THE EQUIVALENCE OF CHEBYSHEV S INEQUALITY TO THE HERMITE-HADAMARD INEQUALITY

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1 Dedicted to Professor Gheorghe Bucur on the occsion of his 70th birthdy THE EQUIVALENCE OF CHEBYSHEV S INEQUALITY TO THE HERMITE-HADAMARD INEQUALITY CONSTANTIN P. NICULESCU nd JOSIP PEČARIĆ We prove tht Chebyshev s inequlity, Jensen s inequlity nd the Hermite- Hdmrd inequlity imply ech other within the frmework of probbility mesures. Some implictions remin vlid for certin clsses of signed mesures. By considering the cse of semiconve functions, we lso prove Kntorovich type vrint of the Hermite-Hdmrd inequlity. AMS 000 Subject Clssifiction: Primry 6A5, 6D5; Secondry 46A55. Key words: Chebyshev s inequlity, Jensen s inequlity, conve function, monotone function. INTRODUCTION The Hermite-Hdmrd inequlity is vluble tool in the theory of conve functions, providing two-sided estimte for the men vlue of conve function with respect to probbility mesure. Its forml sttement is s follows: Theorem. If f : [, b] R is continuous conve function, then ( + b (. f f( + f(b f(d ; b equlity holds in either side only for the ffine functions (i.e., for the functions of the form m + n. The middle point ( + b/ represents the brycenter of the probbility mesure b d (viewed s mss distribution over the intervl [, b], while nd b represent the etreme points of [, b]. Thus the Hermite-Hdmrd inequlity could be seen s precursor of Choquet s theory. See [8] for detils nd further comments. The optiml trnsport theory offers more insights into the mechnism of this inequlity. In fct, from the point of view of tht theory, the brycenter MATH. REPORTS (6, (00, 45 56

2 46 Constntin P. Niculescu nd Josip Pečrić of mss distribution on [, b], represented by Borel probbility mesure µ, is the unique minimizer b µ of the trnsporttion cost, C(y = y dµ(, ssocited to the cost function c(, y = y. See []. The trnsporttion cost being uniformly conve, it ttins its minimum t the unique root of its derivtive, so tht b µ = dµ(. It is useful to formulte the Hermite-Hdmrd inequlity (. in the contet of semiconve functions nd rbitrry mss distributions. Definition. A function f : [, b] R is clled semiconve (of rte k if the function f + k is conve for some rel constnt k, tht is, f(( λ + λy ( λf( + λf(y + k λ( λ y for ll, y [, b] nd ll λ [0, ]. For emple, every twice differentible function f such tht f +k 0 is semiconve of rte k. A semiconve function of negtive rte is usully known s n uniformly conve function. A useful remrk is tht function f is semiconve of rte k if nd only if for some point 0 [, b] (equivlently, for ny point 0 the function h( = f( + k 0 is conve. Bsed on Choquet s theory we cn stte the following generliztion of the Hermite-Hdmrd inequlity in the contet of semiconve functions: Theorem. If µ is Borel probbility mesure on n intervl [, b], then for every semiconve function f : [, b] R of rte k we hve (. f(b µ f(dµ( + k b µ dµ( b b µ b f( + b µ b f(b + k (b µ (b b µ. The term (b µ (b b µ represents the trnsporttion cost of the mss δ bµ to b bµ b δ + bµ b δ b. This proves to be more epnsive thn the trnsporttion cost of µ to δ bµ ( fct which follows from the right hnd side inequlity in

3 3 Chebyshev s inequlity nd the Hermite-Hdmrd inequlity 47 Theorem, when pplied to the function f( = b µ nd k = 0. The difference of the two costs (.3 D(µ = (b b µ (b µ b µ dµ(, is thus nonnegtive nd it reflects both the geometry of the domin nd the mss distribution on it. The im of this pper is to discuss the connection of the Hermite- Hdmrd inequlity to nother clssicl result, Chebyshev s inequlity. Theorem 3 (Chebyshev s inequlity. If f, g : [, b] R re two monotonic functions of the sme monotonicity, then b ( b ( b f(g(d f(d g(d. b b b If f nd g re of opposite monotonicity, then the bove inequlity works in the reverse wy. The proof of Theorem 3 is direct consequence of the property of positivity of the integrl. The bsic remrk is the inequlity (f( f(y (g( g(y 0 for ll, y [, b], which is integrted with respect to nd y. No rgument of such simplicity is known for Theorem. Remrk. In the sttement of Theorem 3, the intervl [, b] cn be replced by ny intervl I, nd the normlized Lebesgue mesure b d cn be replced by ny Borel probbility mesure on the intervl I. The rgument remins the sme! Surprisingly, Theorem cn be derived from Theorem 3 (nd vice-vers. This is shown in the net section. When the normlized Lebesgue mesure is replced by n rbitrry Borel probbility mesure (still defined on compct intervl, the left hnd side inequlity in Theorem is equivlent to Jensen s inequlity (for k = 0, which in turn is equivlent to Chebyshev s inequlity. Steffensen s etension of Jensen s inequlity (see [8], p. 33, strted the importnt subject of generlizing the clssicl inequlities to the frmework of signed Borel mesures. Bsed on previous work done (independently by T. Popoviciu nd A.M. Fink, the first nmed uthor ws ble to provide chrcteriztion of those signed Borel mesures for which Jensen s inequlity remins vlid in full generlity. See [8], Section 4., for detils. A chrcteriztion of the signed Borel mesures defined on compct intervl for which both sides of the Hermite- Hdmrd inequlity still work cn be found in [4]. A tntlizing problem

4 48 Constntin P. Niculescu nd Josip Pečrić 4 is the chrcteriztion of such mesures in the cse of severl vribles. A prticulr result in this direction is presented in [5]. The problem of chrcterizing the signed Borel mesures for which Chebyshev s inequlity remins vlid ws solved by Fink nd Jodeit Jr. [3]. See lso [6], Ch. IX. At bout the sme time Pečrić [9] hs provided n elegnt (prtil solution bsed on n identity describing the precision in Chebyshev s inequlity. His pproch is put here in full generlity, being ccompnied by n nlogue of the Grüss inequlity. See Theorem 5 below. As consequence we re ble to derive the Jensen-Steffensen inequlity s well s n estimte of its precision. The pper ends with generliztion of Kntorovich s inequlity within the frmework of signed mesures.. PROOF OF THEOREM We strt by noticing tht Theorem 3 is strong enough to yield the clssicl inequlity of Jensen. Theorem 4 (Jensen s inequlity. If (X, Σ, µ is finite mesure spce, ϕ : X R is µ-integrble function nd h is continuous conve function defined on n intervl I contining the rnge of ϕ, then (. h ( µ(x X ϕ(dµ( µ(x X h (ϕ(dµ(. Proof. If h : I R is conve function, nd c int I is kept fied, then the function h( h(c c is nondecresing on I\{c}. Let ν be finite positive Borel mesure on I with brycenter b ν = dν(. ν(i I Clerly, b ν I (since otherwise b ν or b ν will provide n emple of strictly positive function whose integrl with respect to ν is 0. According to Chebyshev s inequlity, if b ν int I, then h( h(b ν ( b ν dν( ν(i I ν(i I b ν h( h(b ν dν( b ν ν(i I ( b ν dν( = 0,

5 5 Chebyshev s inequlity nd the Hermite-Hdmrd inequlity 49 which yields (. h(b ν ν([, b] h(dν(. Notice tht the lst inequlity lso works when b ν is n endpoint of I (becuse in tht cse ν = δ bν. Using the technique of pushing-forwrd mesures, we cn infer from (. the generl sttement of the Jensen inequlity. In fct, if µ is positive Borel mesure on X nd ϕ : X I is µ-integrble mp, then the push-forwrd mesure ν = ϕ#µ is given by the formul ν(a = µ(ϕ (A nd its brycenter is According to (., h(ϕ ν(i ϕ = µ(x I X ϕ(dµ(. h(tdν(t = h(ϕ(dµ( µ(x X for ll continuous conve functions h : I R, which ends the proof of Theorem 4. It is cler tht the inequlity of Jensen implies in turn the positivity property of integrl (nd thus ll re equivlent to the inequlity of Chebyshev. The inequlity of Jensen nd the inequlity of Chebyshev re lso dul ech other. See [8], Section.8. Coming bck to the proof of Theorem, we will notice tht the left hnd side inequlity in ( is consequence of the inequlity of Jensen (., pplied to ϕ the identity of [, b], nd to the conve function h( = f(+ k b µ. The right hnd side inequlity in ( cn be obtined in similr mnner, s consequence of the following result: Lemm. For every conve function h : [, b] R nd every Borel probbility mesure µ on [, b], (.3 h(dµ( b b µ b h( + b µ b h(b. Proof. Formlly, this is specil cse of n importnt theorem due to Choquet. See [8], Section 4.4, for detils. A more direct rgument is in order. If µ = n i= λ iδ i is discrete probbility mesure, then its brycenter is b µ = n i= λ i i nd (.3 tkes the form n b n n λ i i λ i i i= i= (.4 λ i h( i h( + h(b. b b i=

6 50 Constntin P. Niculescu nd Josip Pečrić 6 This inequlity follows directly from the property of h of being conve. In fct, which yields i = b i b + i b b h( i b i b h( + i b h(b. Multiplying both sides by λ i nd then summing over i we rrive t (.4. The generl cse of (.3 is now consequence of the following pproimtion rgument: every Borel probbility mesure λ on compct Husdorff spce is the pointwise limit of net of discrete probbility mesures, ech hving the sme brycenter s λ (see [8], Lemm 4..0, p. 83. When dµ = b d is the normlized Lebesgue mesure on [, b], then b µ = (+b/ nd we cn derive the inequlity (.3 directly from Chebyshev s inequlity. In fct, b ( + b h (d b nd b = b ( + b b (( + b = h( + h(b d ( + b h( b b h (d = d b h(d. h (d = 0 3. THE CASE OF SIGNED MEASURES h(d = Is is well known tht the integrl inequlities with respect to signed mesures re considerbly more involving becuse the positivity property of the integrl does not work in tht contet. Notbly, both Chebyshev s inequlity nd Jensen s inequlity dmit etensions tht work for certin signed mesures. The cse of Chebyshev s inequlity is discussed by J. Pečrić in [9], Theorem, in the cse of Stieltjes mesures ssocited to bsolutely continuous functions. His bsic rgument is n identity whose vlidity cn be estblished under less restrictive hypotheses:

7 7 Chebyshev s inequlity nd the Hermite-Hdmrd inequlity 5 Lemm. Suppose tht p, f, g : [, b] R re functions of bounded vrition nd f nd g re lso continuous. Then (3. = (p(b p( f(g(dp( ( p ( p (tdg(t df( + where p ( = p(b p( nd p ( = p( p(. f(dp( g(dp( ( p ( p (tdg(t df(, Proof. We strt by noticing tht the indefinite integrl h(tdp(t of ny continuous function h hs bounded vrition nd verifies the formul ( f(d h(tdp(t = f(h(dp(. Thus for h(t = g(sdp(s (p(b p( g(t = the formul of integrtion by prts leds us to the identity ( h(tdp(t df( = = (p(b p( On the other hnd, h(tdp(t = (p( p( ( =(p( p( g(tdp(t + so tht = = p ( = p ( (p(b p( f(g(dp( f(dp( (g(s g(t dp(s, g(tdp(t (p(b p( g(dp(. g(sdp(s (p(b p( g(tdp (t p ( ( p (tdg(t + p ( f(g(dp( ( p ( p (tdg(t df( + g(tdp (t = p (tdg(t, f(dp( g(tdp(t = g(dp( g(tdp(t = ( p ( p (tdg(t df(.

8 5 Constntin P. Niculescu nd Josip Pečrić 8 An immedite consequence of Lemm is the following stronger form of Chebyshev s inequlity: Theorem 5. Suppose tht p : [, b] R is function of bounded vrition such tht (3. p(b > p( nd p( p( p(b for ll. Then for every pir of continuous functions f, g : [, b] R which re monotonic in the sme sense, the Chebyshev functionl ( T (f, g; p = p(b p( ( f(dp( p(b p( is bounded from bove by p(b p( f(g(dp( p(b p( g(dp(, m{p (, p (}df( m{p (, p (}dg( p(b p( (f(b f((g(b g(, nd is bounded from below by p(b p( min{p (, p (}df( p(b p( min{p (, p (}dg( 0. Proof. In fct, we my ssume tht both functions f nd g re nondecresing (chnging f nd g by f nd g if necessry. Since the integrl of nonnegtive function with respect to nondecresing function is nonnegtive, we hve ( p ( + = p (tdg(t df( + ( p ( p (tdg(t df( df(+ ( min {p (, p (} min {p (t, p (t} dg(t ( min {p (, p (} min {p (, p (} df( min {p (t, p (t} dg(t df( = min {p (, p (} dg(

9 9 Chebyshev s inequlity nd the Hermite-Hdmrd inequlity 53 nd ( p ( p (tdg(t df( + m {p (, p (} df( ( p ( p (tdg(t df( m {p (, p (} dg(. Remrk. As ws noticed by Pečrić [9], very efficient bounds cn be indicted in the frmework of C -differentibility of f nd g. For emple, if f nd g re monotonic in the sme sense, nd f α nd g β (for some positive constnts α nd β, then T (f, g; p αβt (, ; p. The necessity of the condition (3. for the vlidity of Chebyshev s inequlity is discussed in [3]. Theorem 5 yields the following improvement on the Jensen-Steffensen inequlity: Theorem 6. Suppose tht p : [, b] R is function with bounded vrition such tht p(b > p( nd p( p( p(b for ll. Then for every continuous nd strictly monotonic function ϕ : [, b] R nd for every continuous conve function h defined on n intervl I tht contins the imge of ϕ, the inequlity holds. Here 0 h (ϕ(dp( h (b p p(b p( ( h(ϕ(b h(bp h(ϕ( h(b p (ϕ(b ϕ(. ϕ(b b p ϕ( b p b p = p(b p( ϕ(dp( represents the brycenter of the push-forwrd mesure ϕ#dp. Proof. Using n pproimtion rgument we my restrict ourselves to the cse where h is differentible. The functions ϕ( nd h(ϕ( h(bp ϕ( b p being monotonic in the sme sense, Theorem 5 pplies nd it gives us p(b p( p(b p( h(ϕ( h(b p (ϕ( b p dp( ϕ( b p h(ϕ( h(b p dp( ϕ( b p p(b p( (ϕ( b p dp( = 0.

10 54 Constntin P. Niculescu nd Josip Pečrić 0 Thus h(b p p(b p( h (ϕ(dp(. The proof ends by tking into ccount the discrepncy in Chebyshev s inequlity (provided by Theorem 5. Remrk 3. According to Lemm, = (p(b p( + (p(b p( p(b p( ( p ( ( p ( h (ϕ(dp( h (b p = p (tdϕ(t d p (tdϕ(t d ( h(ϕ( h(bp ϕ( b p ( h(ϕ( h(bp ϕ( b p + nd this fct my be used to derive better bounds for the discrepncy in Chebyshev s inequlity. The fct tht Theorem. works outside the frmework of positive Borel mesures is discussed in [4] (under the nonrestrictive ssumption tht k = 0. Let us cll rel Borel mesure µ on [, b], with µ ([, b] > 0, Hermite- Hdmrd mesure if for every conve function f : [, b] R the following two inequlities hold true,, (LHH nd (RHH µ([, b] f(b µ µ([, b] f(dµ( f(dµ( b b µ b f( + b µ b f(b, where b µ = µ([,b] dµ(. According to [4], the mesure ( + d verifies (LHH (respectively (RHH for ll conve functions defined on the intervl [, ] if nd nd only if /3 (respectively /6. As consequence, we cn ehibit emples of functions of bounded vrition s in Theorem 5, which do not give rise to Hermite-Hdmrd mesures. So is the function p( = ( 3 3 4, for [, ], which clerly verifies the condition p(cos = 4 cos 3 [p(, p(]. However, the ssocited mesure dµ = 3( 4 d does not verify (RHH for ll conve functions defined on the intervl [, ].

11 Chebyshev s inequlity nd the Hermite-Hdmrd inequlity AN APPLICATION TO KANTOROVICH S INEQUALITY We strt with the following vrint of Theorem. Theorem 7. Suppose tht µ is Hermite-Hdmrd mesure on [, b], with b µ > 0. Then for every k-conve function f : [, b] R such tht f( > f(b, f(b µ k b µ dµ( f(dµ( [ bf( f(b + k(b D(µ], 4b µ (b (f( f(b where D(µ is given by the formul (.3. Proof. The left hnd side inequlity is equivlent to the left hnd side inequlity in Theorem. As concerns the right hnd side inequlity, it suffices to consider the cse where f(dµ( > 0. In this cse, we mke gin n ppel to Theorem in order to infer tht This yields (b f(dµ( + (b k b µ dµ( bf( f(b + (f(b f(b µ + k (b (b b µ (b µ. bf( f(b + k (b [(b b µ (b µ (b nd the proof is done. [ b µ (b (f( f(b f(dµ( + (f( f(bb µ ] b µ dµ( f(dµ(] /, Corollry. If f : [, b] R is conve function such tht f( > f(b, nd µ is Hermite-Hdmrd mesure on [, b] with b µ > 0, then (bf( f(b f(dµ( 4b µ (b (f( f(b. In the discrete cse, when µ is Borel probbility mesure of the form µ = n i= λ iδ i, the result of Corollry reds s ( n i= λ i i ( n i= λ i f( i (bf( f(b 4 (b (f( f(b.

12 56 Constntin P. Niculescu nd Josip Pečrić If we pply this remrk to the conve function f( = / for [m, M] (where 0 < m < M, we recover clssicl inequlity due to Kntorovich: ( n ( n λ i (M + m λ i i i= i= i 4Mm, for ll,..., n [m, M] nd ll λ,..., λ n [0, ] with n i= λ i =. See []. However, our pproch yields bit more. Precisely, the Kntorovich inequlity still works for those Hermite-Hdmrd mesures n i= λ iδ i (on [m, M] such tht n i= λ i i [m, M]. REFERENCES [] E.F. Beckenbch nd R. Bellmn, Inequlities, nd Ed., Springer-Verlg, Berlin, 983. [] A.M. Fink, A best possible Hdmrd Inequlity. Mth. Inequl. Appl. (998, [3] A.M. Fink nd M. Jodeit Jr., On Chebyšhev s other inequlity. In: Inequlities in Sttistics nd Probbility, IMS Lecture Notes Monogrph Series, Vol. 5, 984, 5 0. [4] A. Flore nd C.P. Niculescu, A Hermite-Hdmrd inequlity for conve-concve symmetric functions. Bull. Soc. Sci. Mth. Roum. 50(98 (007,, [5] M. Mihăilescu nd C.P. Niculescu, An etension of the Hermite-Hdmrd inequlity through subhrmonic functions. Glsgow Mth. J. 49 (007, [6] D.S. Mitrinović, J.E. Pečrić nd A.M. Fink, Clssicl nd New Inequlities in Anlysis. Kluwer, Dordrecht, 993. [7] C.P. Niculescu, An etension of Chebyshev s inequlity nd its connection with Jensen s inequlity. Journl of Inequlities nd Applictions 6 (00, [8] C.P. Niculescu nd L.-E. Persson, Conve Functions nd their Applictions. A Contemporry Approch. CMS Books in Mthemtics, Vol. 3, Springer-Verlg, New York, 006. [9] J.E. Pečrić, On the Ostrowski generliztion of Čebyšev s inequlity. J. Mth. Anl. Appl. 0 (984, [0] J.E. Pečrić, F. Proschn nd Y.C. Tong, Conve functions, Prtil Orderings nd Sttisticl Applictions. Acdemic Press, New York, 99. [] C. Villni, Optiml Trnsport. Old nd New. Springer-Verlg, 009. Received 9 July 009 University of Criov Deprtment of Mthemtics Criov, Romni cniculescu47@yhoo.com nd University of Zgreb Fculty of Tetile Technology Pierottijev Zgreb, Croti pecric@mhzu.hzu.hr