In this paper we establish different refinements and applications of the Hermite–Hadamard inequality for convex functions in the context of NPC global spaces

Conde, Cristian

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Annales Mathematicae Silesianae 32 (2018), 133–144DOI: 10.1515/amsil-2017-0015REFINEMENTS OF THE HERMITE–HADAMARDINEQUALITY IN NPC GLOBAL SPACESCristian CondeAbstract. In this paper we establish different refinements and applications ofthe Hermite–Hadamard inequality for convex functions in the context of NPCglobal spaces.1. IntroductionRecall that f : [a, b]→ R is called a convex function if it holdsf(λx+ (1− λ)y) ≤ λf(x) + (1− λ)f(y)for all x, y ∈ [a, b] and 0 ≤ λ ≤ 1. It is well known that every convex functioncan be modified at the endpoints to become convex and continuous. An im-mediate consequence of this remark is the integrability of f . The integral off can then be estimated by:(1.1) f(a+ b2)≤ 1b− a∫ baf(t) dt ≤ f(a) + f(b)2.This fundamental inequality is well known as the Hermite–Hadamard (H-H)inequality, it was first published by Hermite in 1883 and independently provedReceived: 21.04.2017. Accepted: 21.09.2017. Published online: 31.01.2018.(2010) Mathematics Subject Classification: 39B62, 32F17, 26E60.Key words and phrases: nonpositive curvature metric space, geodesic convexity, shortgeodesic, Hermite–Hadamard inequality, means.134 Cristian Condein 1893 by Hadamard in [12]. One can easily observe that (1.1) is equivalentto the following inequality:f(a+ b2)≤∫ 10f(ta+ (1− t)b) dt ≤ f(a) + f(b)2.From the previous statement, we have concluded in [7] that the notions orconcepts necessaries to obtain an analogue of the H-H inequality in a globalNPC space N are: the existence of an unique geodesic connecting two givenpoints, the notion of convex function associated with this privileged curve andconvex sets. Since these concepts exist in the context of global NPC spaces,we obtained a generalization of the H-H inequality to such spaces.The Hermite–Hadamard inequality has several applications. We would liketo refer the reader to [10, 2, 16, 21, 19, 13, 1, 22, 6, 3] and references thereinfor more information. In this paper we refine one side of H-H inequality inNPC global spaces via sequences.2. PreliminariesThe aim of our paper is to discuss refinements of the H-H inequality to theframework of spaces with global nonpositive curvature. A formal definition ofthese spaces is as follows.Definition 2.1. Let (N, d) be a complete metric space, which is also ageodesic length space in the sense that the distance of N can be computedvia the infimum of the length of the rectifiable arcs joining given endpointsin N (see [14, Section 2.2]). We say that (N, d) is a global NPC space if forx1, x2 ∈ N there exists a point z ∈ N such that for each x ∈ N we haved2(x, z) ≤ 12d2(x, x1) +12d2(x, x2)−14d2(x1, x2).These spaces are also known as the Cat(0) spaces. In a global NPC spaceeach pair of points x1, x2 ∈ N can be connected by a geodesic, i.e. by arectifiable curve γ : [0, 1]→ N such that the length of γ|[s,t] is d(γ(s), γ(t)) forall 0 ≤ s ≤ t ≤ 1. Moreover, this geodesic is unique. The point z that appearsin Definition 2.1 is the midpoint of x1 and x2 and has the propertyd(x1, z) = d(x2, z) =12d(x1, x2).Refinements of the Hermite–Hadamard Inequality in NPC Global Spaces 135Example 2.2. (1) Every Hilbert space is a global NPC space. Its geode-sics are the line segments.(2) A Riemannian manifold (M, g) is a global NPC space if and only if it iscomplete, simply connected, and of nonpositive sectional curvature.(3) Let Sym+(n,R) be the set of all n× n dimensional positive definite ma-trices with real coefficients and for A,B ∈ Sym+(n,R) we consider themetricd(A,B) =(∑klog2 λk)1/2,where λ1, . . . , λn are the eigenvalues of AB−1. Then (Sym+(n,R), d) is aglobal NPC space (see [17]).Definition 2.3. A subset C ⊆ N is called convex if for each geodesicγ : [0, 1]→ N joining two arbitrary points in C holds that γ([0, 1]) ⊆ C.A function f : C → R is called convex if the function f ◦ γ : [0, 1] → R isconvex whenever γ : [0, 1]→ C is geodesic, that is, for all t ∈ [0, 1]f(γ(t)) ≤ (1− t)f(γ(0)) + tf(γ(1)).For sake of completeness, we recall two inequalities obtained in a previouswork (see [7]). These statements formulate different refinements of the H-Hinequality in global NPC spaces.Theorem 2.4 ([7, Theorem 3.3]). Let (N, d) be a global NPC space, C ⊆ Na convex set and f : C → R a convex function. Thenf(γ(1/2))≤ 12[f(γ(2/3)) + f(γ(1/3))]≤ 12[f(γ(1/4)) + f(γ(3/4))]≤∫ 10f(γ(t)) dt≤ 12[f(γ(1/2))+f(γ(0)) + f(γ(1))2]≤ f(γ(0)) + f(γ(1))2(2.1)andf(γ(1/2))≤ l(λ) ≤∫ 10f(γ(t))dt ≤ L(λ) ≤ f(γ(0)) + f(γ(1))2136 Cristian Condefor all geodesic γ : [0, 1]→ C and λ ∈ [0, 1], wherel(λ) = λf(γ(λ2))+ (1− λ)f(γ(1 + λ2))andL(λ) =12[f(γ(λ)) + λf(γ(0)) + (1− λ)f(γ(1))].Remark 2.5. Note that the following inequality is valid for a convexfunction f : C → R, where C ⊆ N is a convex set:∫ 10f(γ(t))dt− l(λ) ≤ L(λ)−∫ 10f(γ(t))dt.The proof is a consequence of the bounds of the normalized Jensen func-tional (see [8, Corollary 1] or [20, Corollary 3.1]), more precisely: let f be aconvex function on [a, b]. Then2λmin(f(x1) + f(x2)2− f(x1 + x22))≤ λf(x1) + (1− λ)f(x2)− f(λx1 + (1− λ)x2)≤ 2λmax(f(x1) + f(x2)2− f(x1 + x22)),where 0 < λ < 1, λmin = min{λ, 1 − λ}, λmax = max{λ, 1 − λ} andx1, x2 ∈ [a, b].3. Refinements of Hermite–Hadamard inequalityWe begin this section with a statement that has motivated the other resultsobtained in this work.Lemma 3.1. Let f : [a, b]→ R be a convex function. Thenf(a+ b2)≤ 1b− a∫ baf(t) dt ≤ 12(f(a+ b2)+12f(a) +12f(b))≤ 116(2f(a+ b2)+ 7f(a) + 7f(b))≤ f(a) + f(b)2.Refinements of the Hermite–Hadamard Inequality in NPC Global Spaces 137Proof. Using convexity of the function f , we can easily verify the lastinequality. Now we will prove that:1b− a∫ baf(t) dt ≤ 12(f(a+ b2)+12f(a) +12f(b))≤ 116(2f(a+ b2)+ 7f(a) + 7f(b)).Let us note that the first inequality is well-known ([11, Theorem 1.1]), but weinclude its proof here for the sake of completeness. We have1b− a∫ baf(t) dt =1b− a(∫ a+b2af(t) dt+∫ ba+b2f(t) dt))≤ 1b− a(f(a) + f(a+b2 )2b− a2+f(b) + f(a+b2 )2b− a2)=14(2f(a+ b2)+ f(a) + f(b))=116(8f(a+ b2)+ 4f(a) + 4f(b))≤ 116(2f(a+ b2)+ 3(f(a) + f(b)) + 4f(a) + 4f(b))=116(2f(a+ b2)+ 7f(a) + 7f(b)). From now on in this section, we assume that (N, d) is a global NPC space,C ⊆ N a convex set, f : C → R a convex function and γ : [0, 1] → C ageodesic connecting γ(0) with γ(1). In this paper we establish a refinement ofinequality (2.1). For this let us first define the following sequences (motivatedby the previous Lemma 3.1) for any n ∈ N with n ≥ 2:xn =12[f(γ( 1n))+ f(γ(1− 1n))]andyn =2n+ 1f(γ(12))+[1− 2n+ 1]f(γ(0)) + f(γ(1))2.138 Cristian CondeRemark 3.2. In the particular case of R with the usual distance we havey3 =12(f(a+ b2)+12f(a) +12f(b))andy15 =116(2f(a+ b2)+ 7f(a) + 7f(b)).The following result is a refinement of the “arithmetic-geometric” inequal-ity.Proposition 3.3. Let xn and yn be sequences defined as above. Then(i) {xn}n≥2 and {yn}n≥2 are non-decreasing sequences,(ii) for n ≥ 2 we havef(γ(1/2))≤ xn ≤ yn ≤f(γ(0)) + f(γ(1))2.Proof. (i) The monotonicity of {xn}n≥2 is a consequence of the convexityof f and the following relations:γ( 1n)= γ([1− 1n(n− 1)] 1n+ 1+1n(n− 1)nn+ 1)andγ(1− 1n)= γ([1− 1n(n− 1)] nn+ 1+1n(n− 1)1n+ 1).On the other hand, {yn}n≥2 is a non-decreasing sequence as consequenceof (2.1).(ii) The first inequality is a consequence of the convexity of f and the factthat for any t0 ∈ [0, 1] the midpoint between γ(t0) and γ(1 − t0) is given byγ(1/2). Then in particular, if we consider t0 = 1/n, we obtain the relationf(γ(1/2)) ≤ 12[f(γ( 1n))+ f(γ(1− 1n))].To prove the second inequality, we use the fact that {xn}n≥2 is a non-decre-asing sequence, the convexity of f and the following equalities:γ(1n+ 1)= γ([1− 2n+ 1]0 +2n+ 112),Refinements of the Hermite–Hadamard Inequality in NPC Global Spaces 139γ(nn+ 1)= γ([1− 2n+ 1]+2n+ 112).Finally, the last inequality is a consequence of (2.1). With the sequences introduced above, we can rewrite the inequality (2.1)as follows:f(γ(1/2))= x2 ≤ x3 ≤ x4 ≤∫ 10f(γ(t)) dt ≤ y3 ≤f(γ(0)) + f(γ(1))2.From the previous statement, we obtain the following improvement of the H-Hinequality and refinement of Lemma 3.1.Theorem 3.4. Let (N, d) be a global NPC space, C ⊆ N a convex set,f : C → R a convex function and n ≥ 3. Thenf(γ(1/2))≤ x3 ≤ x4 ≤∫ 10f(γ(λ)) dλ ≤ y3 ≤ . . . yn ≤ . . .≤ f(γ(0)) + f(γ(1))2for all geodesic γ : [0, 1]→ C.Now we consider the following sequences, for n ≥ 2:zn =12(xn +f(γ(0)) + f(γ(1))2)andwn =12(yn +f(γ(0)) + f(γ(1))2).Proposition 3.5. Let zn, wn be the sequences defined above. Then(i) {zn}n≥2 and {zn}n≥2 are non-decreasing sequences,(ii) for n ≥ 2 we have zn ≤ wn,(iii) for n ≥ 2 we have12[f(γ(1/2))+f(γ(0)) + f(γ(1))2]= z2 ≤ . . . ≤ zn ≤ . . .≤ f(γ(0)) + f(γ(1))2140 Cristian Condeand for n ≥ 312[f(γ(1/2))+f(γ(0)) + f(γ(1))2]= w3 ≤ . . . ≤ wn ≤ . . .≤ f(γ(0)) + f(γ(1))2.Proof. It is a consequence of Proposition 3.3 and the fact that z2 =y3 = w3. Theorem 3.6. Let (N, d) be a global NPC space, C ⊆ N a convex set andf : C → R a convex function. Then for n ≥ 2f(γ(1/2))≤ x3 ≤ x4 ≤∫ 10f(γ(λ)) dλ ≤ z2 ≤ . . . zn ≤ . . .≤ f(γ(0)) + f(γ(1))2and for m ≥ 3f(γ(1/2))≤ x3 ≤ x4 ≤∫ 10f(γ(λ)) dλ ≤ w3 ≤ . . . wm ≤ . . .≤ f(γ(0)) + f(γ(1))2for all geodesic γ : [0, 1]→ C.4. Applications4.1. Let us consider (N, d) = (R, |.|), the global NPC space of the realnumbers with the usual distance, i.e. d(x, y) = |x − y|. If we take γa,b(t) =(1− t)a+ tb, the unique geodesic connecting a to b, and f : [a, b]→ R a convexfunction, then:xn =12[f(a+ (n− 1)bn)+ f((n− 1)a+ bn)],yn =2n+ 1f(a+ b2)+[1− 2n+ 1]f(a) + f(b)2,Refinements of the Hermite–Hadamard Inequality in NPC Global Spaces 141zn =12(xn +f(a) + f(b)2),wn =12(yn +f(a) + f(b)2).In conclusion, we can state the following.Proposition 4.1. Let f : [a, b]→ R be a convex function.(1) For any n ≥ 3 we havef(a+ b2)≤ x3 ≤ x4 ≤1b− a∫ baf(t) dt ≤ 12[f(a+ b2)+f(a) + f(b)2]= y3 ≤ . . . ≤ yn ≤ . . . ≤f(a) + f(b)2.(2) For any n ≥ 2 we havef(a+ b2)≤ x3 ≤ x4 ≤1b− a∫ baf(t) dt ≤ 12[f(a+ b2)+f(a) + f(b)2]= z2 ≤ . . . ≤ zn ≤ . . . ≤f(a) + f(b)2.(3) For any n ≥ 3 we havef(a+ b2)≤ x3 ≤ x4 ≤1b− a∫ baf(t) dt ≤ 12[f(a+ b2)+f(a) + f(b)2]= w3 ≤ . . . ≤ wn ≤ . . . ≤f(a) + f(b)2.Remark 4.2. We can compare the sequences {yn}n≥2, {zn}n≥2 introducedabove with the refinement given by Zabandan in [23]. Recall that the sequenceintroduced by him was defined as follows:Yn(f, a, b) =12n[f(a) + f(b)2+2n−1∑i=1f((1− i2n)a+i2nb)].It is obvious that Y1(f, a, b) = y3 = z2. Then, by the monotonicity of each ofthese sequences, we have that Y1(f, a, b) is a lower bound and consequently142 Cristian Condewe have the following refinement of the H-H inequality:f(a+ b2)≤ x3 ≤ x4 ≤1b− a∫ baf(t) dt ≤ Ym(f, a, b) ≤ . . . ≤ Y1(f, a, b)=12[f(a+ b2)+f(a) + f(b)2]= y3 ≤ . . . ≤ yn ≤ . . .≤ f(a) + f(b)2for all m ≥ 1 and n ≥ 3.4.2. Classical meansRecall that for two positive numbers a and b the following classical meansare defined by:A = A(a, b) =a+ b2arithmetic mean,Fν = Fν(a, b) = (1− ν)√ab+ νa+ b2, ν ∈ [0, 1] Heinz mean,Hν = Hν(a, b) =aνb1−ν + a1−νbν2, ν ∈ [0, 1] Heron mean,G = G(a, b) =√ab geometric mean,L = L(a, b) =b− aln(b)− ln(a)(a 6= b); L(a, a) = a logarithmic mean.Combining the different statements obtained previously we get:Proposition 4.3. For α, β > 0 we haveG(α, β) ≤ F1/n(α, β) ≤ H1− 2n+1 (α, β) ≤ A(α, β),in particular, if α 6= β, thenG(α, β) < H1/3(α, β) < H1/4(α, β) < L(α, β) <(√α+√β2)2< H1− 2n+1 (α, β) < A(α, β),Refinements of the Hermite–Hadamard Inequality in NPC Global Spaces 143G(α, β) < H1/3(α, β) < H1/4(α, β) < L(α, β) <(√α+√β2)2<12(H1/n(α, β) +A(α, β)) < A(α, β),G(α, β) < H1/3(α, β) < H1/4(α, β) < L(α, β) <(√α+√β2)2< H1− 1n+1 (α, β) < A(α, β).Proof. Consider Propositions 3.3 and 4.1 with f(t) = et and α = ea,β = eb. 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Inform. 37 (2010), no. 4, 73–84.[21] Niculescu C.P., Persson L.-E., Convex functions and their applications. A contempo-rary approach, Springer, New York, 2006.[22] Wu S., On the weighted generalization of the Hermite–Hadamard inequality and itsapplications, Rocky Mountain J. Math. 39 (2009), no. 5, 1741–1749.[23] Zabandan G., A new refinement of the Hermite–Hadamard inequality for convex func-tions, J. Inequal. Pure Appl. Math. 10 (2009), no. 2, Art. 45, 7 pp.Instituto de CienciasUniversidad Nacional de Gral. SarmientoJ. M. Gutierrez 1150(B1613GSX) Los PolvorinesandInstituto Argentino de Matemática “Alberto P. Calderón”Saavedra 15 3ž piso(C1083ACA) Buenos AiresArgentinae-mail: cconde@ungs.edu.com

Refinements of the Hermite–Hadamard inequality in NPC global spaces

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Refinements of the Hermite–Hadamard inequality in NPC global spaces

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