Improvements of the Hermite-Hadamard inequality

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1 Pvić Journl of Inequlities nd Applictions (05 05: DOI 0.86/s R E S E A R C H Open Access Improvements of the Hermite-Hdmrd inequlity Zltko Pvić * * Correspondence: Zltko.Pvic@sfsb.hr Mechnicl Engineering Fculty in Slvonski Brod, University of Osijek, Trg Ivne Brlić Mžurnić, Slvonski Brod, 35000, Croti Abstrct The rticle provides refinements nd generliztions of the Hermite-Hdmrd inequlity for convex functions on the bounded closed intervl of rel numbers. Improvements re relted to the discrete nd integrl prt of the inequlity. MSC: 6A5; 6D5 Keywords: convex combintion; convex function; the Hermite-Hdmrd inequlity Introduction Let X be rel liner spce. A liner combintion α + βb of points, b X nd coefficients α, β R is ffine if α + β =.AsetS X is ffine if it contins ll binomil ffine combintions of its points. A function h : S R is ffine if the equlity h(α + βb= αh(+ βh(b ( holds for every binomil ffine combintion α + βb of the ffine set S. Convex combintions nd sets re introduced by restricting to ffine combintions with nonnegtive coefficients. A function h : S R is convex if the inequlity f (α + βb αf (+βf (b ( holds for every binomil convex combintion α + βb of the convex set S. The bove concept pplies to ll n-membered ffine or convex combintions. Jensen (see [] extended the inequlity in eqution ( by relying on induction. Focusing on the set center nd brycenter We use the rel line X = R. Ifκ,...,κ n R re nonnegtive coefficients stisfying n i= κ i =,ndifs = {x,...,x n } is set of points x i R, then the convex combintion point c = κ i x i (3 i= is clled the center of the set S respecting coefficients κ i,orjustthesetcenter. 05 Pvić. This rticle is distributed under the terms of the Cretive Commons Attribution 4.0 Interntionl License ( which permits unrestricted use, distribution, nd reproduction in ny medium, provided you give pproprite credit to the originl uthor(s nd the source, provide link to the Cretive Commons license, nd indicte if chnges were mde.

2 Pvić Journl of Inequlities nd Applictions (05 05: Pge of An integrl version utilizes mesure. If μ is mesure on R,ndifS R is mesurble set of positive mesure, then the integrl men point c = xdμ (4 μ(s S isclledthebrycenter oftheset S respecting mesure μ,orjustthe setbrycenter. In both discrete nd integrl cses, the point c belongs to the convex hull of the set S, s the smllest convex set contining S. Throughout the pper we will use bounded intervl of rel numbers with endpoints < b. Ech point x [, b] cn be presented by the unique binomil convex combintion x = b x + x b. (5 The next three lemms present the properties of convex function f :[, b] R concerning its supporting nd secnt lines. The discrete version refers to intervl points nd intervl endpoints shring the common center. Lemm A Let [, b] R be closed intervl, nd let n i= κ ix i be convex combintion of points x i [, b]. Let α + βb be the unique endpoints convex combintion such tht κ i x i = α + βb. (6 i= Then every convex function f :[, b] R stisfies the double inequlity f (α + βb κ i f (x i αf (+βf(b. (7 i= Proof Tking c = n i= κ ix i, we hve the following two cses. If c {, b},theneqution(7 is reduced to f (c f (c f (c. If c (, b, then using supporting line y = h (xoftheconvexcurvey = f (xtthegrph point C(c, f (c, nd the secnt line y = h (x pssingthroughthegrphpointsa(, f ( nd B(b, f (b,wegettheinequlity f (α + βb=h (α + βb= κ i f (x i i= κ i h (x i i= κ i h (x i =h (α + βb=αf (+βf(b (8 i= contining eqution (7. The discrete-integrl version refers to the connection of the intervl brycenter with intervl endpoints.

3 Pvić Journl of Inequlities nd Applictions (05 05: Pge 3 of Lemm B Let [, b] R be closed intervl, nd let μ be positive mesure on R such tht μ([, b] > 0. Let α + βb be the unique endpoints convex combintion such tht xdμ = α + βb. (9 μ([, b] [,b] Then every convex function f :[, b] R stisfies the double inequlity f (α + βb f (x dμ αf (+βf(b. (0 μ([, b] [,b] Proof The proof coincides with the proof of Lemm A,providedthtweusetheintegrl mens insted of n-membered convex combintions. The integrl version refers to the given intervl nd its subintervl shring the common brycenter. Lemm C Let A, B R be bounded closed intervls such tht A B. Let μ be positive mesure on R such tht 0<μ(A<μ(B nd xdμ = xdμ. ( μ(a A μ(b B Then every convex function f : B R stisfies the double inequlity f (x dμ f (x dμ f (x dμ. ( μ(a A μ(b B μ(b \ A B\A Proof Eqution ( cn be extended with the brycenter of the set B \ A.Usingthesecnt line y = h(xoftheconvexcurvey = f (x respecting the intervl A (f (x h(x, x A nd f (x h(x, x B \A, we firstly prove the inequlity of the left nd right terms of eqution (. Then we express the middle term of eqution ( s the convex combintion of the left nd right terms. A functionl pproch relted to the bove lemms cn be found in []. A more generl version of Lemm C cn be found in [3]. UsingtheRiemnnintegrlinLemmB, the condition in (9 gives the midpoint xdx= + b, (3 nd its use in eqution (0 implies the clssic Hermite-Hdmrd inequlity ( + b f f (x dx f (+f(b. (4 In fct, the bove inequlity holds for every integrble function f :[, b] R tht dmits supporting line t the midpoint c =( + b/, nd fits into the supporting-secnt line inequlity h (x f (x h (x, x [, b]. (5

4 Pvić Journl of Inequlities nd Applictions (05 05: Pge 4 of For exmple, the function f (x= x is integrble on [, ], dmits supporting line t zero, nd stisfies eqution (5, so it stisfies eqution (4. This function is not convex on ny subintervl of [, ]. Moreover, the inequlity in eqution (4 follows by integrting the inequlity in eqution (5overtheintervl[, b]. We finish the section with historic note on the importnt Hermite-Hdmrd inequlity. In 883, studying convex functions, Hermite (see [4] ttined the inequlity in eqution (4. In 893, not knowing Hermite sresult,hdmrd (see [5] got the left-hnd side of eqution (4. For informtion s regrds this inequlity, one my refer to books [6]nd [7], nd ppers [8 ]nd[]. 3 Min results To refine the Hermite-Hdmrd inequlity in eqution (4, we will use convex combintions of points of the closed intervl [, b]. In the next theorem, we will refine the double inequlity in eqution (4 by using two convex combintions of the midpoint x =(+b/. Theorem 3. Let [, b] R be closed intervl, let c, d [, b] be intervl points, nd let α = c, β = b c, γ = d, δ = b d. Then every convex function f :[, b] R stisfies the series of inequlities ( ( ( + b + c c + b f αf + βf f (x dx γ f (+δf (b+f (d f (+f(b. (6 Proof If c, d {, b}, then the inequlity in eqution (6 is ctully reduced to the Hermite-Hdmrd inequlity in eqution (4. Suppose tht c / {, b}. Applying eqution (5 to the inclusion ( + b/ [( + c/, (c + b/], we get the convex combintion equlity + b = α + c + β c + b. (7 Applying the convexity of f to the right-hnd side of eqution (7, nd the left-hnd side of the Hermite-Hdmrd inequlity to midpoints ( + c/ nd (c + b/, we get ( + b f αf = ( + c c ( c + b + βf f (x dx + proving the first hlf of eqution (6. c f (x dx f (x dx (8

5 Pvić Journl of Inequlities nd Applictions (05 05: Pge 5 of Suppose tht d / {, b}. In this cse, we will use the convex combintion equlity γ + δ b + d = + b (9 in terms of eqution (6. Applying the right-hnd side of the Hermite-Hdmrd inequlity to intervls [, d]nd[d, b], nd Lemm A to the combintion in (9, we obtin f (x dx = d f (x dx + γ f (+ δ f (b+ f (d d f (x dx f (+ f (b (0 proving the second hlf of eqution (6. If c = d =( + b/, eqution (6 tkes the form ( + b f [ ( ( ] 3 + b +3b f + f 4 4 f (x dx [ ( + b f + ] f (+f(b f (+f(b. ( The bove improvement of the Hermite-Hdmrd inequlity ws specified in [7]. Now we estimte the double inequlity of eqution (6 contining the integrl term. Tking d = c nd using the rithmetic men form, we cn find tht the following estimtion holds. Corollry 3. Let [, b] R be closed intervl, let c [, b] be n intervl point, nd let α = c, β = b c. Then every convex function f :[, b] R stisfies the inequlity f (x dx ( αf ( + c + βf ( c + b + αf (+βf(b+f(c. ( Proof If c {, b}, the inequlity in eqution ( yields the well-known estimtion f (x dx ( + b f + f (+f(b. (3 We now suppose tht c (, b. From the series of inequlities in eqution (6, we extrct the second lst inequlity referring to the inclusion d [, b] s f (x dx d f (+b d f (b+f(d. (4

6 Pvić Journl of Inequlities nd Applictions (05 05: Pge 6 of Applying the bove inequlity to inclusions ( + c/ [, c]nd(c + b/ [c, b], we get f (x dx = c [ c ] f (x dx + b c [ c b c [ α f (+ ( ] + c f (c+f + β ( ( + c c + b = αf + βf + c ] f (x dx [ f (c+ f (b+f ( ] c + b αf (+βf(b+f(c, (5 nd dividing by, we obtin the inequlity in eqution (. The interpolted terms in eqution (6cn be expnded. Corollry 3.3 Let [, b] R be closed intervl, let c,...,c n [, b] nd d,...,d m [, b] be intervl points, nd let α i = c i n(, β i = b c i n(, γ j = d j m(, δ j = b d j m(. Then every convex function f :[, b] R stisfies the series of inequlities ( + b f [ ( + ci α i f i= + β i f ( ] ci + b f (x dx m j= [γ jf (+δ j f (b+m f (d j ] f (+f(b. (6 Proof The inequlityin eqution (6 cn be chieved by including the convex combintions + b = [ + c i α i i= ] c i + b + β i (7 nd m j= [ γj + δ j b + ] m d j = + b (8 to the procedure of the proof of Theorem 3.. The following theorem presents generliztion of the Hermite-Hdmrd inequlity to ny point of the open intervl (, b. Theorem 3.4 Let [, b] R be closed intervl, nd let α + βb be the endpoints convex combintion whose coefficients re positive.

7 Pvić Journl of Inequlities nd Applictions (05 05: Pge 7 of Then every convex function f :[, b] R stisfies the series of inequlities ( ( + α + βb α + βb + b f (α + βb αf + βf α β( α+βb f (α + βb + f (x dx + αf (+βf (b β α( α+βb f (x dx αf (+βf (b. (9 Proof Applying eqution (5 to the inclusion α + βb [( + α + βb/, (α + βb + b/], we get the convex combintion equlity + α + βb α + βb + b α + βb = α + β. (30 Applying the convexity of f to the right-hnd side of eqution (30, nd the left-hnd side of the Hermite-Hdmrd inequlity to midpoints ( + α + βb/ nd (α + βb + b/, we get the first hlf of eqution (9. Applying the right-hnd side of the Hermite-Hdmrd inequlity to intervls [, α + βb] nd[α + βb, b], nd the convexity inequlity in eqution (, we obtin the second hlf of eqution (9. If α = β = /, then the inequlity in eqution (9 is reduced to the inequlity in eqution (. The integrl refinements of the Hermite-Hdmrd inequlity cn be obtined by pplying Lemm C. Theorem 3.5 Let [, b] R be closed intervl, nd let δ be positive number less thn (/. Then every convex function f :[, b] R stisfies the series of inequlities ( + b f δ δ ( +δ δ +δ f (x dx f (x dx f (x dx + f (x dx b δ f (+f(b. (3 Proof Let A =[ + δ, b δ]ndb =[, b] be observed intervls, nd let A nd B be their lengths, respectively. The brycenter of the sets A, B nd B \ A flls into the midpoint c =( + b/. Let y = h (x be the supporting line of the curve y = f (x tthegrphpoint C(c, f (c, nd let y = h (x be the secnt line pssing through the grph points A(+δ, f (+ δ nd B(b δ, f (b δ. To prove the first inequlity of eqution (3, we use the supporting line ( ( + b + b f = h = h (x dx f (x dx. (3 A A A A

8 Pvić Journl of Inequlities nd Applictions (05 05: Pge 8 of To prove the lst inequlity of eqution (3, we use the secnt line f (x dx B \ A B\A ( + b h (x dx = h = B \ A B\A f (+f(b. (33 The double inequlity of eqution (3 contining the integrls follows from the inequlity in eqution (. Considertions similr to those in Corollry 3. cn be crried out for equtions (6, (9nd(3. 4 Refinements of the most importnt mens Thorough this section we use positive numbers nd b, positive coefficients α nd β whose sum is equl to, nd strictly monotone continuous function ϕ :[, b] R. Thediscretequsi-rithmeticmenofthenumbers nd b with the coefficients α nd β respecting the function ϕ is defined by the number M ϕ (, b; α, β=ϕ ( αϕ(+βϕ(b. (34 Using the identity function ϕ(x=x, we get thegenerlized rithmetic men A(, b; α, β=α + βb, (35 using the hyperbolic function ϕ(x=/x, we hve the generlized hrmonic men H(, b; α, β= ( α + βb, (36 nd using the logrithmic function ϕ(x=ln x, we obtin the generlized geometric men G(, b; α, β= α b β. (37 The bove mens stisfy the generlized hrmonic-geometric-rithmetic men inequlity H(, b; α, β<g(, b; α, β<a(, b; α, β. (38 Applying eqution (9 to the convex function f (x= ln x using substitutions / nd b /b, nd then cting on the rerrnged inequlity with the exponentil function, we cn derive the series of inequlities [ ( H(, b; α, β H, b; α +, β ] α [ ( H, b; α, β + ] β e α β b b b β α b ( α + βb α β αb+β αβ b [ H(, b; α, βg(, b; α, β ] G(, b; α, β (39 refining the generlized hrmonic-geometric men inequlity.

9 Pvić Journl of Inequlities nd Applictions (05 05: Pge 9 of Applying eqution (9totheexponentilfunctionf(x=e x using substitutions ln nd b ln b, we cn obtin the series of inequlities ( G(, b; α, β αg, b; α +, β ( + βg, b; α, β + α β β α b α β αβ α b β ln ln b G(, b; α, β+a(, b; α, β A(, b; α, β (40 refining the generlized geometric-rithmetic men inequlity. To denote the elementry mens with coefficients α = β =/,wewillusethebbrevitions A(, b, H(, b ndg(, b. The integrl qusi-rithmetic men of the numbers nd b respecting the function ϕ is the number M ϕ (, b=ϕ ( Using the identity function, we get the rithmetic men A(, b= ϕ(x dx. (4 xdx= + b, (4 using the hyperbolic function, we hve the logrithmic men ( b L(, b= x dx = ln b ln, (43 nd using the logrithmic function, we obtin the identric men ( I(, b =exp ln xdx = e ( b b The well-known men inequlity sys tht b. (44 H(, b<g(, b<l(, b<i(, b<a(, b. (45 Applying eqution (6 to the function f (x = ln x, using substitutions /, b /b, c /c nd d /d, nd then cting on the rerrnged inequlity with the exponentil function, we cn obtin the refined hrmonic-geometric men inequlity ( b(c ( c(b H(, b c + c [ I (, b ] b c + b (b c c(b b(d d(b b (b d d(b d G(, b. (46

10 Pvić Journl of Inequlities nd Applictions (05 05: Pge 0 of Applying the first hlf of the inequlity in eqution (6 to the exponentil function f (x=e x, using substitutions ln, b ln b nd c ln c,wegettherefinedgeometriclogrithmic men inequlity G(, b ln c ln ln b ln c c + cb L(, b. (47 ln b ln ln b ln To prove the logrithmic-identric men inequlity L(, b<i(, b, we cn pply the integrl form of Jensen s inequlity (see [3] ( f g(x dx f ( g(x dx (48 to the functions f (x= ln x nd g(x=/x, nd then ct on the rerrnged inequlity with the exponentil function. To refine the logrithmic-identric men inequlity, we cn use the procedure pplied in [4]. Applying the first hlf of the reverse inequlity in eqution (6totheconcvefunction f (x = ln x, nd then cting on the rerrnged inequlity with the exponentil function, we hve the refined identric-rithmetic men inequlity I(, b c b c ( + c b (c + b b A(, b. (49 5 Qusi-rithmetic version of the Hermite-Hdmrd inequlity The followingis the generliztion of Lemm B tht includes strictly monotone continuous function. In this generliztion, we use the Riemnn integrl. Lemm 5. Let [, b] R be closed intervl, nd let ϕ :[, b] R be strictly monotone continuous function. Let αϕ(+βϕ(b be the unique convex combintion of endpoints of theintervl ϕ([, b] such tht ϕ(x dx = αϕ(+βϕ(b. (50 Then every convex function f whose domin contins the imge of ϕ stisfies the double inequlity f ( αϕ(+βϕ(b f ( ϕ(x dx αf ( ϕ( + βf ( ϕ(b. (5 Proof We put = ϕ(, b = ϕ(b, nd c = α + βb.thepointc belongs to the interior of the intervl ϕ([, b]. Let z = h (y be supporting line of the curve z = f (ytthegrph point C(c, f (c, nd let z = h (y be the secnt line pssing through the grph points A(, f ( nd B(b, f (b. Applying the procedure of proving eqution (8 withtheuse of equlities h, ( ϕ(x dx = h, ( ϕ(x dx, (5 we obtin the double inequlity in eqution (5.

11 Pvić Journl of Inequlities nd Applictions (05 05: Pge of To present the qusi-rithmetic version of the Hermite-Hdmrd inequlity, we need one more nottion. If ϕ nd ψ re strictly monotone continuous functions on the common intervl, then it is sid tht ψ is ϕ-convex if the composition function ψ ϕ is convex. The sme nottion is used for concvity. Theorem 5. Let [, b] R be closed intervl, nd let ϕ, ψ :[, b] R be strictly monotone continuous functions. Let αϕ(+βϕ(b be the unique binomil convex combintion such tht M ϕ (, b=m ϕ (, b; α, β. (53 If ψ is either ϕ-convex nd incresing or ϕ-concve nd decresing, then M ϕ (, b; α, β M ψ (, b M ψ (, b; α, β. (54 If ψ is either ϕ-convex nd decresing or ϕ-concve nd incresing, then the reverse inequlity is vlid in eqution (54. Proof We prove the cse tht ψ is ϕ-convex nd incresing. The condition in eqution (53 ctully represents the equlity in eqution (50whichenblesustousetheinequlityin eqution (5 with the convex function f = ψ ϕ,ndget ( ψ ϕ ( αϕ(+βϕ(b ψ(x dx αψ(+βψ(b. (55 Acting on the bove inequlity with the incresing function ψ, weobtinthequsi- rithmetic men inequlity in eqution (54. Competing interests The uthor declres tht they hve no competing interests. Acknowledgements This work hs been fully supported by Mechnicl Engineering Fculty in Slvonski Brod, nd Crotin Science Foundtion under the project Received: Jnury 05 Accepted: 4 June 05 References. Jensen, JLWV: Om konvekse Funktioner og Uligheder mellem Middelværdier. Nyt Tidsskr. Mth. 6, (905. Pvić, Z: Functions like convex functions. J. Funct. Spces 04, Article ID (04 3. Pvić, Z, Pečrić, J, Perić, I: Integrl, discrete nd functionl vrints of Jensen s inequlity. J. Mth. Inequl. 5, (0 4. Hermite, C: Sur deux limites d une intégrle définie. Mthesis 3, 8 ( Hdmrd, J: Étude sur les propriétés des fonctions entières et en prticulier d une fonction considerée pr Riemnn. J. Mth. Pures Appl. 58, 7-5 ( Drgomir, SS, Perce, CEM: Selected Topics on Hermite-Hdmrd Inequlities nd Applictions. RGMIA Monogrphs. Victori University, Melbourne ( Niculescu, CP, Persson, LE: Convex Functions nd Their Applictions. Springer, New York ( Chen, F: A note on Hermite-Hdmrd inequlities for products of convex functions. J. Appl. Mth. 03, Article ID (03 9. El Frissi, A: Simple proof nd refinement of Hermite-Hdmrd inequlity. J. Mth. Inequl. 4, (00 0. Lyu, SL: On the Hermite-Hdmrd inequlity for convex functions of two vribles. Numer. Algebr Control Optim. 4, -8 (04. Niculescu, CP, Persson, LE: Old nd new on the Hermite-Hdmrd inequlity. Rel Anl. Exch. 9, (003. Wng, J, Li, X, Fečkn, M, Zhou, Y: Hermite-Hdmrd-type inequlities for Riemnn-Liouville frctionl integrls vi two kinds of convexity. Appl. Anl. 9, 4-53 (03 3. Jensen, JLWV: Sur les fonctions convexes et les inéglités entre les vleurs moyennes. Act Mth. 30,75-93 ( Mićić, J, Pvić, Z, Pečrić, J: The inequlities for qusirithmetic mens. Abstr. Appl. Anl. 0, Article ID0345 (0

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