Appl Mth Inf Sci No 3-5 (26) Applied Mthemtics & Informtion Sciences An Interntionl Journl http://dxdoiorg/2785/mis/bos7-mis A Generl Dynmic Inequlity of Opil Type Rvi Agrwl Mrtin Bohner 2 Donl O Regn 3 Mhmoud Osmn 4 nd Smir Ser 4 Texs A&M University Kingsville Deprtment of Mthemtics Kingsville TX 78363 USA 2 Missouri S&T Deprtment of Mthemtics nd Sttistics Roll MO 6549-2 USA 3 Ntionl University of Irelnd School of Mthemtics Sttistics nd Applied Mthemtics Glwy Irelnd 4 Mnsour University Deprtment of Mthemtics Fculty of Science Mnsour Egypt Received: 28 Jun 25 Revised: 5 Sep 25 Accepted: 3 Oct 25 Published online: My 26 Abstrct: We present new generl dynmic inequlity of Opil type This inequlity is new even in both the continuous nd discrete cses The inequlity is proved by ming use of recently introduced new technique for Opil dynmic inequlities the time scles integrtion by prts formul the time scles chin rule nd clssicl s well s time scles versions of Hölder s inequlity Keywords: Opil s inequlity Hölder s inequlity time scles Introduction In 96 Olech [8] extended n inequlity of Opil [9] nd proved tht if f C ([h]r) h> stisfies f()= then f f dt h 2 f 2 dt () This inequlity creted lot of reserch ctivity which ws summrized in the monogrph [2] both for the continuous nd the discrete cses In [3] (see lso [6 Theorem 623]) the uthors extended () to n rbitrry time scle T nd proved tht if f C rd ([h] TR) h> stisfies f()= then ( f 2) h ( 2 h f ) (2) For extensions nd generliztions of (2) we refer the reder to the monogrph [] Over the lst sixty yers the study of Opil inequlities (continuous nd discrete) or relted Hrdy opertors focused on the investigtions of new inequlities or opertors weighted functions These inequlities hve nturl pplictions in pplied mthemtics especilly in the theory of differentil equtions in elsticity (ordinry or prtil) nd led to mny interesting questions nd connections between different res of mthemticl nlysis For exmple Hrdy opertors re closely relted to qusidditivity properties of cpcities nd were recently used Opil-type inequlities to find the gps between zeros of differentil equtions tht pper in the binding of bems [] Here we will not give n introduction to time scles clculus but insted refer the reder to [6 7] We only remr tht the delt derivtive is the usul derivtive if T=R nd the forwrd difference if T=Z nd the delt integrl is the usul integrl if T = R nd sum if T = Z nd tht the theory cn be pplied to ny nonempty closed set T R the so-clled underlying time scle We note tht pluggingt=r in (2) results in () Using novel technique in [4] the following generliztion of (2) ws estblished involving two different weight functions s nd r see [4 Theorem 52] Theorem Assume tht T b ( ) T rs C rd ([b] T ( )) nd If f()= then K = s ( f 2 ) K f C rd ([b] TR) ( 2 r f ) s 2 (R 2 ) R= r(τ) Corresponding uthor e-mil: shser@mnsedueg c 26 NSP Nturl Sciences Publishing Cor
2 R Agrwl et l : A generl dynmic inequlity We note tht plugging = b = h nd r = s= in Theorem results in (2) Refining the technique from [4] the sme uthors proved in [5] the following generliztion of Theorem Theorem 2 Assume tht T b ( ) T rs C rd ([b] T ( )) nd Let α > nd β If f()= then f C rd ([b] TR) s ( f α ) ( f ) β b K r f α+β K = α(β + ) α+β α+ β R= (s) α (R α+β α+β ) β(α+β) (r) (α )(α+β ) (r(τ)) α+β α α+β We note tht plugging α = 2 nd β = in Theorem 2 results in Theorem The purpose of this pper is to pply the new technique tht ws developed in [45] in order to prove the following generliztion of Theorem 2 Theorem 3 Assume tht T b ( ) T rs C rd ([b] T ( )) nd f C rd ([b] TR) Let α β nd >β + If f()= then nd s ( f α ) ( f ) β K = c (s) { K r f β ( ) R α α β β (r) ( ) β β c=α α α β R= β ( )( β ) ( ) β + α+ β } α+β β We note tht plugging = α + β in Theorem 3 results in Theorem 2 The pper is orgnized s follows: In Section 2 we present the bsic definitions of time scles clculus tht will be used in the sequel In Section 3 we prove Theorem 3 nd give some remrs We prove our min result by using the time scles chin rule the time scles integrtion by prts formul nd clssicl continuous nd discrete s well s time scles versions of Hölder s inequlity 2 Time Scles Preliminries In this section we briefly present some bsic definitions nd results concerning the delt clculus on time scles tht we will use in this rticle A time scletis n rbitrry nonempty closed subset of the rel numbers We define the forwrd jump opertor σ : T T by σ := inf{s T : s > t} for t T For ny function f : T R we put f σ = f σ A function f :T R is clled rd-continuous denoted by f C rd if it is continuous t ech right-dense point (ie σ=t) nd there exists finite left-sided limit t ll left-dense points (ie ρ = t the bcwrd jump ρ is defined in similr wy s the forwrd jump σ) For the definition of the delt derivtive nd the delt integrl we refer to [67] If f C (RR) nd g :T R is delt differentible then the time scles chin rule see [6 Theorem 9] sttes tht ( f g) = g f (hg σ +( h)g ) nd specil cse which we will use in this pper is given by ( f γ ) = γ f (h f σ +( h) f) γ for γ R The time scles Hölder inequlity see [6 Theorem 63] sys { } fg f γ γ { b g ν ν } b T fg C rd ([b] T R) γ > nd ν = γ/(γ ) 3 Proof of the Opil Inequlity In this section we present the proof of our min result Theorem 3 nd give some corollries nd concluding remrs Proof Define g := r(τ) f (τ) c 26 NSP Nturl Sciences Publishing Cor
Appl Mth Inf Sci No 3-5 (26) / wwwnturlspublishingcom/journlssp 3 Then g()= nd f = g = r f { so tht f (τ) f (τ) } = (R) (g) f ( g ) = r { r(τ) f (τ) we hve used the time scles Hölder inequlity conjugte exponents nd > Thus for h [] we obtin h f σ +( h) f h f σ +( h) f h(r σ ) = (hr σ ) (g σ ) +( h)r g (hg σ ) +(( h)r) (( h)g) (hr σ +( h)r) (hg σ +( h)g) we hve used the clssicl Hölder inequlity for sums conjugte exponents nd > Hence (h f σ +( h) f) α h f σ +( h) f α (hr σ +( h)r) ( )(α ) { (hr σ +( h)r) ( )(α ) β { (hg σ +( h)g) α } (hg σ +( h)g) α } β } we hve used the clssicl Hölder inequlity for integrls conjugte exponents β nd > Therefore using the time scles chin rule three times we get ( f α ) ( f ) β =α f (h f σ +( h) f) α = α(g ) r (h f σ +( h) f) α α(g ) r { (hr σ +( h)r) ( )(α ) β } β { } (hg σ +( h)g) α = c r β { α α β β R } β (hr σ +( h)r) α α β β { α+ β β + g (hg σ +( h)g) α+β = c r β { ( ) } β R α α β β nd thus finlly ( β s ( f α ) f ) c s c { { ( ) } β R α α β β (s) (r) β { ( g α+β ) β { } = K g (b) α+β = K(g(b)) α+β } { ( ) } g α+β } ( ) R α α β β (r) β ( β )( ) ( } g ) α+β β we hve used one lst time the time scles Hölder inequlity conjugte exponents β nd > The proof is complete The next result follows from Theorem 3 by choosing β = Corollry Assume tht T b ( ) T rs C rd ([b] T ( )) nd f C rd ([b] TR) Let α nd > If f()= then { s ( f α ) b K r f } α { } K = (s) (R α ) c 26 NSP Nturl Sciences Publishing Cor
4 R Agrwl et l : A generl dynmic inequlity R= The next result follows from Corollry by choosing =α (see lso [5 Corollry 32]) Corollry 2 Assume tht T b ( ) T rs C rd ([b] T ( )) nd Let α > If f()= then References s ( f α ) K f C rd ([b] TR) r f α { } α K = (s) α α (R α ) α R= α [] R P Agrwl D O Regn nd S Ser Dynmic inequlities on time scles Springer Chm 24 [2] R P Agrwl nd P Y H Png Opil Inequlities Applictions in Differentil nd Difference Equtions Kluwer Acdemic Publishers Dordrecht 995 [3] M Bohner nd B Kymçln Opil inequlities on time scles Ann Polon Mth 77(): 2 2 [4] M Bohner R R Mhmoud nd S H Ser Discrete continuous delt nbl nd dimond-lph Opil inequlities Mth Inequl Appl 8(3):923 94 25 [5] M Bohner R R Mhmoud nd S H Ser Improvements of dynmic Opil-type inequlities nd pplictions Dynm Systems Appl 24:229 242 25 [6] M Bohner nd A Peterson Dynmic Equtions on Time Scles: An Introduction Applictions Birhäuser Boston 2 [7] M Bohner nd A Peterson Advnces in Dynmic Equtions on Time Scles Birhäuser Boston 23 [8] Z Olech A simple proof of certin result of Z Opil Ann Polon Mth 8:6 63 96 [9] Z Opil Sur une inéglité Ann Polon Mth 8:29 32 96 [] S H Ser R P Agrwl nd D O Regn New gps between zeros of fourth-order differentil equtions vi Opil inequlities J Inequl Appl pges 22:82 9 22 Rvi Agrwl is the hed of the Deprtment of Mthemtics t Texs A&M University Kingsville in USA He is the uthor of mny boos nd ppers on fixed point theory opertor integrl differentil prtil nd difference equtions oscilltion theory inequlities nd criticl point methods He serves s the mnging editor nd is on the editoril bords of numerous mthemticl journls Mrtin Bohner is the Curtors Professor of Mthemtics nd Sttistics t Missouri University of Science nd Technology in Roll Missouri USA He received the BS (989) nd MS (993) in Econo-mthemtics nd PhD (995) from Universität Ulm Germny nd MS (992) in Applied Mthemtics from Sn Diego Stte University His reserch interests center round differentil difference nd dynmic equtions s well s their pplictions to economics finnce biology physics nd engineering He is the uthor of five textboos nd more thn 2 publictions Editor-in-Chief of two interntionl journls Associte Editor for more thn 5 interntionl journls nd President of ISDE the Interntionl Society of Difference Equtions Professor Bohner s honors t Missouri S&T include five Fculty Excellence Awrds one Fculty Reserch Awrd nd eight Teching Awrds Donl O Regn is Professor of Mthemtics t the Ntionl University of Irelnd in Glwy He is the uthor of mny boos nd ppers on fixed point theory opertor integrl differentil prtil nd difference equtions oscilltion theory inequlities nd criticl point methods He serves on the editoril bord of numerous mthemticl journls c 26 NSP Nturl Sciences Publishing Cor
Appl Mth Inf Sci No 3-5 (26) / wwwnturlspublishingcom/journlssp 5 Mhmoud Osmn is Lecturer of Mthemtics t Mnsour University of Egypt in Mnsour His interests re qulittive nlysis of dynmic equtions nd inequlities on time scles Up to now he hs co-uthored one pper on Opil inequlities in the journl Mthemticl Inequlities nd Applictions Smir Ser is Professor of Mthemtics t Mnsour University of Egypt in Mnsour His interests re qulittive nlysis of dynmic equtions inequlities on time scles nd qulittive behvior of dely models He is n uthor of four boos nd more thn 2 ppers He serves on the editoril bord of mny mthemticl journls c 26 NSP Nturl Sciences Publishing Cor