In this paper, we study the Hadamard’s inequality for midconvex and quasi-midconvex functions

in topological groups. A mapping naturally connected with this inequality and related

result is also pointed out

Morassaei, Ali

English

Victoria University Eprints Repository

ON HADAMARD’S INEQUALITIES FOR THE CONVEX MAPPINGS DEFINED INTOPOLOGICAL GROUPS AND CONNECTED RESULTAli MorassaeiDepartment of Mathematics,Faculty of Sciences,University of Zanjan,Zanjan 45195-313, Iran.morassaei@znu.ac.irAbstractIn this paper, we study the Hadamard’s inequality for midconvex and quasi-midconvex func-tions in topological groups. A mapping naturally connected with this inequality and relatedresult is also pointed out. 11 IntroductionLet f : I −→ R be a convex mapping defined on the interval I of real numbers and a, b ∈ I witha < b. The following double inequality:f(a+ b2)≤ 1b− a∫ baf(x)dx ≤ f(a) + f(b)2(1)is known in the literature as Hadamard’s inequality for convex mapping. Note that some of theclassical inequalities for means can be derived from (1) for appropriate particular selections of themapping f .In the paper [4] (see also [5] and [6]) is considered the following mapping naturally connectedwith Hadamard’s results:H : [0, 1] −→ R, H(t) = 1b− a∫ baf(tx+ (1− t)a+ b2)dx.The following properties of H hold:(i) H is convex and monotonic nondecreasing.12000 AMS Subject Classification: Primary 26A51, 22A10, Secondary 26D07, 26D15.Key words and phrases: convex functions, Hadamard’s inequality.The author is supported in part by the Institute for Advanced Studies in Basic Sciences, Zanjan, Iran.1(ii) One has the boundssupt∈[0,1]H(t) = H(1) =1b− a∫ baf(x)dxandinft∈[0,1]H(t) = H(0) = f(a+ b2).Another mapping also closely connected with Hadamard’s inequality is the following one [5] (seealso [6]):F : [0, 1] −→ R, F (t) = 1(b− a)2∫ ba∫ baf(tx+ (1− t)y)dxdy.The properties of this mapping are the following ones:(i) F is convex and monotonic nonincreasing on [0, 12 ] and nondecreasing on [12 , 1];(ii) F is symmetrical relating the element 12 . That is,F (t) = F (1− t) for all t ∈ [0, 1];(iii) One has the boundssupt∈[0,1]F (t) = F (0) = F (1) =1b− a∫ baf(x)dxandinft∈[0,1]F (t) = F(12)=1(b− a)2∫ ba∫ baf(x+ y2)dxdy ≥ f(a+ b2)(iv) The following inequality holds:F (t) ≥ max{H(t),H(1− t)} for all t ∈ [0, 1].Generalization of (1) for quasi convex functions defined on the real line is also well-known. Itwas established in [2] that for a quasi convex function f defined on [a, b] we havef(a+ b2)≤ 2b− a∫ baf(x)dx.In this paper we shall study generalizations of the left side of (1) inequality for some convex functionsdefined on an open subset of a topological group G.2 A Secondary ResultGeneralization of the left side of (1) for convex functions defined on a convex subset of Rn is well-known. For example, if X ⊂ Rn is a convex bounded symmetrical set (the latter means thatx ∈ X =⇒ −x ∈ X), than (c.f. [8])f(0) ≤ 1µ(X)∫Xf(x)dx (2)2for each lower semicontinuous convex function f : X −→ R, where µ(X) is the volume of the set X.To show (2), consider the transformation of the Rn in itself given by:h : Rn → Rn, h = (h1, h2, · · · , hn),andhi(x1, x2, · · · , xn) = −xi, i = 1, 2, · · · , n.Then h(X) = X and sinceD(h1, h2, · · · , hn)D(x1, x2, · · · , xn) =∣∣∣∣∣∣∣∣∣∣∣−1 0 0 · · · 00 −1 0 · · · 00 0 −1 · · · 0............0 0 0 · · · −1∣∣∣∣∣∣∣∣∣∣∣= (−1)n.we have the change variable:∫Xf(x1, x2, · · · , xn)dx1dx2 · · · dxn=∫Xf (h1(x1, x2, · · · , xn), · · · , hn(x1, x2, · · · , xn))∣∣∣∣D(h1, h2, · · · , hn)D(x1, x2, · · · , xn)∣∣∣∣ dx1dx2 · · · dxn=∫Xf(−x1,−x2, · · · ,−xn)dx1dx2 · · · dxn.Now, by the convexity of f on X we also have:f(0, 0, · · · , 0) = f(x1 − x12,x2 − x22, · · · , xn − xn2)= f((x1, x2, · · · , xn) + (−x1,−x2, · · · ,−xn)2)≤ 12[f(x1, x2, · · · , xn) + f(−x1,−x2, · · · ,−xn)]which gives, by integration of f on X, that:∫Xf(0, 0, · · · , 0)dx1dx2 · · · dxn≤ 12[∫Xf(x1, x2, · · · , xn)dx1dx2 · · · dxn +∫Xf(−x1,−x2, · · · ,−xn)dx1dx2 · · · dxn]=∫Xf(x1, x2, · · · , xn).Consequently, we getf(0, 0, · · · , 0)µ(X) ≤∫Xf(x1, x2, · · · , xn)dx1dx2 · · · dxnand thusf(0, 0, · · · , 0) ≤ 1µ(X)∫Xf(x1, x2, · · · , xn)dx1dx2 · · · dxn.33 Hadamard’s InequalityIn this section we prove the Hadamard’s inequality for midconvex and quasi-midconvex functions intopological groups.Let G be a topological group, and Ω is a nonempty open subset of G and f is a real-valuedfunction on Ω. We say that f is globally (right) midconvex if2f(a) ≤ f(az) + f(az−1) (3)for all a, z ∈ G such that a, az, az−1 ∈ Ω. Also, we say that f is locally (right) midconvex in a ∈ Ωif there exists an open symmetric set V = V −1 from e such that2f(a) ≤ f(az) + f(az−1) (4)for all z ∈ G such that az, az−1 ∈ Ω [1]. Also, f is called quasi-(right)midconvex, iff(az) ≤ max{f(a), f(az2)} (5)for every a, z ∈ G so that a, az, az2 ∈ Ω [7]. Note that a is midpoint of az−1 and az, and az ismidpoint of a and az2.Definition 1. Let Ω be an open subset of topological group G, and a ∈ G. Ω is said tobe symmetric relative to a, if a−1Ω is symmetric and e ∈ a−1Ω.Definition 2. Let G be a topological group and Ω ⊂ G is an open set. A function ω : Ω −→ Ris called symmetric relative to a ∈ G, if∀z ∈ G; az, az−1 ∈ Ω ω(az) = ω(az−1).The following theorems hold:Theorem 1. Let G be a locally compact group and Ω ⊂ G is an open symmetric set relativeto a ∈ G with 0 < µ(Ω) < ∞. If f : Ω −→ R is measurable and locally midconvex in a andalso, f ∈ L1(Ω), and ω : Ω −→ R is nonnegative and symmetric to a and ω ∈ L1(Ω) such thatfω ∈ L1(Ω), thenf(a)∫Ωω(az)dµ(z) ≤∫Ωf(az)ω(az)dµ(z),where that µ is Haar measure on G.Proof. Since f is locally midconvex in a, so2f(a) ≤ f(az) + f(az−1)for any z ∈ Ω, by (4). Since ω is nonnegative and symmetric relative to a, thus2f(a)ω(az) ≤ f(az)ω(az) + f(az−1) + ω(az)= f(az)ω(az) + f(az−1) + ω(az−1).4Integrating this inequality on Ω, we get∫Ω2f(a)ω(az)dµ(z) ≤∫Ωf(az)ω(az)dµ(z) +∫Ωf(az−1)ω(az−1)dµ(z)=∫a−1Ωf(z)ω(z)dµ(z) +∫a−1Ωf(z−1)ω(z−1)dµ(z)=∫Gf(z)ω(z)χa−1Ω(z)dµ(z) +∫Gf(z−1)ω(z−1)χa−1Ω(z)dµ(z)=∫Gf(z)ω(z)χa−1Ω(z)dµ(z) +∫Gf(z−1)ω(z−1)χa−1Ω(z−1)dµ(z)= 2∫Gf(z)ω(z)χa−1Ω(z)dµ(z).= 2∫a−1Ωf(z)ω(z)dµ(z).Consequently, we havef(a)∫Ωω(az)dµ(z) ≤∫Ωf(az)ω(az)dµ(z).Remark 1. If in the above theorem, set a = e and ω ≡ 1 on Ω, we havef(e) ≤ 1µ(Ω)∫Ωf(z)dµ(z).This result is similar result of section 2.Theorem 2. Let G be a locally compact group and Ω ⊂ G is an open symmetric set relativeto a ∈ G with 0 < µ(Ω) < ∞ and e ∈ Ω. If f is measurable and quasi-midconvex real-valuedfunction on Ω such that f ∈ L2(Ω) and also, ω : Ω −→ R is a nonnegative and symetric to a andω ∈ L2(Ω), thenf(a)∫Ωω(az)dµ(z) ≤∫Ωf(az)ω(az)dµ(z) + I(a) (6)whereI(a) =12∫Ω|f(az)− f(az−1)|ω(az)dµ(z).Furthermore, I(a) satisfies the inequalities:0 ≤ I(a) ≤ min{ ∫Ω|f(az)|ω(az)dµ(z),1√2(∫Ωf2(az)dµ(z)− ∫Ωf(az)f(az−1)dµ(z)) 12(∫Ωω2(az)dµ(z)) 12}. (7)Proof. Since Ω is symmetric set relative to a, thus for z in G, by (5), we havef(a) ≤ max{f(az), f(az−1)}where max{f(az), f(az−1)} = f(az)+f(az−1)+|f(az)−f(az−1)|2 and since ω is nonnegative and symmet-ric relative to a, therefore∫Ωf(a)ω(az)dµ(z) ≤ 12∫Ωf(az)ω(az)dµ(z) +12∫Ωf(az−1)ω(az−1)dµ(z)+12∫Ω|f(az)− f(az−1)|ω(az)dµ(z).5So,f(a)∫Ωω(az)dµ(z) ≤∫Ωf(az)ω(az)dµ(z) + I(a)and the inequality (6) is proved.We now observe that, by the Cauchy-Schwartz inequality,0 ≤ I(a) = 12∫Ω|f(az)− f(az−1)|ω(az)dµ(z)≤ 12(∫Ω(f(az)− f(az−1))2dµ(z)) 12(∫Ωω2(az)dµ(z)) 12=12(∫Ω[f2(az)− 2f(az)f(az−1) + f2(az−1)] dµ(z)) 12 (∫Ωω2(az)dµ(z)) 12=12(2∫Ω[f2(az)− f(az)f(az−1)]dµ(z)) 12(∫Ωω2(az)dµ(z)) 12=√22(∫Ωf2(az)dµ(z)−∫Ωf(az)f(az−1)dµ(z)) 12(∫Ωω2(az)dµ(z)) 12.On the other hands,I(a) ≤ 12(∫Ω|f(az)|ω(az)dµ(z) +∫Ω|f(az−1)|ω(az)dµ(z))=∫Ω|f(az)|ω(az)dµ(z)and the inequality (7) is proved. Definition 3. The function f : Ω −→ R is said to be P-function in Ω, iff(a) ≤ f(az) + f(az−1),for all a ∈ Ω and z ∈ G such that az, az−1 ∈ Ω.Theorem 3. Let G be a locally compact group and Ω ⊂ G is an open symmetric set relativeto a ∈ G with 0 < µ(Ω) < ∞. If f is measurable and P-function real-valued on Ω such thatf ∈ L1(Ω) and also, ω : Ω −→ R is nonnegative and symmetric to a and ω ∈ L1(Ω) such thatfω ∈ L1(Ω), thenf(a)∫Ωω(az)dµ(z) ≤ 2∫Ωf(az)ω(az)dµ(z).Proof. Since f is P -function, we havef(a)ω(az) ≤ f(az)ω(az) + f(az−1)ω(az).Integrating this inequality on Ω, we get∫Ωf(a)ω(az)dµ(z) ≤∫Ωf(az)ω(az)dµ(z) +∫Ωf(az−1)ω(az)dµ(z)=∫Ωf(az)ω(az)dµ(z) +∫Ωf(az−1)ω(az−1)dµ(z)= 2∫Ωf(az)ω(az)dµ(z)6Thus, f(a)∫Ωω(az)dµ(z) ≤ 2 ∫Ωf(az)ω(az)dµ(z). Now, for a globally midconvex function f : G −→ R that e ∈ Ω, we can define the mappingH : Ω −→ R,H(x) =1µ(Ω)∫Ωf(xz)dµ(z).The properties of this mapping are embodied in the following theorem:Theorem 4. Suppose that f : G −→ R is globally midconvex and Ω is an open symmetricsubset of G such that e ∈ Ω and 0 < µ(Ω) <∞. Then(i) The mapping H is globally midconvex on G, if G is abelian.(ii) f(e) ≤ H(e).Proof. (i) Assume that a, x, z ∈ G such that ax, ax−1 ∈ Ω, soH(ax) +H(ax−1) =1µ(Ω)∫Ωf(axz)dµ(z) +1µ(Ω)∫Ωf(ax−1z)dµ(z)=1µ(Ω)∫Ω[f(azx) + f(azx−1)]dµ(z)≥ 2µ(Ω)∫Ωf(az)dµ(z)= 2H(a)that is, H is globally midconvex.(ii) Since f is midconvex, by Theorem 1, f(e) ≤ 1µ(Ω)∫Ωf(z)dµ(z) = H(e). 4 ApplicationsIn this section, we study special cases of theorem 1, 2, and 3.Set G = R. Since R is an abelian additive group, thus, for all a, z ∈ R, a− z and a+ z are pointsthat a is its midpoint. Now, if let a− z = x and a+ z = y, then a = x+y2 . Consequently, definitionsof globally midconvex and quasi-midconvex functions are to be written as follows:f(x+ y2)≤ f(x) + f(y)2, (f globally midconvex)f(x+ y2)≤ max{f(x), f(y)}. (f quasi midconvex)Application 1. If in the Theorem 1 let G = R and Ω = [−a, a], we havef(0)∫ a−aω(x)dx ≤∫ a−af(x)ω(x)dx,and if we set ω ≡ 1 on [−a, a], we getf(0) ≤ 12a∫ a−af(x)dx7that is special case of (2), where n = 1.Application 2. If in the Theorem 1, let G = Rn with an additive operation and Ω = X is anopen bounded symmetric and convex subset of Rn, then the result of section 2 holds.References[1] A. Chademan and F. Mirzapour, Midconvex functions in locally compact groups, Proc.Amer. Math. Soc. 127(10), (1999) 2961-2968.[2] S. S. Dragomir and C. E. M. Pearce, Quasi-convex functions and Hadamard’s inequali-ties, Bull. Austral. Math. Soc. 57, (1998) 377-385.[3] S. S. Dragomir and C. E. M. Pearce, Select topics on Hermite-Hadamard inequalities andapplications, Melbourne and Adelaide, 2000.[4] S. S. Dragomir, A mapping in connection to Hadamard’s inequality, An Ostro. Akad. Wiss.Math. -Natur (Wien) 128 (1991) 17-20.[5] S. S. Dragomir, Two mappings in connection to Hadamard’s inequality, J. Math. Anal. Appl.,167 (1992) 49-56.[6] S. S. Dragomir, On Hadamard’s inequality for convex functions, Math. Balkanica, 6 (1992)215-222.[7] F. Mirzapour and A. Morassaei, Quasi-convex functions in topological groups, Int. J. Appl.Math. 16(4) (2004), 281-290.[8] A. M. Rubinov, Abstract convexity and global optimization, Kluwer Academic Publishers,Dordrecht, 2000.8

A mapping in connection to Hadamard’s inequality,

Abstract convexity and global optimization,

Midconvex functions in locally compact groups,

On Hadamard’s inequality for convex functions,

Quasi-convex functions and Hadamard’s inequalities,

Quasi-convex functions in topological groups,

Select topics on Hermite-Hadamard inequalities and applications,

Two mappings in connection to Hadamard’s inequality,

On Hadamard's Inequalities for the Convex Mappings Defined in Topological Groups and Connected Result

In this paper, we study the Hadamard’s inequality for midconvex and quasi-midconvex functions\ud
in topological groups. A mapping naturally connected with this inequality and related\ud
result is also pointed out

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On Hadamard's Inequalities for the Convex Mappings Defined in Topological Groups and Connected Result

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