In this paper a perturbed version of the Median Principle introduced

by the author in 'The median principle for inequalities and applications' is developed. Applications for various Riemann-

Stieltjes integral and Lebesgue integral inequalities are also provided

Dragomir, Sever S

English

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The Perturbed Median Principle for Integral Inequalities with Applications

Victoria University Institutional Repository (VUIR)

In this paper a perturbed version of the Median Principle introduced\ud
by the author in 'The median principle for inequalities and applications' is developed. Applications for various Riemann-\ud
Stieltjes integral and Lebesgue integral inequalities are also provided

A. Ostrowski

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Victoria University Eprints Repository

THE PERTURBED MEDIAN PRINCIPLE FOR INTEGRALINEQUALITIES WITH APPLICATIONSS.S. DRAGOMIRAbstract. In this paper a perturbed version of the Median Principle intro-duced by the author in [1] is developed. Applications for various Riemann-Stieltjes integral and Lebesgue integral inequalities are also provided.1. IntroductionIn Analytic Inequalities Theory, there are many results involving the sup-normof a function or of its derivative. To give only two examples, we recall the Ostrowskiinequality :(1.1)∣∣∣∣∣f (x)− 1b− a∫ baf (t) dt∣∣∣∣∣ ≤14+(x− a+b2b− a)2 (b− a) ‖f ′‖[a,b],∞for any x ∈ [a, b] , where f is absolutely continuous on [a, b] and f ′ ∈ L∞ [a, b] , i.e.,‖f ′‖[a,b],∞ := ess supt∈[a,b]|f ′ (t)| <∞, and the Cˇebysˇev inequality(1.2)∣∣∣∣∣ 1b− a∫ baf (t) g (t) dt− 1b− a∫ baf (t) dt · 1b− a∫ bag (t) dt∣∣∣∣∣≤ 112(b− a)2 ‖f ′‖[a,b],∞ ‖g′‖[a,b],∞ ,provided that f and g are absolutely continuous with f ′, g′ ∈ L∞ [a, b] .Since, in order to estimate∥∥f (r)∥∥[a,b],∞ , in practice it is usually necessary tofind the quantitiesMr := supt∈[a,b]f (r) (t) and mr := inft∈[a,b]f (r) (t)(as, obviously,∥∥f (r)∥∥[a,b],∞ = max {|Mr| , |mr|}), the knowledge of∥∥f (r)∥∥[a,b],∞may be as difficult as the knowledge of Mr and mr.As pointed out in [1], it is natural, therefore, to try to establish inequalities whereinstead of∥∥f (r)∥∥[a,b],∞ one would have the positive quantity Mr −mr. This can bealso useful since for functions whose derivatives f (r) have a “modest variation” thequantity Mr −mr may be much smaller than∥∥f (r)∥∥[a,b],∞ .In order to address this problem, the author has stated in [1] the “Median Prin-ciple” that can be formalised as follows:Date: October 14, 2008.2000 Mathematics Subject Classification. 26D15, 26D10.Key words and phrases. Riemann-Stieltjes integral inequality, Median principle, Ostrowskitype inequality, Gru¨ss type inequality.12 S.S. DRAGOMIRTheorem 1 (Median Principle). Let P◦n be the class of polynomials{Pn|Pn (x) = xn + a1xn−1 + · · ·+ an−1x+ an, ai ∈ R, i = 1, n}and f : [a, b] ⊂ R→ R be a function so that f (n−1) is absolutely continuous andf (n) ∈ L∞ [a, b] . Assume that the following inequality holds:(1.3) L(f, f (1), . . . , f (n−1), f (n); a, b)≤ R(∥∥∥f (n)∥∥∥[a,b],∞; a, b),where L (·, ·, . . . , ·; a, b) : R(n+1) → R (the left hand side) is a general function whileR (·; a, b) : [0,∞)→ R (the right hand side) is monotonic nondecreasing on [0,∞).If g : [a, b]→ R is such that g(n−1) is absolutely continuous and(1.4) −∞ < γ ≤ g(n) ≤ Γ <∞ on [a, b] ,where γ,Γ are real numbers, thensupPn∈P◦nL(g − γ + Γ2Pn, g(1) − γ + Γ2P (1)n , . . . , g(n) − γ + Γ2P (n)n ; a, b)≤ R(Γ− γ2; a, b).In order to exemplify the above principle, the author has given various exam-ples classified as “inequalities of the 0th−degree”, “1st−degree” and, in general,“nth−degree”, where n is the maximal order of the derivative involved in the orig-inal inequality. To motivate our further exploration, we mention here only someresults from the class of “0th−degree” and for the Riemann-Stieltjes integral asfollows:Theorem 2. Let f : [a, b]→ R be a continuous function with the property that(1.5) −∞ < m ≤ f (x) ≤M <∞ for any x ∈ [a, b]and u a function of bounded variation such that u (a) = u (b) . Then(1.6)∣∣∣∣∣∫ baf (t) du (t)∣∣∣∣∣ ≤ 12 (M −m)b∨a(u) .The constant 12 is sharp in (1.6).Theorem 3. Let f, l : [a, b]→ R be two continuous functions on [a, b] . If f satisfies(1.5) and u is of bounded variation such that(1.7)∫ bal (t) du (t) = 0,then(1.8)∣∣∣∣∣∫ baf (t) l (t) du (t)∣∣∣∣∣ ≤ 12 (M −m) ‖l‖[a,b],∞b∨a(u) .The constant 12 is sharp in (1.8).As a corollary of the above, we stated in [1] that the following Gru¨ss type in-equality obtained in [2] can be stated:PERTURBED MEDIAN PRINCIPLE 3Corollary 1. Let f, g be continuous on [a, b] , f satisfies (1.5) and u is of boundedvariation and such that u (b) 6= u (a) . Then(1.9)∣∣∣∣∣ 1u (b)− u (a)∫ baf (t) g (t) du (t)− 1u (b)− u (a)∫ baf (t) du (t) · 1u (b)− u (a)∫ bag (t) du (t)∣∣∣∣∣≤ 12(M −m) 1|u (b)− u (a)|∥∥∥∥∥g − 1u (b)− u (a)∫ bag (s) du (s)∥∥∥∥∥[a,b],∞b∨a(u) .The constant 12 is best possible in (1.9).2. A Perturbed Version of the Median PrincipleFor two real numbers δ,∆ ∈ R with ∆ 6= δ, we consider the set of all functionsf, g : [a, b]→ R given by:(2.1) M[a,b] (δ,∆) :={(f, g)∣∣∣∣∣∣∣∣f (x)− δ +∆2 g (x)∣∣∣∣≤ 12|∆− δ| |g (x)| for any x ∈ [a, b]}.We observe, for instance, if ∆ > δ and g (x) 6= 0 for x ∈ [a, b] , thenM[a,b] (δ,∆)contains the pair of functions (f, g) satisfying the condition(2.2)∣∣∣∣f (x)g (x) − δ +∆2∣∣∣∣ ≤ 12 (∆− δ) , for all x ∈ [a, b]or, equivalently, the condition:(2.3) δ ≤ f (x)g (x)≤ ∆, for all x ∈ [a, b] .Moreover, if we assume that g (x) > 0 for any x ∈ [a, b] , then the condition (2.3)is equivalent with(2.4) δg (x) ≤ f (x) ≤ ∆g (x) , for all x ∈ [a, b] .In practical application, the assumptions (2.3) or (2.4) are natural to impose.However, for the sake of generality, we consider, in the following, the classM[a,b] (δ,∆)which is larger and can be extended to the complex case as well.We are able now to state a perturbed version for the Median Principle of 0th−degree:Lemma 1. Assume that the following inequality holds:(2.5) L(h, h(1), ..., h(n); a, b)≤ R(∣∣∣h(n)∣∣∣ ; a, b) , (n ≥ 0)where L (·, ..., ·; a, b) : Rn+1 → R (the left hand side) is a general function andR (·; a, b) : [0,∞) → R (the right hand side) is monotonic nondecreasing [0,∞).4 S.S. DRAGOMIRIf f, g are such that f (n−1), g(n−1) are absolutely continuous and(f (n), g(n)) ∈M[a,b] (δn,∆n) , for some real numbers δn 6= ∆n, then(2.6) L(f − δn +∆n2g, ..., f (n) − δn +∆n2g(n); a, b)≤ R(12|∆n − δn|∣∣∣g(n)∣∣∣ ; a, b) .Proof. Since(f (n), g(n)) ∈M[a,b] (δn,∆n) , then, by (2.1),(2.7)∣∣∣∣f (n) − δ +∆2 g(n)∣∣∣∣ ≤ 12 |∆n − δn| ∣∣∣g(n)∣∣∣which implies, by making use of (2.5) for h = f − δn+∆n2 g, thatL(f − δn +∆n2g, ..., f (n) − δn +∆n2g(n); a, b)≤ R(∣∣∣∣f (n) − δn +∆n2 g(n)∣∣∣∣ ; a, b) .Further, by utilising the monotonicity property of R (·; a, b) and (2.7) we deducethe desired result (2.6).3. Some Examples for 0th−Degree InequalitiesThe main aim of the results obtained below are to provide various examples forhow the above principle can be utilised in practice in order to derive inequalitieswhen the upper bounds are expressed in terms of the difference between the supre-mum and the infimum of a function. These inequalites are called the ”0th−degree”inequalities.The following result concerning the Riemann-Stieltjes integral holds.Theorem 4. Let (f, g) ∈M[a,b] (δ,∆) and u : [a, b]→ R be such that the Riemann-Stieltjes integrals∫ baf (t) du (t) and∫ bag (t) du (t) exist. Then(3.1)∣∣∣∣∣∫ baf (t) du (t)− δ +∆2∫ bag (t) du (t)∣∣∣∣∣≤ 12|∆− δ| ×‖g‖[a,b],∞∨ba (u) provided g ∈ C [a, b] and u ∈ BV [a, b] ;∫ ba|g (x)| du (x) provided g ∈ C [a, b] and u ∈M↗ [a, b] ;L ‖g‖[a,b],1 provided g ∈ R [a, b] and u ∈ LipL [u, b] ;where C [a, b] is the class of continuous functions on [a, b] , BV [a, b] the class ofbounded variation functions on [a, b] ,M↗ [a, b] is the class of monotonic nonde-creasing functions on [a, b] while LipL [u, b] is the class of Lipschitzian functionswith the constant L > 0. All the inequalities in (3.1) are sharp.Proof. The following result is well known for the Riemann-Stieltjes integral: Ifp : [a, b] → R is continuous on [a, b] and v : [a, b] → R is of bounded variation on[a, b] , then the Riemann-Stieltjes integral∫ bap (x) dv (x) exists and(3.2)∣∣∣∣∣∫ bap (x) dv (x)∣∣∣∣∣ ≤ maxx∈[a,b] |p (x)|b∨a(u) ,PERTURBED MEDIAN PRINCIPLE 5where∨ba (v) denotes the total variation of u on [a, b] .Now, since (f, g) ∈ M[a,b] (δ,∆) , then on applying Lemma 1 for the inequality(3.2) we can state∣∣∣∣∣∫ baf (t) du (t)− δ +∆2∫ bag (t) du (t)∣∣∣∣∣ =∣∣∣∣∣∫ ba[f (t)− δ +∆2g (t)]du (t)∣∣∣∣∣≤ supt∈[a,b]∣∣∣∣f (t)− δ +∆2 g (t)∣∣∣∣ b∨a(u)≤ 12|∆− δ| ‖g‖[a,b],∞b∨a(u)and the first inequality in (3.1) is proved.If p : [a, b] → R is continuous and v : [a, b] → R is monotonic nondecreasing,then the Riemann-Stieltjes integral∫ bap (x) dv (x) obviously exists and(3.3)∣∣∣∣∣∫ bap (x) dv (x)∣∣∣∣∣ ≤∫ ba|p (x)| dv (x) .Also, if p : [a, b] → R is Riemann integrable on [a, b] and v : [a, b] → R isl−Lipschitzian, then the Riemann-Stieltjes integral ∫ bap (x) dv (x) exists and(3.4)∣∣∣∣∣∫ bap (x) du (x)∣∣∣∣∣ ≤ L∫ ba|p (x)| dx.Now, on applying Lemma 1 for these two inequalities we obtain the desired results.To prove the sharpness of the inequalities in (3.1), we assume, for instance, thatf (x) = ∆g (x) , x ∈ [a, b] .In this situation, the inequality (3.1) is obviously equivalent with∣∣∣∣∣∫ bag (x) du (x)∣∣∣∣∣ ≤‖g‖[a,b],∞∨ba (u) ;∫ bag (x) du (x) ;L∫ ba|g (x)| dxwhich are obviously sharp inequalities.The above result has many particular instances of interest.Corollary 2. Let f : [a, b]→ R be such that there exist the constants M > m with(3.5) −∞ < m ≤ f (t) ≤M <∞ for any t ∈ [a, b] .6 S.S. DRAGOMIRThen(3.6)∣∣∣∣∣∫ baf (t) du (t)− m+M2[u (b)− u (a)]∣∣∣∣∣≤ 12(M −m)×∨ba (u) if f ∈ C [a, b] and u ∈ BV [a, b] ;[u (b)− u (a)] if f ∈ C [a, b] and u ∈M↗ [a, b] ;(b− a)L if f ∈ R [a, b] and u ∈ LipL [u, b] .Another simple result is the following one.Corollary 3. Assume that there exist the constants l, L such that:(3.7) −∞ < l ≤ f (t)− f (x)t− x ≤ L <∞for t, x ∈ [a, b] with t 6= x. Then(3.8)∣∣∣∣∣∫ baf (t) du (t)− f (x) [u (b)− u (a)]− l + L2[(x− a)u (a) + (b− x)u (b)−∫ bau (t) dt]∣∣∣∣∣≤ 12(L− l)×[12 (b− a) +∣∣x− a+b2 ∣∣]∨ba (u) if u ∈ BV [a, b] ;[bu (b) + au (a)− x [u (a) + u (b)]+∫ basgn (x− t)u (t) dt]if u ∈M↗ [a, b] ;K[14 +(x− a+b2b−a)2](b− a)2 if u ∈ LipL [u, b] .Proof. The condition (3.7) obviously implies that∣∣∣∣f (t)− f (x)− l + L2 (t− x)∣∣∣∣ ≤ 12 (L− l) |t− x|for t, x ∈ [a, b] .If we fix x ∈ [a, b] and apply Theorem 4 for fx (t) = f (t)− f (x) , gx (t) = t− xand δ = γ, ∆ = Γ, we get∣∣∣∣∣∫ baf (t) du (t)− f (x) [u (b)− u (a)](3.9)− l + L2[(x− a)u (a) + (b− x)u (b)−∫ bau (t) dt]∣∣∣∣∣PERTURBED MEDIAN PRINCIPLE 7≤ 12(L− l)×supt∈[a,b] |t− x|∨ba (u) if u ∈ BV [a, b] ;∫ ba|t− x| du (t) if u ∈M↗ [a, b] ;K∫ ba|t− x| dt if u ∈ LipL [u, b] .=12(L− l)×[12 (b− a) +∣∣x− a+b2 ∣∣]∨ba (u) ;[bu (b) + au (a)− x [u (a) + u (b)]+∫ basgn (x− t)u (t) dt];K[14 +(x− a+b2b−a)2](b− a)2since, a simple integration by parts shows that∫ ba|t− x| du (t) =∫ xa(x− t) du (t) +∫ bx(t− x) du (t)= bu (b) + au (a)− x [u (a) + u (b)] +∫ basgn (x− t)u (t) dt.On the other hand,∫ ba(t− x) du (t) = (b− x)u (b) + (x− a)u (a)−∫ bau (t) dt,which, together with (3.9) produces the desired result (3.8).Remark 1. If f : [a, b]→ R is continuous on [a, b] and differentiable on [a, b] andif γ = inft∈[a,b] f ′ (t) , Γ = supt∈[a,b] f ′ (t) , then by Lagrange’s theorem, we haveγ ≤ f (t)− f (x)t− x ≤ Γfor any x, t ∈ [a, b] with t 6= x, and the inequality (3.8) holds true for these γ andΓ.The following result may be stated as well:Corollary 4. Assume that there exists the constants γ,Γ so that:(3.10)∣∣∣∣f (t)− f (x)− γ + Γ2 |t− x|α∣∣∣∣ ≤ 12 |Γ− γ| |t− x|α8 S.S. DRAGOMIRfor a given x ∈ [a, b] and α > 0 for each t ∈ [a, b] . Then(3.11)∣∣∣∣∣∫ baf (t) du (t)− f (x) [u (b)− u (a)]− γ + Γ2[(x− a)α u (a) + (b− x)α u (b)− α∫ ba|t− x|α−1 u (t) dt]∣∣∣∣∣≤ 12|Γ− γ| ×[12 (b− a) +∣∣x− a+b2 ∣∣]α∨ba (u) if u ∈ BV [a, b] ;[(x− a)α u (a) + (b− x)α u (b)−α ∫ ba(t− x)α−1 u (t) dt]if u ∈M↗ [a, b] ;Lα+1[(b− x)α+1 + (x− a)α+1]if u ∈ LipL [u, b] .Proof. Follows by applying Theorem 4 for δ = γ, ∆ = Γ and gx (t) = |t− x|α withx fixed in [a, b] . The details are omitted.Remark 2. The above result contains some cases of interest, for instance whenx = a, x = b or x = a+b2 . To be more specific, we choose only one case to exemplify.Let us assume that there exist φ,Φ ∈ R such that∣∣∣∣f (t)− f (a+ b2)− φ+Φ2∣∣∣∣t− a+ b2∣∣∣∣α∣∣∣∣ ≤ 12 |Φ− φ|∣∣∣∣t− a+ b2∣∣∣∣αfor any t ∈ [a, b] , where α > 0 is given. Then(3.12)∣∣∣∣∣∫ baf (t) du (t)− f(a+ b2)[u (b)− u (a)]− φ+Φ2[(b− a)α2α[u (b) + u (a)]− α∫ ba∣∣∣∣t− a+ b2∣∣∣∣α−1 u (t) dt]∣∣∣∣∣≤ 12|Φ− φ| ×(b−a)α2α∨ba (u) if u ∈ BV [a, b] ;[(b−a)α2α [u (b) + u (a)]− α · ∫ ba∣∣t− a+b2 ∣∣α−1 u (t) dt] if u ∈M↗ [a, b] ;L(b−a)α+12α(α+1) if u ∈ LipL [u, b] .The following result can be stated as well.Corollary 5. Assume that there exist the constants n,N ∈ R such that:∣∣∣∣∣f (x)− 1b− a∫ baf (s) ds− n+N2· g (x)∣∣∣∣∣ ≤ 12 |N − n| |g (x)|PERTURBED MEDIAN PRINCIPLE 9for any x ∈ [a, b] . Then(3.13)∣∣∣∣∣∫ baf (t) du (t)− u (b)− u (a)b− a∫ baf (s) ds− n+N2∫ bag (t) du (t)∣∣∣∣∣≤ 12|N − n| ×‖g‖[a,b],∞∨ba (u) provided g ∈ C [a, b] and u ∈ BV [a, b] ;∫ ba|g (x)| du (x) provided g ∈ C [a, b] and u ∈M↗ [a, b] ;L ‖g‖[a,b],1 provided g ∈ R [a, b] and u ∈ LipL [u, b] .4. Inequalities of the 1st−DegreeAn inequality that contains at most the first derivative of the involved functionswill be called an inequality of the 1st−degree.If one would like examples of such inequalities for two functions, the followingOstrowski’s inequality obtained in [3] is the most suitable(4.1) |C (h, l; a, b)| ≤ 18(b− a) (M −m) ‖h′‖[a,b],∞ ,where C (h, l; a, b) is the Cˇebysˇev functional given byC (h, l; a, b) :=1b− a∫ bah (x) l (x) dx− 1b− a∫ bah (x) dx · 1b− a∫ bal (x) dxand −∞ < m ≤ h (x) ≤ M <∞ for a.e. x ∈ [a, b] while l is absolutely continuousand such that l′ ∈ L∞ [a, b] . The constant 18 is sharp.Another example of such an inequality is the Cˇebysˇev one(4.2) |C (h, l; a, b)| ≤ 112(b− a)2 ‖h′‖[a,b],∞ ‖l′‖[a,b],∞ ,provided h, l are absolutely continuous and h′, l′ ∈ L∞ [a, b] . The constant 112 hereis sharp.The following result holds:Theorem 5. Let f : [a, b]→ R be a Lebesgue integrable function such that −∞ <m ≤ f (x) ≤ M < ∞ for a.e. x ∈ [a, b] . If g and h are absolutely continous andsuch that there exists the constants δ and ∆ such that (g′, h′) ∈M[a,b] (δ,∆) , thenwe have the following perturbed version of the Ostrowski’s inequality:(4.3)∣∣∣∣C (f, g; a, b)− ∆+ δ2 C (f, h; a, b)∣∣∣∣ ≤ 116 (b− a) (M −m) (∆− δ) ‖h′‖[a,b],∞ .Proof. By applying Lemma 1 to the Ostrowski’s inequality (4.1) we have∣∣∣∣C (f, g − ∆+ δ2 h; a, b)∣∣∣∣ ≤ 18 (b− a) (M −m)∥∥∥∥12 (∆− δ)h′∥∥∥∥[a,b],∞which is clearly equivalent with the desired result (4.3).Finally, we have the following perturbed version of the Cˇebysˇev inequality10 S.S. DRAGOMIRTheorem 6. Let f, g, h, l be absolutely continuous on [a, b] . If there exists theconstants ϕ,Φ, δ and ∆ such that (f ′, l′) ∈M[a,b] (ϕ,Φ) and (g′, h′) ∈M[a,b] (δ,∆) ,then ∣∣∣∣C (f, g; a, b)− ∆+ δ2 C (f, h; a, b)(4.4)−Φ+ ϕ2C (l, g; a, b) +∆+ δ2· Φ+ ϕ2C (h, l; a, b)∣∣∣∣≤ 148(b− a)2 (Φ− ϕ) (∆− δ) ‖h′‖[a,b],∞ ‖l′‖[a,b],∞ .Similar results can be obtained for other inequalities involving the derivativesup to an order n ≥ 1. However, the details are left to the interested reader.References[1] S.S. Dragomir, The median principle for inequalities and applications, in Func-tional Equations, Inequalities and Applications, Ed. by Th.M. Rassias, KluwerAcad.Publ., 2003. Preprint, RGMIA Res.Rep. Coll., 5(2002), Supplement, Article 17.[http://www.staff.vu.edu.au/RGMIA/v5(E).asp].[2] S.S. Dragomir, Improvements of Ostrowski and generalised trapezoid inequality in termsof the upper and lower bounds of the first derivative, Tamkang J. Math., 34(3)(2003), 213-222. Preprint, RGMIA Res.Rep. Coll., 5(2002), Supplement, Article 10.[http://www.staff.vu.edu.au/RGMIA/v5(E).asp].[3] A. Ostrowski, On an integral inequality, Aequat. Math., 4(1970), 358-373.Research Group in Mathematical Inequalities & Applications, School of Engineeringand Science, Victoria University, PO Box 14428, Melbourne City, VIC, Australia. 8001E-mail address: sever.dragomir@vu.edu.auURL: http://www.staff.vu.edu.au/rgmia/dragomir/

The Perturbed Median Principle for Integral Inequalities with Applications

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