = A nd γ = α o obin y() = A α (σ(s))ψ Δ (s)δs { (2.3) = A α (s)ψ(s) ( A α ) } Δ (s)ψ(s)δs = A α ( ()Ψ()+ A α ) Δ (s)ψ(s)δs (2.8) A α ()Ψ()+ A α (σ())ψ()+ A Δ (s)a α 1 (σ(s))ψ(s)δs (s)a α 1 (σ(s))ψ(s)δs, where we hve uilized A A σ in he ls inequliy. Now we use Minkowski s inequliy (2.5) ndtheorem3.3 wih k =, c = α, ndϕ = ( A α 1) σ Ψ o find

12 402 M. J. BOHNER AND S. H. SAKER he esime { ()A α (σ())y ()Δ (2.5) { [ ()A α (σ()) A α (σ())ψ()+ (s)a (σ(s))ψ(s)δs] α 1 Δ { ()A α (σ()) [ A α (σ())ψ() ] Δ { + [ ()A α (σ()) (s)a (σ(s))ψ(s)δs] α 1 Δ { [ + { = ()Ψ () ()Δ A α (σ()) { (3.3) {( ) ()Ψ ()Δ + ()Ψ ()Δ 1 + α ( ){ 1 + α + = ()Ψ ()Δ. 1 + α Rising his inequliy o he -h ower confirms he vlidiy of (4.6). (s)a (σ(s))ψ(s)δs] α 1 Δ EXAMPLE 4.7. Le T = Z nd = 1. As secil cse of Theorem 4.6,wehve from (4.6) he inequliy ( ( 1 + α ( (n) A α (k + 1)x(k)) (n)a n=1 k=n 1 + α + ) α (n + 1) x(n)), n=1 k=n (4.7) where n 1 A(n)= (k), n N. k=1 Noe h (4.7) is he discree inequliy [3, Theorem 10] due o Benne nd Grosse- Erdmnn. EXAMPLE 4.8. Le T = R. As secil cse of Theorem 4.6, we hve from (4.6) he inequliy ( ( ) 1 + α ( () A (s)x(s)ds) α d ()A α () x(s)ds) d, 1 + α + (4.8) where A()= (s)ds, R. Noe h (4.8) is new inequliy.

13 SNEAK-OUT PRINCIPLE 403 REFERENCES [1] R. P. AGARWAL, M. BOHNER, AND S. H. SAKER, Dynmic Lilewood-ye inequliies, Proc. Amer. Mh. Soc., 143 (2): , [2] M. R. S. AMMI AND D. F. M. TORRES, Hölder s nd Hrdy s wo dimensionl dimond-lh inequliies on ime scles, An. Univ. Criov Ser. M. Inform., 37 (1): 1 11, [3] G. BENNETT AND K.-G. GROSSE-ERDMANN, On series of osiive erms, Houson J. Mh., 31 (2): , [4] G. BENNETT AND K.-G. GROSSE-ERDMANN, Weighed Hrdy inequliies for decresing sequences nd funcions, Mh. Ann., 334 (3): , [5] M. BOHNER, R. A. FERREIRA, AND D. F. M. TORRES, Inegrl inequliies nd heir licions o he clculus of vriions on ime scles, Mh. Inequl. Al., 13 (3): , [6] M. BOHNER AND A. PETERSON, Dynmic Equions on Time Scles: An Inroducion wih Alicions, Birkhäuser, Boson, [7] M. BOHNER AND A. PETERSON, Advnces in Dynmic Equions on Time Scles, Birkhäuser, Boson, [8] M. BOHNER AND A. ZAFER, Lyunov ye inequliies for lnr liner Hmilonin sysems on ime scles, Al. Anl. Discree Mh., 7 (1): , [9] E. T. COPSON, Noe on series of osiive erms, J. London Mh. Soc., S1 2 (1): 9 12, [10] E. T. COPSON, Noe on series of osiive erms, J. London Mh. Soc., S1 3 (1): 49 51, [11] E. B. ELLIOTT, A simle exnsion of some recenly roved fcs s o convergency, J. London Mh. Soc., S1 1 (1): 93 96, [12] G. H. HARDY, J. E. LITTLEWOOD, AND G. PÓLYA, Inequliies, Cmbridge Universiy Press, Cmbridge, [13] U. M. ÖZKAN AND H. YILDIRIM, Hrdy-Kno-ye inequliies on ime scles, Dynm. Sysems Al., 17 (3 4): , [14] S. H. SAKER, Hrdy Leindler ye inequliies on ime scles, Al. Mh. Inf. Sci., 8 (6): , [15] S. H. SAKER AND J. GRAEF, A new clss of dynmic inequliies of Hrdy s ye on ime scles, Dynm. Sysems Al., 23 (1): 83 99, [16] S. H. SAKER, D. O REGAN, AND R. AGARWAL, Generlized Hrdy, Coson, Leindler nd Benne inequliies on ime scles, Mh. Nchr., 287 (5 6): , [17] A. TUNA AND S. KÜTÜKÇÜ, Some inegrl inequliies on ime scles, Al. Mh. Mech. (English Ed.), 29 (1): 23 29, (Received Ocober 21, 2014) Mrin J. Bohner Missouri Universiy of Science nd Technology Dermen of Mhemics nd Sisics Roll, Missouri , USA e-mil: bohner@ms.edu Smir H. Sker Mnsour Universiy, Dermen of Mhemics Fculy of Science Mnsour, Egy e-mil: shsker@mns.edu.eg

14 Journl of Mhemicl Inequliies