Theoreticl themtics & pplictions vol.8 no.1 2018 1-8 ISSN: 1792-9687 print 1792-9709 online Scienpress Ltd 2018 Wirtinger s Integrl Inequlity on Time Scle Ttjn irkovic 1 bstrct In this pper we estblish Wirtinger-type inequlity on n rbitrry time scle. We give s specil cses of the time scles new Wirtinger-type inequlity in the continuous nd discrete cses respectively. themtics Subject Clssifiction: 34N05; 26D10 Keywords: Wirtinger-type inequlity; Time Scle; Dynmic Inequlities 1 Introduction time scle we denote it by the symbol T is n rbitrry nonempty closed subset of the rel numbers. For t T we define the forwrd jump opertor σ : T T by σ t := inf {s T : s > t}. If t < T nd σ t = t then t is clled right-dense nd if t > inf T nd ρ t = t then t is clled left-dense. Grininess function µ : T [0 is defined by µ t := σ t t see [2] [3] [6]. 1 High School of pplied professionl studies Vrnje Serbi. E-mil: tmirkovic75@gmil.com rticle Info: Received : Februry 6 2017. Revised : y 5 2017. Published online : Februry 1 2018.
2 Wirtinger s Integrl Inequlity on Time Scle function f : T R is clled rd-continuous provided it is continuous t right-dense points in T nd its left-sided limits exist finite t left-dense points in T. The set of rd-continuous functions f : T R will be denoted by C rd = C rd T = C rd T R. The set of functions f : T R tht re differentible nd whose derivtive is rd-continuous is denoted by C 1 rd = C1 rd T = C1 rd T R. We define the time scle intervl [ b] T by [ b] T = [ b] T. In 2000 Hilscher [8] proved Wirtinger-type inequlity on time scles in the form: Theorem 1.1. Discrete Wirtinger Inequlity [8] If be positive nd strictly monotone such tht exists nd is rd-continuous then t y 2 σ t Ψ 2 b tσt t y t 2 1 for ny y with y = y b = 0 nd such tht y exists nd is rd-continuous where Ψ = t [b] T t σt 1 [ 2 + t [b] T µt t + σt In [4] uthors extended the following theorem: t [b] T t σt ] 1 2. Theorem 1.2. [4] Suppose γ 1 is n odd integer. For positive C 1 rd T stisfying either > 0 or < 0 on T we hve 2 γ tσt t y t γ+1 1 b γ Ψ γ+1 αβγ t y γ+1 t 3 for ny y Crd 1 T with y = y b = 0 where Ψ α β γ is the lrgest root of whereby α := t T k x γ+1 2 γ γ + 1 αx γ 2 γ β = 0 4 σ t t γ γ+1 β := t T k µ t t γ. t
Ttjn irkovic 3 2 in Results Let us prove the following theorem: Theorem 2.1. Let C 1 rd [ b] T k be positive nd strictly monotone such tht stisfying either > 0 or < 0 on [ b] T k. Then for some integer η 1 we hve t y σ t Λ b ω ξ r ψ η tσt t y t 5 η for ny y C 1 rd [ b] T k with y = y b = 0 where Λ ω ξ r ψ is the lrgest root of equlity x = 2 η ωx η + η 2 η r+1 ξ r x r + 2 η ψ 6 whereby ω = ξ r = σ η ψ = t [b] T k t [b] T k µ r t [b] T k σ ηη r r ηη r r µ 1 η η η r = 1... η 1. 7 We denote by = t y b σ t B = η tσt t y t. 8 η Using the integrtion by prts whereby y = y b = 0 left side of inequlity 2.1 become = t y b t = ± t y t { } = ± [ t y t] b b σ t y t = σ t y t = η σ y r y σ η r y r=0 σ y σ η + y y σ η + y 2 y σ η 2 +... + y η y σ + y η y
4 Wirtinger s Integrl Inequlity on Time Scle = σ y + µy η + y y + µy η +... + y η y + µy + y η y σ {2 η y η y + 2 η µ y + 2 η 2 y η y + 2 η 2 µ y y η +...+ + y η y + µ y η y 2 + y η y } = {2 η σ y η y + 2 η 2 σ µ y y η + 2 η 3 σ µ y 2 y η +... + σ µ y η y 2 + 2 η σ µ y } = 2 η b η σ y 1 η σ y η η + 2 η 2 b η σ y η η +2 η 3 b η σ y η η +2 η σ y 3 η + η σ y 2 η +2 η b µ σ η2 y η2 µ 2 σ ηη 2 η σ η y µ η η. ηη 2 1 µ η 2 σ η 2 3η η 2 y η 2 3η µ η σ η 2η η y η 2η y 2 +... pplying Hölder inequlity on ech summnd of the bove inequlity except the lst one it follows { 2 η { +2 η 2 +... + +2 η b { b η σ y } 1 { b η η σ η y } η { { η σ y } 2 η η σ y µ η η η η σ } η y µ σ η2 y η2 µ η σ η 2η y η 2η = 2 η ωb 1 η + 2 η 2 ξ 1 B η 1 + 2 η 3 ξ 2 B η 2 +... +2ξ η 2 B 3 η 2 + ξη B 2 η + 2 η ψb } 1 } η 9
Ttjn irkovic 5 i.e. 2 η ωb 1 η + η 2 η r+1 ξ r B η r r + 2 η ψb. 10 fter some clcultions one obtins it holds the following inequlity 1 2 η ω + 2 η 2 ξ B η B 1 + 2 η 3 ξ B η 2 2 +2ξ η 2 B 2 + ξ η B 1 1 η B 2 η ω + 2 η r+1 ξ r B By introducing C = 1 we get B i.e. + 2 η ψ B η r η C 2 η ω + 2 η r+1 ξ r C r η + 2 η ψ C 2 η ωc η + η whence follows the desired inequlity η +... η B + 2 η ψ. η B 2 η r+1 ξ r C r + 2 η ψ 11 Λ ω ξ r γ B. 3 ppliction Corollry 3.1. In the cse of T = R the inequlity 1.3 reduces to t y t dt 2 η t t η y t dt. 12 Proof: In the cse of T = R it is f t = f t σ t = t nd µ t = 0 so ω = 1 ξ r = 0 nd ψ = 0. By substitute this vlues in the equlities 2.2 we obtin x = 2 η x η. i.e. x η x 2 η = 0. Since inequlity 3.1. f t = f t dt follows
6 Wirtinger s Integrl Inequlity on Time Scle Remrk 3.2. Specilly in the cse of η = 1 the lrgest root of the 1.3 is 2 so the inequlity 1.3 becomes wht ws proved in [6]. t y 2 b t dt 4 2 t t y t 2 dt 13 Theorem 3.3. Let T = hz. For positive sequence { n } 0 n N+1 stisfying either > 0 or < 0 on [0 N] hz we hve N n=0 h n y n Ω η ω ξ r ψ N n=0 η n n+1 h n η h y n for ny sequence {y n } 0 n N+1 with y 0 = y N+1 = 0 where Ω ω ξ r ψ is the smllest root of the inequlity when ω = ξ r = ψ = 0 n N 0 n N 0 n N 1 + 2ω 2 η x η = η n+h n h r η n+h h ηη r n r n h 1η h n n η. 2 η r+1 ξ r x r + 2 η ψ 14 η r = 1... η 1 15 Proof. Strting from the inequlity 1 + C C + η + 1 C η + 2 η C η it is obtined C 1 + C η + 1 C η 2 η C η. Involving this result in 1.2 proves it holds η 1 + C η + 1 C η 2 η C η 2 η ωc η 2 η r+1 ξ r C r 2 η ψ 0. Since 1 + C η + 1 C η
Ttjn irkovic 7 lst inequlity becomes η 1 + 2ω 2 η C η 2 η r+1 ξ r C r + 2 η ψ. Since for T = hz = {hk : k Z} is σ t = t+h µ t = h f t = h f t = f t = µ t f t so tht ft+h ft h = N n=0 t [0N] hz h n y n B = N whence follows the desired inequlity. n=0 η n n+1 h n η hy n 4 Conclusion In this pper we present some new Wirtinger-type inequlities on time scles for function f k. s specil cses some new continuous nd discrete Wirtinger-type inequlities re given. References [1] R. grwl. Bohner Bsic clculus on time scles nd some of its pplictions Results th 351-2 1999 3-22. [2] R. grwl. Bohner. Peterson Inequlities on time scles: survey th. Inequl. ppl 41-2 2001 537-557. [3] R. grwl. Bohner D. O Regn. Peterson Dynmic equtions on time scles: survey Bsic clculus on time scles nd some of its pplictions J. Comput. ppl. th 141 2002 1-26. [4] R. grwl. Bohner D. O Regn S. Sker Some dynmic Wirtingeritype inequlities nd their ppliction Pcific J. th 2521 2011. [5] P. Beesck Integrl inequlities of the Wirtinger type Duke th. J. 25 1958 477-498.
8 Wirtinger s Integrl Inequlity on Time Scle [6]. Bohner. Peterson Dynmic Equtions on time scles n Introduction with pplictions Birkhäuser Boston US 2001. [7] W. Coles generl Wirtinger-type inequlity Duke th. J. 27 1960 133-138. [8] R. Hilscher time scles version of Wirtinger-type inequlity nd pplictionsk J. Comput. ppl. th 141 2002 219-226. [9]. itrinovic P. Vsic n integrl inequlity scribed to Wirtinger nd its vritions nd generliztions Univ. Beogrd. Publ. Elektrotehn. Fk. Ser. t. Fiz. 351-2 1969 247-273.