ON SOME NEW INEQUALITIES OF HADAMARD TYPE INVOLVING h-convex FUNCTIONS. 1. Introduction. f(a) + f(b) f(x)dx b a. 2 a

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1 Act Mth. Univ. Comenine Vol. LXXIX, (00, pp ON SOME NEW INEQUALITIES OF HADAMARD TYPE INVOLVING h-convex FUNCTIONS M. Z. SARIKAYA, E. SET nd M. E. ÖZDEMIR Abstrct. In this pper, we estblish some inequlities o Hdmrd type or h convex unctions.. Introduction Let : I R R be convex mpping deined on the intervl I o rel numbers nd, b I with < b. The ollowing double inequlity (. ( + b ( + (b (xdx b is known in the literture s Hdmrd inequlity or convex mpping. Note tht some o the clssicl inequlities or mens cn be derived rom (. or pproprite prticulr selections o the mpping. Both inequlities hold in the reversed direction i is concve. In [8], Fejér gve generliztion o the inequlity (. s ollows. I : [, b] R is convex unction nd g : [, b] R is nonnegtive, integrble nd symmetric bout +b, then (. ( b + b g(xdx (xg(xdx ( + (b For some results which generlize, improve nd extend the inequlities (. nd (., we reer the reder to the recent ppers (see [6], [7], [], [5]. Deinition ([9]. We sy tht : I R R is Godunov-Levin unction or tht belongs to the clss Q(I i is nonnegtive nd or ll x, y I nd α (0,, we hve (αx + ( αy (x α + (y α. Received Februry 9, 00; revised June 8, Mthemtics Subject Clssiiction. Primry 6D07, 6D5. Key words nd phrses. Hdmrd s inequlity; Convex onction; h-convex unction.

2 66 M. Z. SARIKAYA, E. SET nd M. E. ÖZDEMIR The clss Q(I ws irstly described in [9] by Godunov nd Levin. Some urther properties o it re given in [6], [3] nd [4]. Among the others, it is noted tht nonnegtive monotone nd nonnegtive convex unctions belong to this clss o unctions. Deinition ([]. Let s be rel number, s (0, ]. A unction : [0, [0, is sid to be s-convex (in the second sense or belongs to the clss K s, i or ll x, y [0, nd α [0, ]. (αx + ( αy α s (x + ( α s (y In 978, Breckner introduced s-convex unctions s generliztion o convex unctions []. Also, in the pper Breckner proved the importnt ct tht the set-vlued mp is n s-convex only i the ssocited support unction is s-convex unction [3]. A number o properties nd connections with s-convexity in the irst sense is discussed in pper []. O course, s-convexity mens just convexity when s =. In [] nd [4], Berstein-Doetsch type results were proved on rtionlly s-convex unctions, moreover, or the s-hölder property o s-convex unctions. Deinition 3 ([6]. We sy tht : I R is P -unction or tht belongs to the clss P (I i is nonnegtive nd or ll x, y I nd α [0, ], we hve (αx + ( αy (x + (y. Deinition 4 ([6]. Let h : J R R be nonnegtive unction. We sy tht : I R R is h-convex unction, or belongs to the clss SX(h, I, i is nonnegtive nd or ll x, y I nd α (0,, we hve (.3 (αx + ( αy h(α(x + h( α(y. I inequlity (.3 is reversed, then is sid to be h-concve, i.e. SV (h, I. Obviously, i h(α = α, then ll nonnegtive convex unctions belong to SX(h, I nd ll nonnegtive concve unctions belong to SV (h, I; i h(α = α, then SX(h, I = Q(I; i h(α =, then SX(h, I P (I; nd i h(α = α s, where s (0,, then SX(h, I K s. Proposition ([6]. Let nd g be similrly ordered unctions on I, i.e. ( (x (y (g (x g (y 0 or ll x, y I. I SX (h, I, g SX (h, I nd h(α + h( α c or ll α (0,, where h (t = mx {h (t, h (t} nd c is ixed positive number, then the product g belongs to SX (ch, I. For recent results or h-convex unctions, we reer the reder to the recent ppers (see [], [5], [0], [5]. In [7], Drgomir nd Fitzptrick proved vrint o Hdmrd s inequlity which holds or s-convex unctions in the second sense.

3 HADAMARD TYPE INEQUALITY 67 Theorem ([7]. Suppose tht : [0, [0, is n s-convex unction in the second sense, where s (0,, nd let, b [0,, < b. I L ([, b], then the ollowing inequlities hold (.4 ( + b s (xdx b ( + (b. s + The constnt k = s+ is the best possible in the second inequlity in (.4. In [6], Drgomir et. l. proved two inequlities o Hdmrd type or clsses o Godunov-Levin unctions nd P -unctions. (.5 (.6 Theorem ([6]. Let Q(I,, b I with < b nd L ([, b]. Then ( + b 4 (xdx. b Theorem 3 ([6]. Let P (I,, b I with < b nd L ([, b]. Then ( + b (xdx [( + (b]. b In [5], Sriky et. l. estblished new Hdmrd-type inequlity or h- convex unctions. Theorem 4 ([5]. Let SX(h, I,, b I with < b nd L ([, b]. Then (.7 h( ( + b (xdx [( + (b] b 0 h(αdα. The min purpose o this pper is to estblish new inequlities like those given the in bove theorems, but now or the clss o h-convex unctions.. Min Results In the sequel o the pper, I nd J re intervls on R, (0, J nd unctions h nd re rel nonnegtive unctions deined on J nd I, respectively. Throughout this pper, we suppose tht h( 0. Lemm. Let SX(h, I. Then or ny x in [, b], (. ( + b x (h(α + h( α [( + (b] (x, α [0, ]. I is n h-concve unction, then lso the reversed inequlity holds.

4 68 M. Z. SARIKAYA, E. SET nd M. E. ÖZDEMIR Proo. Any x in [, b] cn be represented s α + ( αb, 0 α. Thus, we obtin ( + b x = ( + b α ( αb = (( α + αb h( α( + h(α(b = (h(α + h( α [( + (b] [h(α( + h( α(b] (h(α + h( α [( + (b] (α + ( αb = (h(α + h( α [( + (b] (x. Theorem 5. Let SX(h, I,, b I with < b, L ([, b] nd g : [, b] R is nonnegtive, integrble nd symmetric bout ( + b /. Then (. (xg(xdx ( + (b ( h ( b x + h b ( x b Proo. Since SX(h, I nd g is nonnegtive, integrble nd symmetric bout ( + b /, we ind tht (xg(xdx = (xg(xdx + ( + b xg( + b xdx = = + h = The proo is complete. ((x + ( + b x g(xdx [ { h ( x b ( + (b ( b x b + x ( x b b + b + b x ] b b g(xdx ( b x b ( + h ( + h ( h ( x (b b ( } b x (b g(xdx b ( b x + h b ( x b Remrk. In Theorem 5, i we choose h(α = α nd g(x =, then (. reduces the second inequlity in (., nd i we tke h(α = α, then (. reduces the second inequlity in (..

5 HADAMARD TYPE INEQUALITY 69 Theorem 6. Let SX(h, I,, b I with < b, L ([, b] nd g : [, b] R is nonnegtive, integrble nd symmetric bout ( + b /. Then (.3 h( ( + b ( + (b b g(xdx (h(α + h( α (xg(xdx Proo. Since SX(h, I nd g : [, b] R is nonnegtive, integrble nd symmetric bout ( + b /, we hve h( ( + b b g(xdx = h( = h( h( = = ( + b g(xdx ( + b x + x g(xdx h( (( + b x + (xg(xdx ( + b xg( + b xdx + (x (xg(xdx This proves the irst inequlity in (.3. On the other hnd, rom Lemm, we hve (xg(xdx = = = ( + b xg( + b xdx + ( + b xg(xdx + (xg(xdx (xg(xdx [(h(α+h( α [(+(b] (x] g(xdx + ( + (b (h(α + h( α (xg(xdx

6 70 M. Z. SARIKAYA, E. SET nd M. E. ÖZDEMIR Remrk. In Theorem 6, i we tke h(α = α, then the inequlity (.3 reduces inequlity to (.. Remrk 3. In Theorem 6, i we tke g(x =, then the inequlity (.3 reduces to the ollowing inequlity h( ( + b ( + (b (xdx (h(α + h( α. b Integrting both sides o the bove inequlity over [0, ] with α, we hve the inequlity (.7. Remrk 4. In Theorem 6, i we tke h(α = α s, s (0, nd g(x =, then the inequlity (.3 reduces to the ollowing inequlity ( + b s ( + (b (xdx (α s + ( α s. b Integrting both sides o the bove inequlity over [0, ] with α, we hve the inequlity (.4. (.4 Theorem 7. Let g SX(ch, I,, b I with < b nd g L ([, b]. Then ( + b ch( (g where c is ixed positive number. b (g(xdx c[(g( + (g(b] Proo. Since g SX(ch, I, α (0,, then 0 h(αdα, (.5 (g (αx + ( α y ch (α (g (x + ch ( α (g (y. For x = t + ( tb, y = ( t + tb nd α = (g( + b ( ch (g (t + ( tb + ch we obtin (g (( t + tb. ( Integrting both sides o the bove inequlity over [0, ], we obtin ( + b (g b ch ( b (g(xdx, which completes the proo o the irst inequlity in (.4.

7 HADAMARD TYPE INEQUALITY 7 The proo o the second inequlity ollows by using (.5 with x = nd y = b nd integrting with respect to α over [0, ]. Tht is, (.6 b (g(xdx c[(g( + (g(b] 0 h(αdα. We obtin inequlities (.4 rom (.5 nd (.6.The proo is complete. Remrk 5. In Theorem 7, i we choose c = nd g(x =, then inequlities o (.4 reduce to inequlities (.7. Reerences. Bombrdelli M. nd Vrošnec S., Properties o h-convex unctions relted to the Hermite- Hdmrd-Fejér inequlities, Comput. Mth. Appl. 58(9 (009, Breckner W. W., Stetigkeitsussgen ür eine Klsse verllgemeinerter konvexer unktionen in topologischen lineren Rumen, Pupl. Inst. Mth. 3 (978, , Continuity o generlized convex nd generlized concve set-vlued unctions, Rev. Anl. Numér. Thkor. Approx. (993, Breckner W. W. nd Orbán G., Continuity properties o rtionlly s -convex mppings with vlues in ordered topologicl liner spce, Bbes-Bolyi University, Kolozsvár, Buri P. nd Házy A., On pproximtely h-convex unctions, Journl o Convex Anlysis 8( (0. 6. Drgomir S. S., Pečrić J. nd Persson L. E., Some inequlities o Hdmrd type, Soochow J. Mth. (995, Drgomir S. S. nd Fitzptrik S., The Hdmrd s inequlity or s-convex unctions in the second sense, Demonstrtion Mth. 3(4, (999, Fejér L., Über die Fourierreihen, II. Mth. Nturwiss, Anz. Ungr. Akd. Wiss., 4 (960, , (In Hungrin. 9. Godunov E. K. nd Levin V. I., Nervenstv dlj unkcii sirokogo klss, soderzscego vypuklye, monotonnye i nekotorye drugie vidy unkii, in: Vycislitel. Mt. i. Fiz. Mezvuzov. Sb. Nuc. Trudov, MGPI, Moskv, 985, Házy A., Bernstein-Doetsch-type results or h-convex unctions, ccepted to Mthemticl Inequlities nd Applictions, (see e.g. pre.pd. Hudzik H. nd Mligrnd L., Some remrks on s-convex unctions, Aequtiones Mth. 48 (994, 00.. Kirmci U. S., Bkul M. K., Ozdemir M. E. nd. Pečrić J., Hdmrd-type inequlities or s-convex unctions, Appl. Mth. nd Compt. 93 (007, Mitrinovic D. S. nd Pečrić J., Note on clss o unctions o Godunov nd Levin, C. R. Mth. Rep. Acd. Sci. Cn. (990, Mitrinovic D. S., Pečrić J. nd Fink A. M., Clssicl nd new inequlities in nlysis, Kluwer Acdemic, Dordrecht, Sriky M. Z., Sglm A. ndyıldırım H., On some Hdmrd type inequlities or h- convex unctions, Jour. Mth. Ineq. (3 (008, Vrošnec S., On h-convexity, J. Mth. Anl. Appl. 36 (007,

8 7 M. Z. SARIKAYA, E. SET nd M. E. ÖZDEMIR M. Z. Sriky, Deprtment o Mthemtics, Fculty o Science nd Arts, Düzce University, Düzce-Turkey, e-mil: srikymz@gmil.com, sriky@ku.edu.tr E. Set, Attürk University, K.K. Eduction Fculty, Deprtment o Mthemtics, 540, Cmpus, Erzurum, Turkey, e-mil: erhnset@yhoo.com M. E. Özdemir, Attürk University, K.K. Eduction Fculty, Deprtment o Mthemtics, 540, Cmpus, Erzurum, Turkey, e-mil: emos@tuni.edu.tr

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