Frctionl Differentil Clculus Volume 6, Number 2 (216), 275 28 doi:1.7153/fdc-6-18 CHEBYSHEV TYPE INEQUALITY ON NABLA DISCRETE FRACTIONAL CALCULUS SERKAN ASLIYÜCE AND AYŞE FEZA GÜVENILIR (Communicted by F. Atici) Abstrct. In this pper, we estblish some Chebyshev type inequlities on discrete frctionl clculus with nbl opertor (or bckwrd difference opertor). 1. Introduction By the nineteenth century, efforts of number of mthemticins, most notedly Riemnn, Grünwld, Letnikov, nd Liouville, led to consistent theory of frctionl clculus for rel vrible functions. Although there re mny definitions of frctionl derivtives, the most known definitions re Riemnn-Liouville nd Cputo derivtives. When it comes to the theory of discrete frctionl clculus, we mention the pper presented by Diz nd Osler in 1974 [11]. In this pper, the uthors introduced frctionl difference opertor using n infinite series. In 1988, Gry nd Zhng [15]introduced new definition of frctionl difference opertor nd they proved Leibniz formul, composition rule nd power rule. Wheres Diz et l. gve definition for the delt (forwrd) difference opertor, Gry et l. gve their definition for the nbl (bckwrd) difference opertor. Mthemticins hve begn to py ttention to this theory for lst three decdes. As pioneering work, Atici nd Eloe [3] presented properties of generlized flling function tht plys mjor role s n exponentil function in difference clculus, power rule nd commuttivity of frctionl sums. For more results we refer to [2, 4, 5, 6, 8, 14, 17]. Inequlities re useful tools in mthemtics. In order to see the use of inequlities s mthemticl tools, we refer to [13]. In this work, to show the continuous dependence of solutions of initil vlue problems on initil conditions uthor shows nd uses frctionl discrete nlogue of Gronwll inequlity. For more inequlities on discrete frctionl clculus (delt or nbl cse) see [1, 2, 7, 1, 12, 13, 16]. In this pper, our min purpose is to mke contribution to this re with estblishing the discrete frctionl nlogue of Chebyshev s inequlity using nbl opertor. Here, we will estblish discrete frctionl nlogue of Chebyshev s inequlity given below [9]. Mthemtics subject clssifiction (21): Primry 26D15, 26A33; Secondry 39A12, 26D1. Keywords nd phrses: Chebyshev inequlity, nbl opertor, discrete frctionl clculus. c D l,zgreb Pper FDC-6-18 275
276 S. ASLIYÜCE AND A. FEZA GÜVENILIR Let f nd g be two integrble functions in [,1]. If both functions re simultneously incresing or decresing for sme vlues of x in [,1],then f (x)g(x)dx f (x)dx g(x)dx. If one function is incresing nd the other decresing for the sme vlues of x in [,1],then f (x)g(x)dx f (x)dx g(x)dx. 2. Preliminries on discrete nbl frctionl clculus In this section, we introduce the reder to bsic concepts nd results bout discrete frctionl clculus with nbl opertor. The rising function is defined by t n = t(t + 1)(t + 2)...(t + n 1), for n N. Using the Gmm function we cn generlize the rising function s t v = Γ(t + v), v R nd t R \{..., 2, 1,}. Γ(t) REMARK 1. Using the properties of the Gmm function, it is esily seen tht for t ndv, we get t v. For R,wedefine the set N = {, + 1, + 2,...}. Also, we use the nottion ρ(s) =s 1 for the shift opertor nd ( f )(t) = f (t) f (t 1) for the bckwrd difference opertor. For function f : N R, discrete frctionl sum of order v isdefined s f )(t)= 1 ( f )(t)= f (t), t N, t s= (t ρ(s)) v 1 f (s), t N, v >. REMARK 2. If v = 1, we get the summtion opertor ( 1 f )(t)= t s= f (s). The following result will be used in the sequel.
CHEBYSHEV TYPE INEQ. ON NABLA DISCRETE FRACT. CALC. 277 LEMMA 1. (See [4, Lemm 2.1]) If R nd μ, μ + ν R \{..., 2, 1}, then ( v (s + 1) μ) Γ(μ + 1) (t)= Γ(μ + ν + 1) (t + 1)μ+ν, t N ν. REMARK 3. The function t (t ) v defined on N, R nd v > isincresing. Indeed, we hve tht (t ) v = v(t ) v 1 nd (t ) v 1. DEFINITION 1. Two functions f nd g re clled synchronous, respectively synchronous, on N if for ll τ, s N,wehve( f (τ) f (s))(g(τ) g(s)), respectively ( f (τ) f (s))(g(τ) g(s)). 3. Discrete frctionl Chebyshev type inequlities We strt by proving the min result of this pper. THEOREM 1. If v > nd f,g re two synchronous functions on N, then fg)(t) (t ) v ( v f )(t)( v g)(t), t N. (1) Proof. Since the functions f nd g re synchronous on N, then for ll τ, s N, we hve ( f (τ) f (s))(g(τ) g(s)), i.e. f (τ)g(τ)+ f (s)g(s) f (τ)g(s)+ f (s)g(τ). (2) Now, multiplying both sides of (2) by (t ρ(τ))v 1, t N nd τ {, + 1,...,t},we obtin (t ρ(τ)) v 1 f (τ)g(τ)+ (t ρ(τ))v 1 f (τ)g(s)+ (t ρ(τ))v 1 (t ρ(τ))v 1 f (s)g(s) Now, tking the sum of both sides of (3) for τ {, + 1,...,t}, we get fg)(t)+ f (s)g(s) Multiplying both sides of (4) by (t ρ(s))v 1 (t ρ(s)) v 1 (t ρ(s))v 1 f (s)g(τ). (3) 1)(t) g(s)( v f )(t)+ f (s) g)(t). (4) (t ρ(s))v 1 fg)(t)+ g(s), t N nd s {, + 1,...,t}, we obtin (t ρ(s))v 1 f )(t)+ f (s)g(s) 1)(t) f (s) g)(t), (5)
278 S. ASLIYÜCE AND A. FEZA GÜVENILIR nd gin, tking sum on both sides of (5) for s {, + 1,...,t}, nd using Lemm 1, we get i.e. This shows (1). 1)(t)( v g)(t) f )(t) fg)(t)+( v f )(t)+ g)(t) fg)(t)( v 1)(t) f )(t) g)(t), 1)(t) fg)(t) (t )v = ( v fg)(t). REMARK 4. The inequlity sign in (1) is reversed if the functions re synchronous on N. EXAMPLE 1. Let α,β nd consider the functions f (t)=(t ) α, f (t)=(t ) β, t N. From Remrk 3, we sy tht the functions f nd g re incresing, so f nd g re synchronous. Therefore, we cn pply Theorem 1 with functions f nd g for ν. Using Lemm 1, we obtin fg)(t) (t ) v ( v f )(t) g)(t) = (t ) v ( v (t )α ) (t )β ) = (t ) v Γ(α + 1) Γ(α + v + 1) (t )α+v Γ(β + 1) Γ(β + v + 1) (t )β +v. THEOREM 2. If v, μ > nd f,g re two synchronous functions on N,then (t ) v (t ( μ )μ fg)(t)+ Γ(μ + 1) ( v fg)(t) f )(t)( μ g)(t)+( μ f )(t) g)(t), t N. (6) Proof. For the proof, we continue s in the proof of Theorem 1 nd using inequlity (4), we cn write (t ρ(s)) μ 1 (t ρ(s))μ 1 ρ(s))μ 1 fg)(t)+(t f (s)g(s) 1)(t) g(s) (t ρ(s))μ 1 f )(t)+ f (s) g)(t). (7) Now, tking the sum of both sides of (7) for s {, + 1,...,t}, we obtin the desired inequlity (6).
CHEBYSHEV TYPE INEQ. ON NABLA DISCRETE FRACT. CALC. 279 REMARK 5. If we let v = μ in Theorem 2, we obtin Theorem 1. Finlly, we give generliztion of Theorem 1. THEOREM 3. Assume tht f i, 1 i n, re n N functions on N stisfying k 1 f i nd f k re synchronous for ll k {2,...,n}, (8) f i for 3 i n. (9) Suppose tht v >. Then, for ll t N, we hve ( v n f i )(t) ( ) n 1 n (t ) v ( v f i ) (t). (1) Proof. In view of (8) nd (9), pplying Theorem 1 repetedly, we hve ( ) ( ) v n f i (t) (t ) v v n 1 f i (t) f n)(t) ( ) ( 2 (t ) v v n 2 n f i )(t) f i )(t) i=n 1... ( ) n 1 n ( (t ) v v f i) (t). REMARK 6. If the functions f i,1 i n, in Theorem 3 re either ll nonnegtive incresing or nonnegtive decresing, then both (8) nd (9) re stisfied. REFERENCES [1] E. AKIN, S. ASLIYÜCE, A. F. GÜVENİLİR, B. KAYMAKÇALAN, Discrete Grüss type inequlity on frctionl clculus, J. Ineq. Appl. 174, (215), 7 pp. [2] G.A.ANASTASSIOU, Nbl discrete frctionl clculus nd nbl inequlities, Mth. Comput. Modelling 51, (21), 562 571. [3] F. M. ATICI P. W. ELOE, A trnsform method in discrete frctionl clculus, Int. J. Difference Equ. 2, (27), 165 176. [4] F. M. ATICI P. W. ELOE, Initil vlue problems in discrete frctionl clculus, Proc. Amer. Mth. Soc. 137, (29), 981 989. [5] F. M. ATICI P. W. ELOE, Discrete frctionl clculus with the nbl opertor, Electron J. Qul. Theory Differ. Equ. 3, (29), 12pp. [6] F. M. ATICI S. ŞENGÜL, Modeling with frctionl difference equtions, J. Mth. Anl. Appl. 369, (21), 157 159. [7] F. M. ATICI P. W. ELOE, Gronwll s inequlity on discrete frctionl clculus, Comput. Mth. Appl. 64, (212), 3193 32. [8] N. R. O. BASTOS, R. A. C. FERREIRA, D. F. M. TORRES, Necessry optimlity conditions for frctionl difference problems of the clculus of vritions, Discrete Contin. Dyn. Syst. 29, (211), 417 437.
28 S. ASLIYÜCE AND A. FEZA GÜVENILIR [9] P. L. CHEBYSHEV, Sur les expressions pproximtives des integrles definies pr les utres prises entre les memes limites, Proc.Mth.Soc.Chrkov2, (1882), 93 98. [1] G. V. S. R. DEEKSHITULU, J. J. MOHAN, Frctionl difference inequlities of Bihri type, Commun. Appl. Anl. 14, (21), 343 354. [11] J. B. DIAZ, T. J. OSLER, Differences of frctionl order, Mth. Comp. 28, (1974), 185 22. [12] Q. FENG, Some new generlized Gronwll-Bellmn type discrete frctionl inequlities, Appl. Mth. Comput. 259, (215), 43 411. [13] R. A. C. FERREIRA, A discrete frctionl Gronwll inequlity, Proc. Amer. Mth. Soc. 14, (212), 165 1612. [14] C. S. GOODRICH, Continuity of solutions to discrete frctionl initil vlue problems, Comput. Mth. Appl. 59, (21), 3489 3499. [15] H. L. GRAY, N. F. ZHANG, On new definition of frctionl difference, Mth. Comp. 5, (1988), 513 529. [16] A. F. GÜVENİLİR, B. KAYMAKÇALAN, A. C. PETERSON, K. TAŞ, Nbl discrete frctionl Grüss type inequlity, J. Ineq. Appl. 86, (214), 9 pp. [17] M. HOLM, Sum nd difference compositions in discrete frctionl clculus, Cubo 13, (211), 153 184. (Received Mrch 18, 216) Serkn Asliyüce Amsy University Fculty of Sciences nd Arts, Deprtment of Mthemtics 51, Amsy, Turkey nd Ankr University, Fculty of Sciences Deprtment of Mthemtics 61, Ankr, Turkey e-mil: ssliyuce@nkr.edu.tr, serkn.sliyuce@msy.edu.tr Ayşe Fez Güvenilir Ankr University, Fculty of Sciences Deprtment of Mthemtics 61, Ankr, Turkey e-mil: guvenili@science.nkr.edu.tr Frctionl Differentil Clculus www.ele-mth.com fdc@ele-mth.com