FRACTIONAL DYNAMIC INEQUALITIES HARMONIZED ON TIME SCALES

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1 FRACTIONAL DYNAMIC INEQUALITIES HARMONIZED ON TIME SCALES M JIBRIL SHAHAB SAHIR Accepted Mnuscript Version This is the unedited version of the rticle s it ppered upon cceptnce by the journl. A finl edited version of the rticle in the journl formt will be mde vilble soon. As service to uthors nd reserchers we publish this version of the ccepted mnuscript (AM) s soon s possible fter cceptnce. Copyediting, typesetting, nd review of the resulting proof will be undertken on this mnuscript before finl publiction of the Version of Record (VoR). Plese note tht during production nd pre-press, errors my be discovered which could ffect the content. 208 The Author(s). This open ccess rticle is distributed under Cretive Commons Attribution (CC-BY) 4.0 license. Publisher: Cogent OA Journl: Cogent Mthemtics & Sttistics DOI:

2 Click here to downlod Mnuscript - with uthor detils Jibriel Article.pdf FRACTIONAL DYNAMIC INEQUALITIES HARMONIZED ON TIME SCALES M JIBRIL SHAHAB SAHIR Abstrct. We present here some symmetric frctionl Rogers-Hölder s inequlities by using Riemnn-Liouville integrl on time scles. We consider nd impose different conditions on three nonzero rel numbers p, q nd r when p + q + r = 0.. Introduction Rogers-Hölder s inequlity is n importnt nd well known inequlity. This inequlity hs lot of pplictions. We give its symmetric form here. When f i g i h i = for ll i =, 2,...n, where f i, g i, h i re sets of positive vlues, then ( n ( p n ( q n r (.), f p i g q i s given in [, 3] for three nonzero rel numbers p, q nd r, where ll but one of p, q nd r re positive. Inequlity (.) is reversed if ll but one of p, q nd r re negtive for sets of positive vlues of f i, g i nd h i. We unify nd extend (.) on time scles. Time scle clculus ws initited by Stefn Hilger s in []. This inequlity nd its inverse cn be proved in weighted form by using Riemnn- Liouville integrl on time scles. Their different versions cn be expressed in sme mnner. 2. Preliminries We need here bsic concepts of delt clculus. The results of delt clculus re dpted from [3, 8, 9]. A time scle is n rbitrry nonempty closed subset of the rel numbers. It is denoted by T. For t T, forwrd jump opertor σ : T T is defined by h r i σ(t) := inf{s T : s > t}. The mpping µ : T R + 0 = [0, ) such tht µ(t) := σ(t) t is clled the grininess. When T = R, we see tht σ(t) = t nd µ(t) 0 for ll t T nd when T = N, we hve tht σ(t) = t + nd µ(t) for ll t T. The bckwrd jump opertor ρ : T T is defined by ρ(t) := sup{s T : s < t}. The mpping ν : T R + 0 such tht ν(t) := t ρ(t) is clled the bckwrd grininess. If σ(t) > t, we sy tht t is right-scttered, while if ρ(t) < t, we sy tht t is leftscttered. Also, if t < sup T nd σ(t) = t, then t is clled right-dense, nd if Key words nd phrses. Riemnn-Liouville integrl, Rogers-Hölder s inequlity, time scles.

3 2 M JIBRIL SHAHAB SAHIR t > inf T nd ρ(t) = t, then t is clled left-dense. If T hs left-scttered mximum M, then T k = T {M}. For function f : T R, the derivtive f is defined s follows. Let t T k, if there exists f (t) R such tht for ll ɛ > 0, there exists neighborhood U of t with f(σ(t)) f(s) f (t)(σ(t) s) ɛ σ(t) s, for ll s U, then f is sid to be differentible t t, nd f (t) is clled the delt derivtive of f t t. A function f : [, b] R is sid to be right-dense continuous (rd-continuous) if it is right-continuous t ech right-dense point nd there exists finite left limit t ll left-dense points. The next definition is given in [3, 8, 9]. Definition. For, b T, nd delt differentible function f, the Cuchy integrl of f is defined by b f (t) t = f(b) f(). The following results of nbl clculus re tken from [3, 7, 8, 9]. The function f : T R is clled nbl differentible t t T k, if there exists n f (t) R with the following property: For ny ɛ > 0, there exists neighborhood U of t, such tht f(ρ(t)) f(s) f (t)(ρ(t) s) ɛ ρ(t) s, for ll s U. For T = R, we hve f (t) = f (t) nd for T = Z, bckwrd difference opertor is defined s f (t) = f(t) = f(t) f(t ). A function f : T R is left-dense continuous or ld-continuous provided it is continuous t left-dense points in T nd its right-sided limits exist (finite) t right-dense points in T. If T = R, then f is ld-continuous if nd only if f is continuous. If T = Z, then ny function is ld-continuous. If T hs right-scttered minimum m, then T k = T {m}. The next definition is given in [3, 7, 8, 9]. Definition 2. A function F : T R is clled nbl ntiderivtive of f : T R provided F (t) = f(t) holds for ll t T. Then nbl integrl of f is defined by f(s) s = F (t) F (). Every ld-continuous hs nbl ntiderivtive. We need the following definitions of -Riemnn-Liouville type frctionl integrl nd -Riemnn-Liouville type frctionl integrl. The following definition is tken from [4, 6]. Definition 3. For α, the time scle -Riemnn-Liouville type frctionl integrl is defined by (, b T) (2.) I α f(t) = h α (t, σ(τ))f(τ),

4 3 where t [, b] T. Notice I f(t) = f(τ), where h α : T T R, α 0 re coordinte wise rd-continuous functions. The following definition is tken from [5, 6]. Definition 4. For α, the time scle -Riemnn-Liouville type frctionl integrl is defined by (, b T) (2.2) J α f(t) = where t [, b] T. Notice J f(t) = ĥ α (t, ρ(τ))f(τ) τ, f(τ) τ, where ĥα : T T R, α 0 re coordinte wise ld-continuous functions. 3. Min Results The following result cn be proved for different conditions imposed on p, q nd r s p > 0, q > 0 but r < 0; p > 0, r > 0 but q < 0 or q > 0, r > 0 but p < 0. Theorem. Let w, f i C rd ([, b] T, R) for i =, 2, 3 nd p, q nd r be three nonzero rel numbers with p + q + 3 r = 0. Further ssume tht t [, b] T. If p > 0, q > 0 but r < 0, then for α f i (t) =, where (3.) (I α ( w(t) f (t) p ) p (I α ( w(t) f 2 (t) q ) q (I α ( w(t) f 3 (t) r ) r. Proof. Given condition p + q + r = 0 cn be rerrnged s ( p r ) + ( q r ) =. Set P = p r > nd Q = q r >. Now we pply Rogers-Hölder s inequlity for F (x) nd G(x), s So, ( w(τ) F (τ)g(τ) ( w(τ) F (τ) P P t w(τ) G(τ) Q Q. (3.2) w(τ) F (τ)g(τ) ( w(τ) F (τ) p r ) r p ( w(τ) G(τ) q r ) r q.

5 4 M JIBRIL SHAHAB SAHIR Replcing F (τ) by (h α (t, σ(τ))) r p f r (τ) nd G(τ) by (h α (t, σ(τ))) r q f2 r (τ) where τ [, t) T nd tking power r > 0, then (3.2) tkes the form (3.3) ( ) h α (t, σ(τ)) w(τ) f (τ)f 2 (τ) r r ( ( h α (t, σ(τ)) w(τ) f (τ) p p t As f (t)f 2 (t)f 3 (t) =, then (3.3) tkes the form ( ) h α (t, σ(τ)) w(τ) f 3 (τ) r r ( ( h α (t, σ(τ)) w(τ) f (τ) p p t Then the proof is cler. h α (t, σ(τ)) w(τ) f 2 (τ) q q. h α (t, σ(τ)) w(τ) f 2 (τ) q q. Remrk. If we tke α = nd w(t) = nd lso T = Z in Theorem, then we get discrete version of Rogers-Hölder s inequlity s given in (.), where f i, g i nd h i for ll i =, 2,..., n re sets of positive vlues. Remrk 2. If we tke α =, w(t) =, r =, p >, f i g i h i = for ll i =, 2,..., n nd lso T = Z, then we get discrete version of Rogers-Hölder s inequlity from Theorem, s ( n n ( p n q f i g i f i p g i q. Hölder proved this inequlity s given in [2]. Remrk 3. If we tke α =, w(t) =, r =, p > nd lso f (t)f 2 (t)f 3 (t) =, then we get integrl version of Rogers-Hölder s inequlity from Theorem, s ( ( f (t) f 2 (t) t f (t) p q t t f 2 (t) q q t, s given in [2, 3, 8]. When r =, w(t) =, p > nd f (t)f 2 (t)f 3 (t) =, then (3.) tkes the form s given in [0]. I α ( f (t)f 2 (t) ) (I α f (t) p p (I α f 2 (t) q q, Similrly nbl version of Theorem cn be written s: Corollry. Let w, f i C ld ([, b] T, R) for i =, 2, 3 nd p, q nd r be three nonzero rel numbers with p + q + r = 0. Further ssume tht 3 f i (t) =. If p > 0, q > 0 but r < 0, then for α (3.4) (J α ( w(t) f (t) p ) p (J α ( w(t) f 2 (t) q ) q (J α ( w(t) f 3 (t) r ) r. Proof. Similr proof to the Theorem.

6 5 Result given in (3.) is reversed if q < 0, r < 0 but p > 0; p < 0, r < 0 but q > 0 or p < 0, q < 0 but r > 0. Corollry 2. Let w, f i C rd ([, b] T, R) for i =, 2, 3 nd p, q nd r be three nonzero rel numbers with p + q + r = 0. Further ssume tht 3 f i (t) =. If q < 0, r < 0 but p > 0; p < 0, r < 0 but q > 0 or p < 0, q < 0 but r > 0, then for α (3.5) (I α ( w(t) f (t) p ) p (I α ( w(t) f 2 (t) q ) q (I α ( w(t) f 3 (t) r ) r. Proof. Similr proof to the Theorem. Similrly nbl version of inequlity (3.4) cn be written s: Corollry 3. Let w, f i C ld ([, b] T, R) for i =, 2, 3 nd p, q nd r be three nonzero rel numbers with p + q + r = 0. Further ssume tht 3 f i (t) =. If q < 0, r < 0 but p > 0; p < 0, r < 0 but q > 0 or p < 0, q < 0 but r > 0, then for α (3.6) (J α ( w(t) f (t) p ) p (J α ( w(t) f 2 (t) q ) q (J α ( w(t) f 3 (t) r ) r. Proof. Similr proof to the Theorem. Now we generlize (3.) nd (3.5), s Theorem 2. Let w, f i C rd ([, b] T, R) for i =, 2,..., n nd p i be nonzero rel numbers with n n p i = 0. Further ssume tht f i (t) =. If ll p i for i =, 2,..., n re positive but one is negtive, then for α n (3.7) (I α ( w(t) f i (t) pi ) p i. Proof. Similr proof to the Theorem. Theorem 3. Let w, f i C rd ([, b] T, R) for i =, 2,..., n nd p i be nonzero rel numbers with n n p i = 0. Further ssume tht f i (t) =. If ll p i for i =, 2,..., n re negtive but one is positive, then for α n (3.8) (I α ( w(t) f i (t) pi ) p i. Proof. Similr proof to the Theorem. Remrk 4. Similrly we cn generlize our results (3.) nd (3.5) on nbl clculus.

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