Research Article On The Hadamard s Inequality for Log-Convex Functions on the Coordinates

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1 Hindwi Publishing Corportion Journl of Inequlities nd Applictions Volume 29, Article ID 28347, 3 pges doi:.55/29/28347 Reserch Article On The Hdmrd s Inequlity for Log-Convex Functions on the Coordintes Mohmmd Alomri nd Mslin Drus School of Mthemticl Sciences, Universiti Kebngsn Mlysi, UKM, Bngi, 436 Selngor, Mlysi Correspondence should be ddressed to Mslin Drus, mslin@ukm.my Received 5 Jnury 29; Revised 3 My 29; Accepted 2 July 29 Recommended by Sever Silvestru Drgomir Inequlities of the Hdmrd nd Jensen types for coordinted log-convex functions defined in rectngle from the plne nd other relted results re given. Copyright q 29 M. Alomri nd M. Drus. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited.. Introduction Let f : I R R be convex mpping defined on the intervl I of rel numbers nd, b I, with <b, then b f 2 b f x dx f f b 2. holds, this inequlity is known s the Hermite-Hdmrd inequlity. For refinements, counterprts, generliztions nd new Hdmrd-type inequlities, see 8. A positive function f is clled log-convex on rel intervl I, b, if for ll x, y, b nd λ,, f λx λ y f λ x f λ y..2 If f is positive log-concve function, then the inequlity is reversed. Equivlently, function f is log-convex on I if f is positive nd log f is convex on I. Also, if f>ndf exists on I, then f is log-convex if nd only if f f f 2.

2 2 Journl of Inequlities nd Applictions The logrithmic men of the positive rel numbers, b, / b, is defined s L, b b log b log..3 A version of Hdmrd s inequlity for log-convex concve functions ws given in 9, s follows. Theorem.. Suppose tht f is positive log-convex function on, b,then b b f x dx L f,f b..4 If f is positive log-concve function, then the inequlity is reversed. For refinements, counterprts nd generliztions of log-convexity see 9 3. A convex function on the coordintes ws introduced by Drgomir in 8. Afunction f : Δ R which is convex in Δ is clled coordinted convex on Δ if the prtil mpping f y :, b R, f y u f u, y nd f x : c, d R, f x v f x, v, re convex for ll y c, d nd x, b. An inequlity of Hdmrd s type for coordinted convex mpping on rectngle from the plne R 2 estblished by Drgomir in 8, is s follows. Theorem.2. Suppose tht f : Δ R is coordinted convex on Δ, then b f 2, c d 2 b d c b d c f x, y dy dx f, c f, d f b, c. 4.5 The bove inequlities re shrp. The mximum modulus principle in complex nlysis sttes tht if f is holomorphic function, then the modulus f cnnot exhibit true locl mximum tht is properly within the domin of f. Chrcteriztions of the mximum principle for sub super hrmonic functions re considered in 4, s follows. Theorem.3. Let G R 2 be region nd let f : G R be subsuperhrmonic function. If there is point λ G with f λ f x, for ll x G then f x is constnt function. Theorem.4. Let G R 2 be region nd let f nd g be bounded rel-vlued functions defined on G such tht f is subhrmonic nd g is superhrmonic. If for ech point G lim sup f x lim inf g x,.6 x x then f x <g x for ll x G or f g nd f is hrmonic.

3 Journl of Inequlities nd Applictions 3 In this pper, new version of the mximum minimum principle in terms of convexity, nd some inequlities of the Hdmrd type re obtined. 2. On Coordinted Convexity nd SubSuperHrmonic Functions Consider the 2-dimensionl intervl Δ :, b c, d in R 2.Afunctionf : Δ R is clled convex in Δ if f λx λ y λf x λ f y 2. holds for ll x, y Δ nd λ,. As in 8, we define log-convex function on the coordintes s follows: function f : Δ R will be clled coordinted log-convex on Δ if the prtil mppings f y :, b R, f y u f u, y nd f x : c, d R, f x v f x, v, re log-convex for ll y c, d nd x, b. A forml definition of coordinted log-convex function my be stted s follows. Definition 2.. A function f : Δ R will be clled coordinted log-convex on Δ, for ll t, s, nd x, y, u, v Δ, if the following inequlity holds, f tx t y, su s w f ts x, u f s t y, u f t s x, w f t s y, w. 2.2 Equivlently, we cn determine whether or not the function f is coordinted logconvex by using the following lemm. Lemm 2.2. Let f : Δ R.Iff is twice differentible then f is coordinted log-convex on Δ if nd only if for the functions f y :, b R, defined by f y u f u, y nd f x : c, d R, defined by f x v f x, v, we hve f x f x f x 2 2, fy f y f y. 2.3 Proof. The proof is stright forwrd using the elementry properties of log-convexity in one vrible. Proposition 2.3. Suppose tht g :, b R is twice differentible on, b nd log-convex on, b nd h : c, d R is twice differentible on c, d nd log-convex on c, d. Letf : Δ, b c, d R be twice differentible function defined by f x, y g x h y, thenf is coordinted log-convex on Δ. Proof. This follows directly using Lemm 2.2. The following result holds. Proposition 2.4. Every log-convex function f : Δ, b c, d R is log-convex on the coordintes, but the converse is not generlly true.

4 4 Journl of Inequlities nd Applictions Proof. Suppose tht f : Δ R is convex in Δ. Consider the function f x : c, d R, f x v f x, v, then for λ,, ndv, w c, d, we hve f x λv λ w f x, λv λ w f λx λ x, λv λ w f λ x, v f λ x, w fx λ v fx λ w, 2.4 which shows the log-convexity of f x. The proof tht f y :, b R, f y u f u, y, is lso log-convex on, b for ll y c, d follows likewise. Now, consider the mpping f :, 2 R given by f x, y e xy. It is obvious tht f is log-convex on the coordintes but not log-convex on, 2. Indeed, if u,,,w, 2 nd λ,, we hve: log f λ u, λ,w log f λu, λ w λ λ uw, λ log f u, λ log f,w. 2.5 Thus, for ll λ, nd u, w,, we hve log f λ u, λ,w >λlog f u, λ log f,w 2.6 which shows tht f is not log-convex on, 2. In the following, Jensen-type inequlity for coordinted log-convex functions is considered. Proposition 2.5. Let f be positive coordinted log-convex function on the open set, b c, d nd let x i, b, y j c, d.ifα i,β j > nd n i α i, m j β j, then n m log f α i x i, β j y j i i n m α i β j log f x i,y j. i j 2.7 Proof. Let x i, b, α i > be such tht m j α i, nd let y i c, d, β j > be such tht m j β j, then we hve, f n m α i x i, β j y j i j n f α i i x i, m β j y j j n m f α iβ j xi,y j, i j 2.8 nd, since f is positive, log f n m α i x i, β j y j i j n m α i β j log f x i,y j, i j 2.9 which is s required.

5 Journl of Inequlities nd Applictions 5 Remrk 2.6. Let f x, y xy, then the following inequlity holds: n log α i x i m β j y j i j n m α i β j log x i y j. i j 2. The bove result my be generlized to the integrl form s follows. Proposition 2.7. Let f be positive coordinted log-convex function on the Δ:, b c, d, nd let x t : r,r 2 R be integrble with <x t <b, nd let y t : s,s 2 R be integrble with c<y t <d.ifα : r,r 2 R is positive, r 2 r α t dt, nd αx t is integrble on r,r 2 nd β : s,s 2 R is positive, s 2 s β t dt, nd βy t is integrble on s,s 2, then log f r2 r α t x t dt, r2 s2 r s2 s β u y u du s α t β u log f x t,y u du dt. 2. Proof. Applying Jensen s integrl inequlity in one vrible on the x-coordinte nd on the y-coordinte we get the required result. The detils re omitted. Theorem 2.8. Let f : Δ R be positive coordinted log-convex function in Δ, then for ll distinct x,x 2,x 3, b, such tht x <x 2 <x 3 nd distinct y,y 2,y 3 c, d such tht y < y 2 <y 3, the following inequlity holds: f x 2y 2 y 3 x 3 x,y f y x 2 y 2 x 3 x,y 3 f x y 2 x 2 y 3 x3,y f x y y 2 x 2 x3,y 3 f x y 3 x 3 y x2,y 2 f x 2y 3 y 2 x 3 x,y f y x 3 x 2 y 2 x,y 3 f x y 3 x 2 y 2 x3,y 2.2 f x y 2 y x 2 x3,y 3 f x y x 3 y 3 x2,y 2. Proof. Let x,x 2,x 3 be distinct points in, b nd let y,y 2,y 3 be distinct points in c, d. Setting α x 3 x 2 / x 3 x, x 2 αx α x 3 nd let β y 3 y 2 / y 3 y, y 2 βy β y 3, we hve log f x 2,y 2 log f αx α x 3,βy β y 3 αβ log f x,y α β log f x,y 3 β α log f x 3,y α β log f x3,y 3 x 3 x 2 x 3 x y 3 y 2 y 3 y log f x,y x 3 x 2 x 3 x y 2 y y 3 y log f x,y x 2 x x 3 x y 3 y 2 y 3 y log f x 3,y x 2 x x 3 x y 2 y y 3 y log f x 3,y 3,

6 6 Journl of Inequlities nd Applictions nd we cn write log f x 2y 2 y 3 x 3 x,y f y x 2 y 2 x 3 x,y 3 f x y 2 x 2 y 3 x3,y f x y y 2 x 2 x3,y 3 f x y 3 x 3 y x2,y 2 f x 2y 3 y 2 x 3 x,y f y x 3 x 2 y 2 x,y 3 f x y 3 x 2 y 2 x3,y f x y 2 y x 2 x3,y 3 f x y x 3 y 3 x2,y From this inequlity it is esy to deduce the required result 2.2. The subhrmonic functions exhibit mny properties of convex functions. Next, we give some results for the coordinted convexity nd sub super hrmonic functions. Proposition 2.9. Let f : Δ R 2 R be coordinted convex concve on Δ. Iff is twice differentible on Δ,thenf is subsuperhrmonic on Δ. Proof. Since f is coordinted convex on Δ then the prtil mppings f y :, b R, f y u f u, y nd f x : c, d R, f x v f x, v, re convex for ll y c, d nd x, b. Equivlently, since f is differentible we cn write f x 2 f 2 y 2.5 for ll y c, d,nd f y 2 f 2 x 2.6 for ll x, b, which imply tht f x f y 2 f 2 x 2 f 2 y 2.7 which shows tht f is subhrmonic. If f is coordinted concve on Δ, replce by bove, we get tht f is superhrmonic on Δ. We now give two version s of the Mximum Minimum Principle theorem using convexity on the coordintes. Theorem 2.. Let f : Δ R 2 R be coordinted convex concve function on Δ. Iff is twice differentible in Δ nd there is point, 2 Δ with f, 2 f x, y, for ll x, y Δ then f is constnt function. Proof. By Proposition 2.9, wegetthtf is sub super hrmonic. Therefore, by Theorem.3 nd the mximum principl the required result holds see 4, pge 264.

7 Journl of Inequlities nd Applictions 7 Theorem 2.. Let f nd g be two twice differentible functions in Δ. Assume tht f nd g re bounded rel-vlued functions defined on Δ such tht f is coordinted convex nd g is coordinted concve. If for ech point, 2 Δ lim sup f x, y lim inf g x, y, x,y, 2 x,y, then f x, y <g x, y for ll x, y Δ or f g nd f is hrmonic. Proof. By Proposition 2.9, we get tht f is subhrmonic nd g is superhrmonic. Therefore, by Theorem.4 nd using the mximum principl the required result holds, see 4, pge 264. Remrk 2.2. The bove two results hold for log-convex functions on the coordintes, simply, replcing f by log f, to obtin the results. 3. Some Inequlities nd Applictions In the following we develop Hdmrd-type inequlity for coordinted log-convex functions. Corollry 3.. Suppose tht f : Δ, b c, d R is log-convex on the coordintes of Δ, then b log f 2, c d 2 b d c b d c log f x, y dy dx log 4 f, c f, d f b, c. 3. For positive coordinted log-concve function f, the inequlities re reversed. Proof. In Theorem.2, replce f by log f nd we get the required result. Lemm 3.2. For A, B, C R with A, B, C >, the function ψ β C β Aβ B ln, β 3.2 A β B is convex for ll β,. Moreover, ψ β dβ ψ ψ, for ll A, B, C >. Proof. Since ψ is twice differentible for ll β, with A, B, C >, we note tht for ll < β β 2 <, ψ β ψ β 2, which shows tht ψ is incresing nd thus ψ is nonnegtive which

8 8 Journl of Inequlities nd Applictions is equivlent to sying tht ψ is incresing nd hence ψ is convex. Now, using inequlity., weget C β Aβ B ln dβ A β B ψ β dβ ψ ψ 2 2 [ B ln B ] AB C, 3.4 ln AB which completes the proof. Theorem 3.3. Suppose tht f : Δ, b c, d R is log-convex on the coordintes of Δ. Let A f, c f b, c f, d, f, d B, f b, c C, 3.5 then the inequlities d b I f x, y dx dy b d c c, A B C, B C, A, ln B ln C H C, B, H B, C, C, A B, ln C B, A C, ln B γ ln Ei,, B C, [ ] B AB C, A,B,C >, 2 ln B ln AB C β Aβ B ln A β B dβ, otherwise 3.6 hold, where γ is the Euler constnt, Ei, ln x ln ln x Ei, ln Ax ln ln Ax H x 2ln ln ln x ln x, <, <,, otherwise, e t Ei x V.P. x t dt 3.7

9 Journl of Inequlities nd Applictions 9 is the exponentil integrl function. For coordinted log-concve function f, the inequlities re reversed. Proof. Since f : Δ, b c, d R is log-convex on the coordintes of Δ, then f α α b, βc β d f αβ, c f β α b, c f α β, d f α β b, d f αβ, c f β b, c f αβ b, c f α, d f αβ, d 3.8 f β b, d f α b, d f αβ b, d [ f, c f b, c f, d ] αβ [ f, d ] α [ ] f b, c β. Integrting the previous inequlity with respect to α nd β on, 2, we hve, f α α b, βc β d dα dβ [ f, c f b, c f, d ] αβ [ f, d ] α [ ] f b, c β dα dβ. 3.9 Therefore, by 3.9 nd for nonzero, positive A, B, C, we hve the following cses. If A B C, the result is trivil. 2 If A, then f α α b, βc β d dα dβ [ f, d ] α [ ] f b, c β dα dβ B α dα C β dβ B C. ln B ln C 3.

10 Journl of Inequlities nd Applictions 3 If B, then f α α b, βc β d dα dβ A αβ C β dα dβ A α C ln A α C dα 2ln ln ln C ln C, <, <, otherwise Ei, ln C ln ln C Ei, ln AC ln ln AC If C, then f α α b, βc β d dα dβ A αβ B α dα dβ A β B ln A β B dβ 2ln ln ln B ln B, <, <, otherwise Ei, ln B ln ln C Ei, ln AB ln ln AB If A B, then f α α b, βc β d dα dβ A αβ B α C β dα dβ C β dβ C ln C. 3.3

11 Journl of Inequlities nd Applictions 6 If A C, then f α α b, βc β d dα dβ A αβ B α C β dα dβ B α dα B ln B If B C, then f α α b, βc β d dα dβ A αβ B α C β dα dβ A β α dα dβ A α α dα γ ln Ei, If A, B, C >, then A αβ B α C β dα dβ [ ] A C β β B ln A β B dβ. 3.6 Therefore, by Lemm 3.2, we deduce tht A αβ B α C β dα dβ [ B 2 ln B ] AB C. 3.7 ln AB 9 If A, B, C /, we hve A αβ B α C β dα dβ [ ] A C β β B ln A β B dβ, 3.8 which is difficult to evlute becuse it depends on the vlues of A, B, nd C. Remrk 3.4. The integrls in 3, 4,nd 7 in the proof of Theorem 2. re evluted using Mple Softwre.

12 2 Journl of Inequlities nd Applictions Corollry 3.5. In Theorem 3.3, if f x, y f x,then b b f x dx L f,f b, 3.9 nd for instnce, if f x e xp, p we deduce b b e xp dx L e p,e bp f x, y f x f 2 y,then I L f,f b L f 2 c,f 2 d, 3.2 nd for instnce, if f x, y e xp y q, p, q, we deduce b d c b d c e xp y q dx dy L e p,e bp L e cp,e dp Proof. Follows directly by pplying inequlity.4. Acknowledgment The uthors cknowledge the finncil support of the Fculty of Science nd Technology, Universiti Kebngsn Mlysi UKM GUP TMK References S. S. Drgomir, Two mppings in connection to Hdmrd s inequlities, Journl of Mthemticl Anlysis nd Applictions, vol. 67, no., pp , S. S. Drgomir nd R. P. Agrwl, Two inequlities for differentible mppings nd pplictions to specil mens of rel numbers nd to trpezoidl formul, Applied Mthemtics Letters, vol., no. 5, pp. 9 95, S. S. Drgomir, Y. J. Cho, nd S. S. Kim, Inequlities of Hdmrd s type for Lipschitzin mppings nd their pplictions, Journl of Mthemticl Anlysis nd Applictions, vol. 245, no. 2, pp , 2. 4 S. S. Drgomir nd C. E. M. Perce, Selected Topics on Hermite-Hdmrd Inequlities nd Applictions, RGMIA Monogrphs, Victori University, Melbourne City, Austrli, 2. 5 S. S. Drgomir nd S. Wng, A new inequlity of Ostrowski s type in L norm nd pplictions to some specil mens nd to some numericl qudrture rules, Tmkng Journl of Mthemtics, vol. 28, no. 3, pp , S. S. Drgomir nd S. Wng, Applictions of Ostrowski s inequlity to the estimtion of error bounds for some specil mens nd for some numericl qudrture rules, Applied Mthemtics Letters, vol., no., pp. 5 9, 998.

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