The extension of classical inequalities into time scale aims to generalize these inequalities to make themapplicable to dynamic systems that exhibit both continuous and discrete characteristics. Time-scale calculusis becoming an unavoidable area as it is gaining ground in most areas of study. In this work, we presentedsome Chebyshev and other type inequalities on time scales by applying the Cauchy-Schwarz and ArithmeticGeometric Mean inequalit

Ajega-Akem, Stephen

Muniru Iddrisu, Mohammed

Nantomah, Kwara

English

SABA Publishing

Journal of Mathematical Analysis and Modeling J Math Anal & Modeljmam.sabapub.com (2024)5(2) : 53-63ISSN 2709-5924 doi:10.48185/jmam.v5i2.1068Some Chebyshev type inequalities on time scalesSTEPHEN N. AJEGA-AKEM ∗,a,c , MOHAMMED M. IDDRISUb AND KWARA NANTOMAHaa Department of Mathematics, School of Mathematical Sciences, C. K. Tedam University ofTechnology and Applied Sciences, Navrongo , Ghanab Department of Mathematics, Faculty of Physical Sciences, University for DevelopmentStudies, Tamale , Ghanac Department of Mathematics, St. John Bosco College of Education, Navrongo, Ghana• Received: 09 March 2024 • Accepted: 07 June 2024 • Published Online: 29 June 2024AbstractThe extension of classical inequalities into time scale aims to generalize these inequalities to make themapplicable to dynamic systems that exhibit both continuous and discrete characteristics. Time-scale calculusis becoming an unavoidable area as it is gaining ground in most areas of study. In this work, we presentedsome Chebyshev and other type inequalities on time scales by applying the Cauchy-Schwarz and Arithmetic-Geometric Mean inequality.Keywords: : Chebyshev inequality, Cauchy-Schwarz inequality, time scales.2010 MSC: 2010 (MSC): 26D15, 26E70, 26A33.1. IntroductionThe inequality1b− a∫bax(t)dt1b− a∫bay(t)dt ⩽1b− a∫bax(t)y(t)dt, (1.1)where x,y are both non-increasing or non-decreasing function on [a,b] is the Chebyshevinequality[8]. Inequality (1.1) is well known in literature as one of the foundationalinequality in statistical estimation and in mathematical analysis. There are some otherliterature on (1.1) in terms of applications, extensions and generalizations [2, 7, 10, 11,14, 15, 16, 17, 18, 20, 21].According to[19], we need to develop and refine our capabilities to generalize recentresults directly related to the topic of fractional calculus because fractional calculus is in-teresting and has many applications in modeling natural phenomena in the world. Time∗Corresponding author: akemesteve@gmail.comAjega-Akem S.N. et al. / Some Chebyshev type inequalities on time scales 54Scale calculus is also another area of importance that we should also consider. A timescale is an arbitrary nonempty closed subset of the real numbers and is usually denotedby T. Time scale calculus gives an efficient tool to unify continuous and discrete prob-lems in one theory. Time scale calculus provides an integrated framework that can handlea broader collection of systems and data types and this makes time scale calculus morevaluable in various applications where traditional calculus falls short. When T = R, TimeScale reduces to real analysis and when T = Z, Time Scale reduces to discrete analysis.Time scale calculus is an area of mathematics that has recently received a lot of attentionfrom researchers. This area was born by Stefan Hilger in 1988 [13]. Since the discovery oftime scale calculus till date, time scale has made a lot of steady inroads in explaining theinterconnections that exist among the various calculi. The birth of time scale by StefanHilger was therefore to unify discrete and continuous analysis. The extension of classi-cal inequalities into time scale in recent times among researchers is on the increase dueto influence of time scale in mathematical analysis. The extension of classical inequalitiesinto time scale is an aim of generalizing these inequalities and well-established mathemat-ical concepts to make them applicable to dynamic systems that exhibit both continuousand discrete characteristics. Time scale calculus is becoming unavoidable as it is gaininggrounds in most areas of study. Since its inception, time scale calculus has found applica-tions in entomology, computer science, medical sciences, macroeconomics and other keyareas of life [5].2. PreliminariesDefinition 2.1. [1] Let t ∈ T, then t is called right dense if σ(t) = t. It is called rightscattered if σ(t) > t. It is called left scattered if ρ(t) < t. If t < sup T and σ(t) = t, then tis called right-dense, and if t > inf T and ρ(t) = t, then t is called left-dense. Points thatare right-dense and left-dense at the same time are said to be dense.Definition 2.2. Let sup T be finite and left-scattered, then we defined T⋆ = T\{sup T}.Lemma 2.3. [1] Let T be an arbitrarily given time scale. Also, let f be delta differentiable att ∈ T⋆. Thenf∆(t) =f(σ(t) − f(t))µ(t). (2.1)Definition 2.4. Let f : f ∈ T. When T = R, then we havef∆ = f▽ = f ′ (2.2)and ∫baf(t)∆t =∫baf(t)dt. (2.3)Ajega-Akem S.N. et al. / Some Chebyshev type inequalities on time scales 55Definition 2.5. [12] Let a and b be two arbitrary points in T. Then every constant functionf(t) = c, for t ∈ T is ∆- integrable from a to b and is given by∫bac∆t = c(b− a). (2.4)Definition 2.6. [4] The graininess function µ : T −→ [0,∞) is defined byµ(t) = σ(t) − t. (2.5)Definition 2.7. [4] Let t be an arbitrary point in T. Every function f defined on T is∆-integrable from t to σ(t) and is expressed as∫σ(t)tf(x)∆x = (σ(t) − t)f(t). (2.6)Definition 2.8. [1] Let f,g be two differentiable functions on T, then(fg)∆(t) = f∆(t)g(t) + f(σ(t))g∆(t). (2.7)Theorem 2.9. [9] Let T be a bounded time scales. Let also f,g : T → R be integrable func-tions, and α ∈ R a constant. Then the Constant Multiple Rule, Sum rule and Combinationrule are defined respectively by (2.8), (2.9) and (2.10).∫baαf(t)∆t = α∫baf(t)∆t, (2.8)∫ba(f(t) + g(t))∆t =∫baf(t)∆t+∫bag(t)∆t (2.9)and∫baf(t)∆t =∫caf(t)∆t+∫bcf(t)∆t. (2.10)Definition 2.10. [6] A function F : T → R is called a delta antiderivative of f : T → R ifF∆(t) = f(t) for all t ∈ Tk. The delta indefinite and definite integrals are defined by thefollowing equations respectively.Ajega-Akem S.N. et al. / Some Chebyshev type inequalities on time scales 56∫f(t)∆t = F(t) +C, (2.11)∫baf(s)∆s = F(b) − F(a). (2.12)Definition 2.11. Let x,y be random variables defined to assumed values in I ∈ R. Thenthe two variable x,y are identically distributed ifF(x) = F(y), (2.13)for all x,y ∈ I and F : T → R.Lemma 2.12. Let f be continuous on [a,b] and differentiable on (a,b), then there existc ∈ (a,b)) such thatf ′(c) =f(b) − f(a)b− a. (2.14)The integral form of (2.14) is given by∫baf(t)dt = f(c)(b− a). (2.15)The classical discrete and integral forms of the Cauchy-Schwarz inequality are definedrespectively below by the following definitions.Definition 2.13. Let a1,a2...an and b1,b2...bn be real numbers then,(n∑i=1|aibi|)2⩽n∑i=1|ai|2n∑i=1|bi|2. (2.16)Definition 2.14. Let f,g be continuous functions on [a,b] then∫ba|f(t)g(t)|dt ⩽√∫ba|f(t)|2dt.∫ba|g(t)|2dt. (2.17)Lemma 2.15. Let ξ and µ positve real numbers. Then the Arithmetic-Geometric Mean In-equality is defined asξ+ µ2⩾√ξµ, (2.18)for ξ > µ, equality occurs when ξ = µ.Ajega-Akem S.N. et al. / Some Chebyshev type inequalities on time scales 57Lemma 2.16. [3] For the positive integrable functions η, ζ on [0,∞) and for the positive inte-grable functions G1,G2,H1,H2, let ν ∈ (0, 1], γ ∈ C, ℜ(γ) ⩾ 0, and ψ be a strictly increas-ing continuous function. If the following conditions: (Z1)0 < G1(µ) ⩽ η(µ) ⩽ G2(µ), 0 <H1(µ) ⩽ ζ(µ) ⩽ H2(µ), (µ ∈ [0,ω],ω > 0), hold, then the following inequality also holds:(k,ψ)Iγ,υ{H1H2η2}(ω)(γ,υ)Iγ,υ{G1G2ζ2}(ω)((k,ψ)Iγ,υ {(G1H1 +G2H2)ηζ} (ω))2 ⩽ 14. (2.19)The above inequality was obtained through the application of Lemma 2.15.3. Results and DiscussionTheorem 3.1. Let f and g are real-valued integrable functions over [a,b] and that both arenon-increasing or non-decreasing, h ⩾ 0 for all a,b ∈ R. Then(b− a)∫baf(s)g(s)h(s)∆s+∫baf(s)g(s)∆s∫bah(s)∆s ⩾∫baf(s)h(s)∆s∫bag(s)∆s+∫baf(s)∆s∫bag(s)h(s)∆s,(3.1)with the inequality reversed if one is non-increasing and the other is non-decreasing.Proof. Since f,g are both non-increasing or non-decreasing, then for all s, t ⩾ 0 we have[f(s) − f(t)][g(s) − g(t)] ⩾ 0 (3.2)andh(s) + h(t) ⩾ 0 (3.3)since h ⩾ 0.Using (3.2) and (3.3) we have∫ba∫ba[f(s)g(s)h(s) + f(s)g(s)h(t) + f(t)g(t)h(s) + f(t)g(t)h(t)−f(s)g(t)h(s) − f(t)g(s)h(t) − f(s)g(t)h(t) − f(t)g(s)h(t)]∆s∆t ⩾ 0.(3.4)Which further gives(b− a)∫baf(s)g(s)h(s)∆s+∫baf(s)g(s)∆s∫bah(t)∆t+∫baf(t)g(t)∆t∫bah(s)∆s+(b− a)∫baf(t)g(t)h(t)∆t−∫baf(s)h(s)∆s∫bag(t)∆t−∫baf(t)h(t)∆t∫bag(s)∆s−∫baf(s)∆s∫bag(t)h(t)∆t−∫baf(t)h(t)∆t∫bag(s)∆s ⩾ 0.(3.5)Applying (2.13) to (3.5) and simplifying, yields the required result.Ajega-Akem S.N. et al. / Some Chebyshev type inequalities on time scales 58Corollary 3.2. Let f,g be two positive non-decreasing functions on the interval I = [0,a]Tsuch that f,g are time scale integrable and s, t are identically distributed. Then(∫a0f(s)∆s)(∫a0g(s)∆s)⩽ a∫a0f(s)g(s)∆s. (3.6)Proof. Let f,g be synchronous functions for all s, t ∈ [0,a]T. Then∫a0∫a0[f(s) − f(t)][g(s) − g(t)]∆s∆t ⩾ 0.Thus ∫a0∫a0[f(s)g(s) − f(t)g(s) − f(s)g(t) + f(t)g(t)]∆s∆t ⩾ 0,anda∫a0f(s)g(s)∆s−∫a0f(t)∆t∫a0g(s)∆s−∫a0f(s)∆s∫a0g(t)∆t+ a∫a0f(t)g(t)∆t ⩾ 0.Since s, t are identically distributed, then by (2.13) we havea∫a0f(s)g(s)∆s ⩾(∫a0f(s)∆s)(∫a0g(s)∆s)(3.7)Corollary 3.3. If f and g are real-valued integrable functions over [a,b], both non-increasingor non-decreasing and h > 0 for all a,b in R. Then∫baf(s)g(s)∆s ⩾1(b− a)∫baf(s)∆s1(b− a)∫bag(s)∆s (3.8)Proof. From (3.1) when h(s) = 1 and by (2.4),we have∫baf(s)g(s)∆s+∫baf(s)g(s)∆s ⩾1(b− a)∫baf(s)∆s1(b− a)∫bag(s)∆s+1(b− a)∫baf(s)∆s1(b− a)∫bag(s)∆s.(3.9)Hence we obtained (3.8) from (3.9) by simplification.Remark 3.4. Let T = R . Then (3.8) becomes1(b− a)∫baf(s)g(s)ds ⩾1(b− a)∫baf(s)ds1(b− a)∫bag(s)ds (3.10)which is the classical Chebyshev inequality in (1.1).Ajega-Akem S.N. et al. / Some Chebyshev type inequalities on time scales 59Theorem 3.5. Let f and g be two real-valued integrable functions over [a,b]. Also, let ϕ ⩾ 0for all [t,σ(t)] in T such that ϕ(t) ⩽ f(x)g(x) ⩽ ϕ(σ(t)), g(x) ̸= 0. Then1(ϕ(t) +ϕ(σ(t)))∫ba(f2(x) +ϕ(σ(t))ϕ(t)(b− a)g2(x))∆x ⩽(∫baf2(x)∆x) 12(∫bag2(x)∆x) 12.(3.11)Proof. Sinceϕ(t) ⩽f(x)g(x)⩽ ϕ(σ(t)),we have [f(x)g(x)−ϕ(t)] [ϕ(σ(t)) −f(x)g(x)]g2(x) ⩾ 0. (3.12)Which further yields∫ba(f(x)g(x)ϕ(σ(t)) − f2(x) −ϕ(σ(t))ϕ(t)g2(x) +ϕ(t)f(x)g(x))∆x ⩾ 0,andϕ(t) +ϕ(σ(t))b− a∫baf(x)g(x)∆x ⩾1b− a∫baf2(x)∆x+ϕ(σ(t))ϕ(t)b− a∫bag2(x)∆x.By (2.17)we haveϕ(t) +ϕ(σ(t))b− a(∫baf2(x)∆x) 12(∫bag2(x)∆x) 12⩾1b− a∫baf2(x)∆x+ϕ(σ(t))ϕ(t)b− a∫bag2(x)∆x,(3.13)ϕ(t) +ϕ(σ(t))b− a(∫baf2(x)∆x) 12(∫bag2(x)∆x) 12⩾1b− a∫ba(f2(x) +ϕ(σ(t))ϕ(t)(b− a)g2(x))∆x,and(∫baf2(x)∆x) 12(∫bag2(x)∆x) 12⩾1ϕ(t) +ϕ(σ(t))∫ba(f2(x) +ϕ(σ(t))ϕ(t)(b− a)g2(x))∆x.Corollary 3.6. Let f and g be two real-valued integrable functions over [a,b]. Also, let ϕ ⩾ 0for all [t,σ(t)] in T such that ϕ(t) ⩽ f(x)g(x) ⩽ ϕ(σ(t)),g(x) ̸= 0. ThenAjega-Akem S.N. et al. / Some Chebyshev type inequalities on time scales 60ϕ(t) +ϕ(σ(t))b− a(∫baf2(x)∆x) 12(∫bag2(x)∆x) 12⩾2ϕ(σ(t))ϕ(t)∫ba f2(x)∆x∫ba g2(x)∆x(b− a)(ϕ(σ(t))ϕ(t)∫ba g2(x)∆x+∫ba f2(x)∆x) .(3.14)Proof. Recalling (3.13) and applying AM-GM inequality yields the desired result.Theorem 3.7. Let f : T → R be differentiable and T a given time scale such that σ(t) isforward jump operator and f is ∆ differentiable at t ∈ T⋆. Then(∫ 10f∆(t2)∆σ(t)∆t4)2⩽ f(µ(t))f(t). (3.15)and ∫10 f∆(∫σ(t)t f(τ)∆τ)σ(t)∆τ42 ⩽ f(µ(t))f(t). (3.16)Proof. Let ϕ(τ) = f (t+ τσ(t)).When τ = 0, we haveϕ(0) = f(t). (3.17)When τ = 1 we haveϕ(1) = f(t+ σ(t)). (3.18)Subtracting (3.17) from (3.18) yieldsϕ(1) −ϕ(0) = f(t+ σ(t)) − f(t) =∫ 10ϕ∆(τ)∆τ. (3.19)Butϕ∆(τ) = f∆(t+ τσ(t))σ(t). (3.20)Thusf(t+ σ(t)) − f(t) =∫ 10f∆(t+ τσ(t))σ(t)∆τ,which further becomesf(σ(t)) =∫ 10f∆(t+ τσ(t))σ(t)∆τ. (3.21)Ajega-Akem S.N. et al. / Some Chebyshev type inequalities on time scales 61When τ = 1, (3.21) becomesf(µ(t) + t) =∫ 10f∆(t2)∆σ(t)∆τwhich further yields ∫ 10f∆(t2)∆σ(t)∆τ ⩽ 2√f(µ(t))f(t). (3.22)That ends the proof for 3.15.Let τ = [σ(t)−t]f(t)−tσ(t) in (3.21). Then we have∫ 10f∆(∫σ(t)tf(t)∆t)f(σ(t))∆τ ⩽ 2√f(µ(t))f(t). (3.23)Ajega-Akem S.N. et al. / Some Chebyshev type inequalities on time scales 624. ConclusionThe extension of classical inequalities into time scale is aiming at generalizing suchwell-established mathematical concepts to make them applicable to dynamic systems thatexhibit both continuous and discrete characteristics. This paper presented some Cheby-shev type integral inequalities on time scales by applying Cauchy-Schwarz inequality andarithmetic-geometric inequality. The Chebyshev type integral inequalities presented inthis work reduces to the classical Chebyshev type inequalities when the time scale (T) ison the real (R). In future work, we are going consider studying Hermite–Hadamard-typeinequalities on time scale which are equally important in the research community.References[1] Agarwal, R. P., Bohner, M., and Peterson, A. (2001). Inequalities on time scales: a survey. MathematicalInequalities & Applications, Element, 4:535–558.[2] Ajega-Akem, S. N., Iddrisu, M. M., and Nantomah, K. (2019). On chebyshev and riemann-liouvillefractional inequalities in q-calculus. 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Some Chebyshev type inequalities on time scales

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Some Chebyshev type inequalities on time scales

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