Weighted Hardy-Type Inequalities on Time Scales with Applications

Transcription

1 Medierr J Mh DOI 0007/s y c Sringer Bsel 204 Weighed Hrdy-Tye Ineuliies on Time Scles wih Alicions S H Sker, R R Mhmoud nd A Peerson Absrc In his er, we will rove some new dynmic Hrdy-ye ineuliies on ime scles wih wo differen weighed funcions The sudy is o deermine condiions on which he generlized ineuliies hold using some known hyohesis The min resuls will be roved by emloying Hölder s ineuliy, Minkowski s ineuliy nd chin rule on ime scles As secil cses of our resuls, when he ime scle is he rel numbers, we will derive some well-known resuls due o Coson, Bliss, Fle nd Benne by suible choice of he weighed funcions We will ly he resuls o invesige he oscillion nd nonoscillion of hlf-liner second order dynmic euion on ime scles Mhemics Subjec Clssificion 26A5, 26D0, 26D5, 39A3, 34A40, 34N05 Keywords Hrdy s ineuliy, Minkowski s ineuliy, ime scles, oscillion, hlf-liner dynmic euions Inroducion Hrdy [8] roved he discree ineuliy n ) i)) n), >, ) n n= i= where n) 0forn This ineuliy hs been discovered in his em o give n elemenry roof of Hilber s ineuliy for double series h ws known h ime Hrdy [9] roved he coninuous ineuliy, using he clculus of vriions, which ses h for f 0 nd inegrble over ny finie inervl 0,)ndf is inegrble nd convergen over 0, )nd>, hen ) f)d d f )d 2) 0 0 n= 0

2 n= S H Sker e l MJOM The consn / )) in ) nd 2) is he bes ossible So he nurl uesion now is: is i ossible o give unified roof for hese wo ineuliies on ime scles? In his er we will sudy his nd he sme ime we will generlize hese wo resuls wih weighed funcions on ime scles Since he discovery of hese wo ineuliies vrious ers which del wih new roofs, generlizions nd eensions hve ered in he lierure, we refer he reder o he ers [3 7,3,9,20,23,26,27], nd he books [22,24,25,29] In he following, we resen some of hese resuls h serve nd moive he conens of his er Hrdy s ineuliy ) hs been generlized lso by Hrdy himself in [9, Theorem B] In riculr, Hrdy roved h if >, λn) > 0, Λn) = n i= λi), nd An) = n i= i)λi), hen ) λn) Λ n) A n) λn) n) 3) Hrdy s ineuliy 2) hs been generlized by Hrdy himself in [20] There he showed, for ny inegrble funcion f) > 0on0, ) nd>, h ) m f)d d m m f )d, m > 0 0 Cosonin[3] generlized 3) nd roved h if >, i) 0, λi) 0, nd c>, hen ) λn) An)) Λ c λn)λ c n) n) 4) n) c n= n= Also he roved h if > nd 0 c<, hen ) λn) An)) Λ c λn)λ c n) n) 5) n) c n= Coson in [4, Theorems nd 3] showed h he coninuous counerrs of he ineuliies 4) nd 5) lso hold In riculr he roved h if, c>, Λ) = 0 λs)ds nd Φ) = λs)gs)ds, hen n= n= ) b λ) Λ)) c Φ)) λ) d c Λ)) c g )d, 0 nd if >, 0 c<, hen λ) Λ)) c Φ)) d ) c λ) Λ)) c g )d, where Φ) = λs)gs)ds Leindler [26] nd Benne [5] roved n ineresing vrin of Coson ineuliies 4) nd 5) Leindler roved h if

3 Weighed Hrdy s Ineuliy i=n λi) <, > nd 0 c<, hen ) λn) Λ n)) c A n)) λn)λ n)) c n), c n= where A n) = i=n i)λi) nd Λ n) = i=n λi) Benne [5] roved h if i=n λi) < nd <c, hen ) λn) Λ n)) c An)) λn)λ n)) c n), c n= n= where An) = n i= i)λi) During he ls decdes, hese ineuliies were eended o he following generlized form wih wo differen weigh funcions u) f)d d C f )v)d, 6) where <b,u,vre mesurble osiive funcions in, b) nd < < In [29] Oic e l roved h he ineuliy 6) holds if nd only if he following condiion holds r A M := su u)d v )d < <r<b r n= Moreover, he esime for he consn C in 6) is given by C + ) / ) + A M, where := Insired by he develomen in he coninuous cse i is nurl o sk bou he chrcerizion of he discree version of weighed Hrdy ineuliy 6) which is given by n= u n n k= k ) ) C v n n n= ), 7) where { n } n= is nonnegive seuence nd {u n} n=, {v n} n=, re osiive seuences, 0 << nd 0 << The firs conribuion regrding his issue is due o Andersen nd Heinig [4, Theorem 4] They roved h if < nd ) n ) su u k v k <, where := n N, k=n k= hen he ineuliy 7) holds Moreover, Heinig [23, Theorem 3] roved h if <<, r = nd H := n= k=n u k ) r n k= v k ) r v n r,

4 S H Sker e l MJOM hen he ineuliy 7) holds wih C / ) H, where := Benne resened full chrcerizion of he weighed ineuliy 7) in his ers [5 7] Since he coninuous nd discree ineuliies re imorn in he nlysis of uliive roeries of soluions of differenil nd difference euions, we lso believe h he Hrdy dynmic ineuliies on ime scles will ly he sme effecive role in he nlysis of uliive roeries of soluions of dynmic euions To he bes of he uhors knowledge here re few ineuliies in he lierure [2, 28, 3, 32, 35 44] which sudied dynmic ineuliies of Hrdy s ye on ime scles The generl ide is o rove resul for dynmic ineuliy where he domin of he unknown funcion is so-clled ime scle T, which is nonemy closed subse of he rel numbers R, o void roving resuls wice, once in he coninuous cse which leds o differenil ineuliy nd once gin on discree ineger cse which leds o difference ineuliy In he following, we recll some of hese resuls h moive he conens of his er In [32], Řehák lied echniue used by Ellio [6] nd roved ime scle version of he Hrdy ineuliy 2) In riculr, i ws roved h if > ndg is nonnegive rd-coninuous funcion nd he del inegrl g)) Δ eiss s finie number, hen ) g)δ Δ g )Δ 8) If in ddiion μ)/ 0s, hen he consn is he bes ossible In he roof of he ineuliy 8) he uhor ssumed h ϕ Δ ) > 0 where ϕ) = gs)δs/ ) In [38] he uhors roved h if >, γ>nd here eiss consn K>0, wih σ), for 0 K hen ) K γ γ g)δ Δ γ γ g )Δ, 9) where g is nonnegive rd-coninuous funcion nd he inegrl γ g ) Δ converges In [42] he uhors imroved he ineuliy 9) nd roved h if, γ>, hen ) σ) ) γ ) σ) ) γ g)δ Δ g )Δ γ ) γ ) 0)

5 Weighed Hrdy s Ineuliy In [38] he uhors roved h if >ndγ<, hen ) σ γ gs)δs g ) Δ ) γ σ γ Δ, ) ) where g is nonnegive rd-coninuous funcion such h he del inegrl σ)) γ g )Δ eiss In [40] he uhors roved h if k c>, hen where λ) Λ σ )) c Φ σ )) k Δ Λ) := k c λs)δs, ndφ) := ) k Λ σ )) k )c Λ)) kc ) λ)gk )Δ, 2) λs)gs)δs, for ny [, ) T Also in [40] hey roved h if 0 c<ndk>, hen ) k λ) Ω)) c Ψσ )) k k λ) Δ c Ω)) c k gk )Δ, where Ω) := λs)δs, nd Ψ) := λs)gs)δs, for ny [, ) T Following hese rends nd o develo he sudy of dynmic ineuliies of Hrdy s ye on ime scles, we will rove he ime scle version of 6) nd is comnion on n rbirry ime scle T This er is orgnized s follows: in Sec 2, we resen some reliminries bou he heory of ime scles nd rove he bsic lemms h we will need in he roofs In Sec 3, we will rove he min resuls In Sec 4, we will give n licion of he weighed Hrdy s ineuliy obined in Sec 3 o emine he oscillory roeries of hlf-liner second order dynmic euion on ime scle T 2 Preliminries nd Bsic Lemms A ime scle T is n rbirry nonemy closed subse of he rel numbers R We ssume hroughou h T hs he oology h i inheris from he sndrd oology on he rel numbers R The forwrd nd bckwrd jum oerors re defined by: σ) :=inf{s T : s>} nd ρ) :=su{s T : s<} resecively, where su =inft A oin T, is sid o be lef-dense if ρ) = nd > inf T, is righ-dense if σ) =, is lef-scered if ρ) <nd righ-scered if σ) >A funcion g : T R is sid o be righ-dense coninuous rd-coninuous) rovided g is coninuous righdense oins nd lef-dense oins in T, lef hnd limis eis nd re finie The se of ll such rd-coninuous funcions is denoed by C rd T) If f is iecewise rd-coninuously differenible, we wrie f C T)

6 S H Sker e l MJOM The grininess funcion μ for ime scle T is defined by μ) :=σ) 0, nd for ny funcion f : T R he noion f σ ) denoes fσ)) We ssume h su T =, nd define he ime scle inervl [, b] T by [, b] T := [, b] T The hree mos oulr emles of clculus on ime scles re differenil clculus, difference clculus, nd unum clculus, ie, when T = R, T = N nd T = N0 = { : N 0 } where > For more deils of ime scle nlysis we refer he reder o he wo books by Bohner nd Peerson [, 2] which summrize nd orgnize much of he ime scle clculus In his er, we will refer o he del) inegrl which we cn define s follows If G Δ ) =g), hen he Cuchy del) inegrl of gs)δs := G) G) I cn be shown see []) h if g C rd T), hen he Cuchy inegrl G) := 0 gs)δs eiss, 0 T, nd sisfies G Δ ) =g), T An imroer inegrl is defined by f)δ = g is defined by lim b f)δ A simle conseuence of Keller s chin rule [, Theorem 90] is given by γ )) Δ = γ 0 [h σ )+ h))] γ dh Δ ), nd he inegrion by rs formul on ime scles is given by b u)v Δ )Δ =[u)v)] b u Δ )v σ )Δ The Hölder ineuliy, see [, Theorem 63], on ime scles is given by γ ν f)g) Δ f) γ Δ g) ν Δ, 2) where, b T nd f, g C rd I, R), γ>nd γ + ν = Throughou he er, we will ssume h he funcions in he semens of he heorems re nonnegive nd rd-coninuous funcions nd he inegrls re ssumed o eis Now, we rove he bsic lemms h will be needed in he roofs of he min resuls The roofs deend on he licion of he ime scles chin rule see [, Theorem 87]) given by g δ) Δ ) =g δ d)) δ Δ ), where d [, σ )], 22) where i is ssumed h g : R R is coninuously differenible nd δ : T R is del differenible Lemm 2 Le T be ime scle wih, T such h If0 <, hen σ) f)δ f) fs)δs Δ, 23)

7 Weighed Hrdy s Ineuliy Proof Le F ) := fs)δs Alying he chin rule 22), we see h F )) Δ = F d)f Δ ) = F f), d) 24) where d [, σ )] Since F Δ ) =f) 0, nd σ ) d, we see h F σ ) F d), nd so F d) f) F σ f), )) since 0 < 25) Subsiuing 25) ino24), we ge h F )) Δ F σ )) f) 26) Inegring boh sides of 26) from o, we ge h 23) holds For he cse, he ineuliies in 25) nd 26) will be reversed nd hen we ge he following resul Lemm 22 Le T be ime scle wih, T such h If, hen σ) f)δ f) fs)δs Δ 27) Remrk 2 If we choose T = N in Lemms 2 nd 22, we ge he following discree ineuliies h ered in [8, 5, 34], n ) n k k, for 0 <, nd k= k= k= k n ) n k k= k j j= k j j=, for The following vrin of he ineuliies 23) nd 27) will be useful Lemm 23 Le T be ime scle wih, b T such h b If0 <, hen f)δ f) fs)δs Δ 28) Proof Le F ) := fs)δs, T, b

8 S H Sker e l MJOM Alying he chin rule 22), we see h F )) Δ = F d) F Δ ) ) = F f)), d) 29) where d [, σ )] Since, F Δ ) = f) 0, nd d, we ge h F d) f) F f) ) since 0 < 20) Subsiuing 20) ino29), we ge h F )) Δ F )f) 2) Inegring boh sides of 2) from o b, we ge 28) holds For he cse, he ineuliies in 20) nd 2) will be reversed nd hen we ge he following ineuliy Lemm 24 Le T be ime scle wih, b T such h b If, hen f)δ f) fs)δs Δ, Remrk 22 If we choose T = N in Lemms 23 nd 24, we ge he following discree ineuliies h ered in [8], N ) N N k, for 0 <, nd k=n k=n k=n k N ) N k k=n k j j=k N j j=k, for We ne se he following Minkowski inegrl ineuliy on ime scles due o Bibi e l [9, Theorem 2] Lemm 25 Le X, M, μ Δ ) nd Y,L,ν Δ ) be ime scles mesure sces nd le u, v nd f be nonnegive rd-coninuous funcions on X, Y nd X Y resecively If, hen f, y)vy)dν Δ y) u)dμ Δ ) X Y f, y)u)dμ Δ ) holds rovided ll inegrls in 22) eiss Y X vy)dν Δ y), 22)

9 Weighed Hrdy s Ineuliy In he roof of he ne lemm, which is he secil form of he Minkowski inegrl ineuliy 22), we will use he chrcerisic funcion over he inervl [s, b] T which is defined by χ [s,b]t χ [s,b]t ) := {, if [s, b]t, 0, if / [s, b] T 23) Lemm 26 Le T be ime scle wih, b T nd le f nd g be nonnegive rd coninuous funcions on [, b] T Ifm, hen m m σ) m f) gs)δs Δ gs) f)δ Δs 24) Proof The ide of he roof is bsed rimrily on he coninuous counerr roved in [27, Theorem 2] Now he lef-hnd side of 24) cn be wrien in he form m m σ) f) gs)δs Δ = f)) m gs)χ [s,b]t )Δs s m Δ m, 25) where χ [s,b]t is he chrcerisic funcion defined in 23) Alying he Minkowski ineuliy 22) wih f, y), dν Δ =Δs nd dμ Δ =Δ o he righ-hnd side of 25), we hve m m σ) m f) gs)δs Δ f)gs)) m χ [s,b]t )Δ Δs nd hence 24) holds = = s gs) f)gs)) m Δ s f)δ m m Δs, Δs 3 Min Resuls In his secion, we will rove our min resuls using Hölder s ineuliy, Minkowski s ineuliy, nd mny of he resuls in Sec 2 For convenience,

10 S H Sker e l MJOM we use he noions B := u)δ su <<b v )Δ, 3) nd k, ) := + ) / ) / + 32) Theorem 3 Le T be ime scle wih, b T, < <, f C rd [, b] T, R) is nonnegive funcion nd le u, v be osiive rd-coninuous funcions on, b) T Then he ineuliy u) f)δ Δ C f )v)δ, 33) holds, if nd only if B< 34) Moreover, for he consn C in 33) he following esime is sisfied B C k, )B, 35) where k, ) is defined s in 32) Proof i) Sufficiency of condiion 34) Assume firs h B < This imlies h σ) for [, b] T This imlies h 0 <h) := v y)δy <, σ) v s,, b] T, 36) where s> is fied number Alying Hölder s ineuliy 2), we hve h f)δ = f)v / )h)v / )h )Δ f )v)h )Δ v )h )Δ 37)

11 Weighed Hrdy s Ineuliy Using he definiion of h) which is given in 36), we see h 38) h )v )Δ = σ) v y)δy /s v )Δ Alying he ineuliy 23) o h righ-hnd side of 38) wih f) = v, we obin σ) v y)δy /s v )Δ s s v y)δy = s s hs ) ) 39) Subsiuing 39) ino37), we see h ) s f)δ f )v)h )Δ h s ) ), s nd conseuenly, we obin u) f)δ Δ s ) s f )v)h )Δ s s h s ) )u)δ 30) Using he Minkowski inegrl ineuliy 24) on he righ hnd side of 30), we ge h u) f)δ Δ s ) s b f )v)h ) h s ) )u)δ Δ 3)

12 S H Sker e l MJOM Bu h s ) ) =[h s )] s )/s B s )/s s )/s u)δ This llows us o wrie, h s ) )u)δ B s )/s u)δ + s u)δ 32) Alying ineuliy 28) o h righ-hnd side of 32) wih f = u nd =/s since 0 < /s < ), we ge h s u)δ u)δ s Subsiuing he ls ineuliy ino 32), we obin h s ) )u)δ sb s )/s u)δ s u)δ From his nd he definiion of B [see 3)], we obin h s h s ) )u)δ s B s )/s s u)δ 33) Reclling 3), we ge h u)δ B σ) v )Δ / 34) Subsiuing 34) ino33), we hve h s B s )/s B σ) v )Δ / s = s B h ) 35)

13 Weighed Hrdy s Ineuliy From 3) nd 35), we ge h u) f)δ Δ s ) s = Ns)B s B f )v)δ f )v)h)h )Δ, 36) where ) Ns) :=s s s Since s>, we ge h inf Ns) =k, ) 37) s> Then, we ge from 36) nd 37) h u) f)δ Δ k, )B f )v)δ, which is he desired ineuliy ii) Necessiy of condiion 34) Suose h he ineuliy 33) holds for ll nonnegive rd-coninuous funcions f :, b) T R :ndc< Le ξ [, b] T be fied, hen u) f)δ Δ u) f)δ Δ ξ ξ ξ u)δ σξ) This ineuliy ogeher wih 33), gives h σξ) u)δ f)δ C f )v)δ Assume h f)δ 38) v )Δ < for every [, b] T, 39)

14 S H Sker e l MJOM nd se We hve nd moreover 0 < f) = σξ) { v ), for, ξ) T, 0, for [ξ,b) T f)δ = f )v)δ σξ) = v )Δ, σξ) v )Δ < The ls inegrl is finie due o 39) nd osiive Conseuenly, from 38), we hve σξ) u)δ v )Δ C, ξ which is he desired ineuliy 35) The roof is comlee As in he roof of Theorem 3, we cn esily rove he following dul heorem by using Lemms 23 nd 24 insed of Lemms 2 nd 22 Theorem 32 Le T be ime scle wih, b T, < <, f C rd [, b] T, R) is nonnegive funcion nd le u, v be osiive rd-coninuous funcions on, b) T Then he ineuliy u) f)δ Δ C f )v)δ, 320) holds, if nd only if B := su <<b u)δ v )Δ < Moreover, for he consn C in 320) he following esime is sisfied B C k, )B, where k, ) is defined s in 32) Remrk 3 In Theorem 3 if we ke T =R, we ge he following weighed Hrdy-ye ineuliy u) f)d d C f )v)d, 32)

15 Weighed Hrdy s Ineuliy nd if we ke T = N, we ge he discree nlogue of ineuliy 32) n ) ) ) u n k C v n n n= k= By mking suible subsiuions for he wo weigh funcions u) nd v), we could obin conseuences of he dynmic Hrdy-ye ineuliies For emle if =, u) = σ) ) nd v) =, hen Theorem 3 reduces o he following conseuence of he Hrdy-ye ineuliy due o Řehák [32] g)δ Δ R g )Δ, n= where R sisfies he following esime ) R ) su Δ σ ) ) σ) <<b If we choose u) =/ γ nd v) =/ γ, we ge he following, which is conseuence of 9), Hrdy s ineuliy γ g)δ Δ S γ g )Δ, due o Sker nd O Regn [38], where S sisfies he following esime ) S ) su γ Δ γ Δ If we choose <<b )γ ) u) = ) γ, nd v) =, ) γ ) we ge he following conseuence of 0) ) γ g)δ Δ S 2 ) γ ) ) γ ) g )Δ, due o Sker e l [42], where S 2 sisfies he following esime ) S 2 ) su σ) ) γ Δ σ) ) γ ) ) γ ) Δ <<b

16 S H Sker e l MJOM If we choose u) = σ γ ) nd v) = σ γ ), we ge he following conseuence of ) σ γ ) gs)δs Δ S 3 g ) σ γ ) Δ, due o Sker nd O Regn [38], where S 3 sisfies he following esime ) S 3 ) su σ γ ) Δ σ γ Δ ) <<b As secil cse of Theorem 3, if we ke λ) u = Λ σ )),v= c) λ)λσ )) c nd f = λs)gs), we ge he following resul Corollry 3 Le T be ime scle wih, b T nd < < If c<, Λ) = λs)δs,nd hen Φ) = λs)gs)δs, λ)λ σ )) c) Φ σ )) Δ K λ)λ σ )) c g )Δ where K is osiive consn deends on c, nd nd sisfies he following esime K k, ) su <<b λ)λ σ )) c) Δ λ)λ σ )) c) Δ Remrk 32 If we ke T = R, hen Λ σ ) =Λ), nd Corollry 3 gives us he following coninuous ineuliy of Benne Coson ye λ)λ)) c) Φ )d K 2 λ)λ)) c g )d,

17 Weighed Hrdy s Ineuliy where K 2 sisfies he following esime K 2 k, ) su <<b λ)λ)) c) d λ)λ)) c) d If T = N, hen Corollry 3 reduces o he following conseuence of Benne s resul [5, Corollry 7] N n= k λ n Λ c) n n= where K 3 sisfies he following esime K 3 k, ) su <<b N n=k λ k k ) K 3 N λ n Λ c) n n= ) k n= λ n Λ c n n ) ) λn Λ c ) n If = = k, hen Corollry 3 reduces o following conseuence reled o he ineuliy 2) due o Sker e l [40] Corollry 32 Le T be ime scle wih, b T nd c 0 <k< Le Λ) = λs)δs, nd Then Φ) = λ) b Λ σ )) c Φ k )Δ S 4 where S 4 sisfies he following esime S 4 k k ) k su λs)gs)δs <<b λ)λ σ )) k c g k )Δ, λ)λ)) c Δ λ)λ)) k c) k Δ k, As secil cse of Theorem 3 when u = λ obin he following resul nd v = λ,we

18 S H Sker e l MJOM Corollry 33 Le T be ime scle wih, b T, λ nd < < If F ) = fs)δs, hen ) F ) λ Δ K 4 λ f )Δ, where K 4 sisfies he esime K 4 su <<b λ Δ λ ) Δ Remrk 33 Corollry 33 cn be considered s he ime scle version of Hrdy-ye ineuliy due o Fle [7] The secil cse of Fle s ineuliy wih λ = is due o Bliss [0] nd Hrdy nd Lilewood [2] 4 Alicions In his secion, we will ly he resuls of Sec 3 o invesige he oscillion nd nonoscillion of hlf-liner second order dynmic euion on ime scles More recisely, we will emloy he weighed Hrdy s ineuliy 33) o he following hlf-liner euion in order o emine is oscillory roeries, r)ϕα Δ )) Δ + s ) ϕα σ )=0, 4) where <α<, ϕ α y) = y α 2 y nd r, s C rd [, b], R) wih r) 0 for nd [, b] T When α =2E4) becomes he liner Surm Liouville euion r) Δ ) ) Δ + s ) σ ) =0 When α 2E4) is clled hlf-liner becuse he se of is soluions hs he roery of homogeneiy bu no ddiiviy By soluion of 4) onninervli, we men nonrivil rel-vlued funcion C rd I, R), which hs he roery h r ) Δ ) α 2 Δ ) Crd I, R) nd sisfies 4) oni We sy h soluion of 4) hs generlized zero if ) = 0 nd hs generlized zero in, σ)) in cse ) σ ) < 0ndμ) > 0 To invesige he oscillion roeries of 4) i is roer o use he noions such s conjugcy nd disconjugcy of he euion 4) Euion 4) is disconjuge on he inervl [ 0,b] T,if here is no nonrivil soluion of 4) wih wo or more) generlized zeros in [ 0,b] T Euion 4) is sid o be nonoscillory on [ 0, ] T if here eiss c [ 0, ] T such h his euion is disconjuge on [c, d] T for every d>c In he oosie cse 4) is sid o be oscillory on [ 0, ] T A soluion ) of4) is sid o be oscillory if i is neiher evenully osiive nor evenully negive, oherwise i is oscillory We sy h 4) is righ disfocl lef disfocl) on [, b] T if he soluions of 4) such h Δ ) =0 Δ b) = 0) hve no generlized zeros in [, b] T

19 Weighed Hrdy s Ineuliy Ne, we will use he vriionl rincile esblished in [33] o emine oscillory roeries of 4) using he weighed Hrdy-ye ineuliy 33) To do h, we should recll he fundmenl resul in he uliive heory of hlf-liner dynmic euions of form 4) is he so-clled Roundbou Theorem on ime scles Consider E 4), where r, C rd I, R) wih r) 0 Along wih 4) consider he generlized Ricci dynmic euion R[w] :=w Δ + s)+s[w, r]) =0, 42) where ) w r S[w, r] = lim λ μ λ ϕ ϕ r)+λϕ w)) Theorem 4 [, 33] The following semens re euivlen: i) Euion 4) is disconjuge on I ii) Euion 4) hs soluion such h r σ ) > 0 for I k ie, soluion hving no generlized zeros on I) iii) Euion 42) hs soluion w wih { ϕ r)+μϕ w) } ) > 0, for I k iv) The funcionl F g,, b) = r) g Δ ) α s ) g σ ) α) Δ, is osiive definie on he so-clled dmissible funcions U defined by U, b) = { g C r I, R) :g) =gb) =0 } Roundbou Theorem gives wo mehods of invesigion of E 4) The firs one is bsed on he euivlence of i) nd iii) nd is clled Ricci echniue The second one is bsed on he euivlence of i) nd iv) nd cn be invesiged by he vriionl mehod Our im is o develo he vriionl mehod of sudying euion 4) by lying he weighed Hrdy-ye ineuliy 33) We would like o menion he ioneering work of Oelbev [30] on licion of Muckenhou s resuls on Hrdy ineuliies o he oscillion heory of he Surm Liouville euion From he euivlence of i) nd iv) he following semen cn be deduced [33] Lemm 4 Euion 4) is nonoscillory if nd only if here eiss, b T such h F g) = r) g Δ ) α s ) g σ ) α) Δ>0, 43) for every nonrivil g U) he clss of he so-clled dmissible funcions), where U) := { g C r I, R) : wih b> wih g) =0 if /, b) }

20 S H Sker e l MJOM Noe h he ineuliy 43) is euivlen o he Hrdy ineuliy 33) for he cse = = α, his α s)g σ )) α Δ <C r) g Δ ) ) α α Δ, wih C = Unforunely, in he heory of weighed Hrdy ineuliies he ec vlues of he consn C were found only in some riculr cses However, in mny cses wo-sided esimes for he consn C were obined Alying Theorem 3 ogeher wih Lemm 4 gives us he following resul Theorem 42 If B α < α) α α ) α, hen euion 4) is nonoscillory, where B α := su <<b s)δ α α r α )Δ, α = α α References [] Agrwl, RP, Bohner, M, Řehák, P: Hlf-liner dynmic euions Nonliner Anlysis nd Alicions: To V Lkshmiknhm on his 80h Birhdy, vol Kluwer Acdemic Publishers, Dordrech 2003) [2] Agrwl, RP, Bohner, M, Sker, SH: Dynmic Lilewood-ye ineuliies Proc Am Mh Soc 432), ) [3] Anderson, K, Muckenhou, B: Weighed wek ye Hrdy ineuliies wih licions o Hilber rnsforms nd miml funcions Sud Mh 72, ) [4] Andersen, KF, Heinig, HP: Weighed norm ineuliies for cerin inegrl oerors SIAM J Mh 4, ) [5] Benne, G: Some elemenry ineuliies Q J Mh Of 382), ) [6] Benne, G: Some elemenry ineuliies II Q J Mh 39, ) [7] Benne, G: Some elemenry ineuliies III Q J Mh Of Ser 422), ) [8] Benne, G, Grosse-Erdmnn, KG: On series of osiive erms Hous J Mh 3, ) [9] Bibi, R, Bohner, M, Pečrić, J, Vrosnec, S: Minkowski nd Beckenbch- Dresher ineuliies nd funcionl on ime scles J Mh Ineul 3, ) [0] Bliss, GA: An inegrl ineuliy J Lond Mh Soc 5, ) [] Bohner, M, Peerson, A: Dynmic Euions on Time Scles: An Inroducion wih Alicions Birkhäuser, Boson 200) [2] Bohner, M, Peerson, A: Advnces in Dynmic Euions on Time Scles Birkhäuser, Boson 2003)

21 Weighed Hrdy s Ineuliy [3]Coson,ET:NoeonseriesofosiiveermsJLondMhSoc3, ) [4] Coson, ET: Some inegrl ineuliies Proc R Soc Edinb Sec A 75, ) [5] Dvies, GS, Peersen, GM: On n ineuliy of Hrdy s II) Q J Mh Of Ser 2 5), ) [6] Ellio, EB: A simle ension of some recenly roved fcs s o convergency J Lond Mh Soc, ) [7] Fle, TM: A noe on some ineuliies Proc Glsgow Mh Assoc 4, ) [8] Hrdy, GH: Noes on heorem of Hilber Mh Z 6, ) [9] Hrdy, GH: Noes on some oins in he inegrl clculus LX) An ineuliy beween inegrls Messenger Mh 54, ) [20] Hrdy, GH: Noes on some oins in he inegrl clculus LXIV) Messenger Mh 57, ) [2] Hrdy, GH, Lilewood, JE: Noes on he heory of series XII): on cerin ineuliies conneced wih he clculus of vriions J Lond Mh Soc 5, ) [22] Hrdy, GH, Lilewood, JE, Poly, G: Ineuliies, 2nd edn Cmbridge Universiy Press, Cmbridge 952) [23] Heinig, HP: Weighed norm ineuliies for cerin inegrl oerors II Proc Am Mh Soc 95, ) [24] Kufner, A, Persson, LE: Weighed Ineuliies of Hrdy Tye World Scienific Publishing Co, Singore, New Jersy, London, Hong Kong 2003) [25] Kufner, A, Mligrnd, L, Persson, LE: The Hrdy Ineuliies: Abou is Hisory nd Some Reled Resuls Vydvelski Servis Publishing House, Pilsen 2007) [26] Leindler, L: Generlizion of ineuliies of Hrdy nd Lilewood Ac Sci Mh Szeged) 3, ) [27] Mz j, VG: Sobolev Sces Sringer Series in Sovie Mhemics, Sringer- Verlg, Berlin 985) [28] Ogunuse, JA, Persson, L-E: Time scles Hrdy-ye ineuliies vi suerudrciy Ann Func Anl 52), ) [29] Oic, B, Kufner, A: Hrdy-Tye Ineuliies, Pimn Reserch Noes in Mhemics Series Longmn Scienific nd Technicl, Hrlow 990) [30] Oelbev, M: Esimes of he secrum of he Surm-Liouville oeror Gylym, Alm-A 990) in Russin) [3] Ozkn, UM, Yildirim, H: Hrdy-Kno-ye ineuliies on ime scles Dyn Sys Al 7, ) [32] Řehák, P: Hrdy ineuliy on ime scles nd is licion o hlf-liner dynmic euions J Ineul Al 5, ) [33] Řehák, P: Hlf-liner dynmic euions on ime scles: IVP nd oscillory roeries Nonliner Func Anl Al 7, ) [34] Rudin, W: Problem E338 [989, 254] Am Mh Mon 97, ) [35] Sker, SH: Hrdy Leindler ye ineuliies on ime scles Al Mh Inf Sci 86), )

22 S H Sker e l MJOM [36] Sker, SH, O Regn, D: Eensions of dynmic ineuliies of Hrdy s ye on ime scles Mh Slovc cceed) [37] Sker, SH, Gref, J: A new clss of dynmic ineuliies of Hrdy s ye on ime scles Dyn Sys Al 23, ) [38] Sker, SH, O Regn, D: Hrdy nd Lilewood ineuliies on ime scles Bull Mlys Mh Sci Soc cceed) [39] Sker, SH, O Regn, D, Agrwl, RP: Some dynmic ineuliies of Hrdy s ye on ime scles Mh Ineul Al 7, ) [40] Sker, SH, O Regn, D, Agrwl, RP: Generlized Hrdy, Coson, Leindler nd Benne ineuliies on ime scles Mh Nchr ), ) [4] Sker, SH, O Regn, D, Agrwl, RP: Dynmic ineuliies of Hrdy nd Coson yes on ime scles Anlysis 34, ) [42] Sker, SH, O Regn, D, Agrwl, RP: Lilewood nd Benne ineuliies on ime scles, Medierr J Mh 5 204) [43] Sidi, MR, TorresA, FM: Hölder s nd Hrdy s wo dimensionl dimondlh ineuliies on ime scles Ann Univ Criv, Mh Com Ser 37, 200) [44] Tun, A, Kuukcu, S: Some inegrls ineuliies on ime scles Al Mh Mech Engl Ed 29, ) S H Sker Dermen of Mhemics, Fculy of Science Mnsour Universiy Mnsour, Egy e-mil: shsker@mnsedueg R R Mhmoud Dermen of Mhemics, Fculy of Science Fyoum Universiy Fyoum, Egy e-mil: rrm00@fyoumedueg A Peerson Dermen of Mhemics Universiy of Nebrsk Lincoln Lincoln, NE , USA e-mil: eerson@mhunledu Received: Seember, 204 Revised: December 2, 204 Acceed: December 6, 204