A NOTE ON THE HERMITE HADAMARD INEQUALITY FOR CONVEX FUNCTIONS ON THE CO ORDINATES FEIXIANG CHEN. 1. Introduction. f (t)dt. b a a

Size: px

Start display at page:

Download "A NOTE ON THE HERMITE HADAMARD INEQUALITY FOR CONVEX FUNCTIONS ON THE CO ORDINATES FEIXIANG CHEN. 1. Introduction. f (t)dt. b a a"

Angelica Cobb
5 years ago
Views:

1 Journal o Mathematial Inequalities Volume 8 Number ( doi:.753/jmi-8-69 A NOTE ON THE HERMITE HADAMARD INEQUALITY FOR CONVEX FUNCTIONS ON THE CO ORDINATES FEIXIANG CHEN (Communiated by S. Abramovih Abstrat. In this paper we obtain some new Hermite-Hadamard-type inequalities or onvex untions on the o-ordinates. We onlude that the results obtained in this work are the reinements o the earlier results.. Introdution Let : I R R be a onvex untion ab I with a < b wehavethe ollowing double inequality b (tdt b a a (a+ (b. ( This remarkable result is well known in the literature as the Hermite-Hadamard inequality or onvex mapping. Sine then some reinements o the Hermite-Hadamard inequality or onvex untions have been extensively investigated by a number o authors (e.g. [3 [5 [6 [7 [8. In ([7 A. E. Farissi established a simple proo a new generalization o the Hermite-Hadamard inequality as ollows: THEOREM.. ([7 Assume that : I R is a onvex untion on I. Then or all λ [ we have where l(λ b (xdx L(λ b a a ( λ b +( λ a l(λ =λ +( λ L(λ = (a+ (b ( ( + λ b +( λ a ( (λ b +( λ a+λ (a+( λ (b. Mathematis subjet lassiiation (: 6D5 5A. Keywords phrases: Hermite-Hadamard s inequality onvex untion onvex untions on the oordinates. This work is supported by Youth Projet o Chongqing Three Gorges University o China. D l Zagreb Paper JMI

2 96 FEIXIANG CHEN Let us onsider the bidimensional interval Δ := [ab [d in R with a < b < d a mapping : Δ R is said to be onvex on Δ i the inequality (λ x +( λ zλ y +( λ w λ (xy+( λ (zw holds or all (xy (zw Δ λ [. A untion : Δ R is said to be o-ordinated onvex on Δ i the partial mappings y : [ab R y (u= (uy x : [d R x (v= (xv are onvex or all y [d x [ab. Aormaldeinition or o-ordinated onvex untions may be stated as ollows: DEFINITION.. A untion : Δ R is said to be onvex on o-ordinates on Δ i the inequality (λ x +( λ zty+( tw λt(xy+λ ( t (xw +( λ t(zy+( t( λ (zw holds or all (xy (xw (zy (zw Δ t [ λ [. S. S. Dragomir in [ established the ollowing Hadamard-type inequalities or o-ordinated onvex untions in a retangle rom the plane R. THEOREM.3. Suppose that : Δ =[ab [d R is onvex on the oordinateson Δ. Then one has the inequalities + d [ b ( x + d dx+ d b a a d y dy b d (xydydx (b a(d a [ b [ (x+ (xddx+ d [ (ay+ (bydy b a a d (a+ (ad+ (b+ (bd. Some new integral inequalities that are related to the Hermite-Hadamard type or o-ordinated onvex untions are also established by many authors. In ([ M. E. Özdemir deined a new mapping assoiated with o-ordinated onvexity proved the ollowing inequalities based on the properties o this mapping. THEOREM.. ([ Suppose that : Δ R R is onvex on the o-ordinates

3 HERMITE-HADAMARD INEQUALITY FOR CONVEX FUNCTIONS ON THE CO-ORDINATES 97 on Δ =[ab [d. Then: + b d (xydydx (b a(d a [ (a+ (ad+ (b+ (bd ( + d + a + d ( + b + d + + d. In ([ 8 M. Alomari Darus deined o-ordinated s-onvex untions proved some inequalities based on this deinition. In ([9 9 analogous results or h-onvex untions on the o-ordinates were proved by M. A. Lati M. Alomari. In ([ 9 Alomari et al. established some Hadamard type inequalities or o-ordinated log-onvex untions. In ([ M. A. Lati S. S. Dragomir obtained some new Hadamard type inequalities or dierentiable o-ordinated onvex onave untions. For reent results generalizations onerning Hermite-Hadamard type inequality or o-ordinated onvex untions see ([ the reerenes given therein. In this paper we obtain some new Hermite-Hadamard-type inequalities or untions on the o-ordinates. These results reine the Hermite-Hadamard-type inequalities given in Theorem.3.. Main results THEOREM.. Suppose that : Δ =[ab [d R is o-ordinated onvex on Δ. Then or all λ [ t [ one has the inequalities + d l(λ t b d (xydydx (b a(d a L(λ t (a+ (ad+ (b+ (bd where ( ( λ a + λ b ( t + td l(λ t =tλ ( ( λ a +( + λ b ( t + td +λ ( t ( ( λ a + λ b ( t +( + td +t( λ ( ( λ a +( + λ b ( t +( + td +( λ ( t

4 98 FEIXIANG CHEN L(λ t = tλ t (a+λ( (ad+ t( λ (b+ (( λ a + λ b( t +td + + λ (a( t + td + λ (b( t + td+ t (( λ a + λ b + t (( λ a + λ bd. Proo. For λ [ t [welet Δ =[ab [d=δ Δ Δ 3 Δ ( λ ( t (bd where Δ =[a( λ a + λ b [( t +td Δ =[a( λ a + λ b [( t +tdd Δ 3 =[( λ a + λ bb [( t +td Δ =[( λ a + λ bb [( t +tdd. Now by applying Theorem.3 to Δ Δ Δ 3 Δ with λ t respetively we obtain ( ( λ a + λ b ( t + td ( λ a+λ b ( t+td (xydydx tλ (b a(d a ( [ (a+ (a( t + td + (( λ a + λ b+ (( λ a + λ b( t +td ( ( λ a + λ b ( t +( + td ( λ a+λ b d (xydydx λ ( t(b a(d a ( t+td [ (a( t + td+ (ad + (( λ a + λ b( t +td+ (( λa + λbd (3

5 HERMITE-HADAMARD INEQUALITY FOR CONVEX FUNCTIONS ON THE CO-ORDINATES 99 ( ( λ a +( + λ b ( t + td b ( t+td (xydydx ( λ t(b a(d ( λ a+λ b [ (( λ a + λ b+ (( λ a + λ b( t +td + (b+ (b( t + td ( ( ( λ a +( + λ b ( t +( + td b d (xydydx ( λ ( t(b a(d ( λ a+λ b ( t+td [ (( λ a + λ bd+ (( λ a + λb( t +td + (b( t + td+ (bd. (5 Multiplying ( by tλ (3 by λ ( t ( by t( λ (5 by ( λ ( t respetively adding the resulting inequalities we get ( ( λ a+λ b tλ ( t+td ( ( λ a+λ b +λ ( t ( t+(+td ( ( λ a +( + λ b ( t + td +t( λ ( ( λ a +( + λ b ( t +( + td +( λ ( t = l(λ t b d (xydydx (b a(d a tλ [ (a+ (a( t + td + (( λ a + λ b+ (( λ a + λ b( t +td λ ( t [ + (a( t + td+ (ad + (( λ a + λ b( t +td+ (( λ a + λbd + t( λ [ (( λ a + λ b+ (( λ a + λ b( t +td + (b+ (b( t + td

6 9 FEIXIANG CHEN ( λ ( t [ + (( λ a + λ bd+ (( λ a + λb( t +td + (b( t + td+ (bd = tλ t (a+λ( (ad+ t( λ (b+ (( λ a + λ b( t +td + + λ + t t (( λ a + λ b+ (( λ a + λ bd = L(λ t. ( λ ( t (bd (a( t + td+ λ (b( t + td Using the at that (xy is a o-ordinated onvex untion we obtain + d ( ( λ a+λ b ( λ a+(+λ b = λ +( λ t ( t+td +( t ( t+(+td ( ( λ a+λ b tλ ( t+td ( ( λ a+(+λ b +t( λ ( t+td ( ( λ a + λ b ( t +( + td +λ ( t ( ( λ a +( + λ b ( t +( + td +( λ ( t = l(λ t. On the other h we have L(λ t = tλ t (a+λ( (ad+ t( λ (b+ (( λ a + λ b( t +td + + λ (a( t + td ( λ ( t (bd + λ (b( t + td+ t (( λ a + λ b + t (( λ a + λ bd tλ t (a+λ( (ad+ t( λ ( λ ( t (b+ (bd ( λ ( t + (a+ t( λ λ ( t (ad+ (b++ tλ (bd λ ( t + (a+ tλ λ ( t (ad+( (b++ t( λ (bd + t( λ (a+ λ λ ( t λ ( t (b+( (ad++ (bd (a+ (ad+ (b+ (bd =.

7 HERMITE-HADAMARD INEQUALITY FOR CONVEX FUNCTIONS ON THE CO-ORDINATES 9 We have ompleted the proo. REMARK. Applying Theorem. or λ = t = ( L [ (a+ (ad+ (b+ (bd ( + + d + a + d ( + b + d we get the result o Theorem d COROLLARY.. With notations above we have the ollowing inequality + d sup l(λ t λ t b d (b a(d a L(λ t in λ t (xydydx (a+ (ad+ (b+ (bd where l(λ t L(λ t are deined in Theorem.. COROLLARY.3. With notations above we have the ollowing inequality { } + d max sup l(λ ta λ t b d (b a(d { min a (xydydx } in L(λ tb λ t (a+ (ad+ (b+ (bd where l(λ t L(λ t are deined in Theorem. B = b A = [ ( x + d dx+ b a a d [ b [ (x+ (xddx+ b a a d d d y dy [ (ay+ (bydy.

8 9 FEIXIANG CHEN EXAMPLE. Let Δ =[ [ (xy=x 3 y 3. We get that (xy=x 3 y 3 is onvex on the o-ordinates on Δ. Now (xydydx = 6 Thereore ( l = [ A = [ ( x ( + dx+ ( 3 + ( L = [ ( + B = [ [ (x+ (xdx+ ( y dy ( 3 + ( ( + ( = 6 A = 3 ( l = 9 sup l(λ t λ t (xydydx = 6 in L(λ t λ t ( L = 5 56 = 3 ( 3 3 = 9 ( + = 5 56 [ (y+ (ydy = 8. B = 8 (+ (+ (+ ( = we get an estimation better than Theorem.3. Aknowledgements. This work is supported by Youth Projet o Chongqing Three Gorges University o China (No. 3QN.

9 HERMITE-HADAMARD INEQUALITY FOR CONVEX FUNCTIONS ON THE CO-ORDINATES 93 REFERENCES [ M. ALOMARI AND M. DARUS The Hadamard s inequality or s -onvex untions o -variables on the o-ordinates Int. J. Math. Anal. 3 ( [ M. ALOMARI AND M. DARUS On The Hadamard s inequality or log-onvex untions on the oordinates J. Inequal. Appl. 9 9: 837. [3 M. BESSENYEI AND ZSOLT PÁLES Hadamard-type inequalities or generalized onvex untions Math. Inequal. Appl. 6 3 ( [ S. S. DRAGOMIR On the Hadamard s inequality or onvex untions on the o-ordinates in a retangle rom the plane Taiwanese J. Math. 5 ( [5 S. S. DRAGOMIR Hermite-Hadamard s type inequalities or operator onvex untions Appl. Math. Comput. 8 3 ( [6 S. S. DRAGOMIR Hermite-Hadamard s type inequalities or onvex untions o seladjoint operators in Hilbert spaes Linear Algebra Appl ( [7 ABDALLAH EL FARISSI Simple proo reinement o Hermite-Hadamard inequality J. Math. Inequal. 3 ( [8 XIANG GAO A note on the Hermite-Hadamard inequality J. Math. Inequal. ( [9 M. A. LATIF AND M. ALOMARI On Hadamard-type inequalities or h-onvex untions on the oordinates Int.J.Math.Anal.3 33 ( [ M. A. LATIF AND S. S. DRAGOMIR On some new inequalities or dierentiable o-ordinated onvex untions J. Inequal. Appl. :8. [ M. E. ÖZDEMIR H. KAVURMACI A. O. AKDEMIR AND M. AVCI Inequalities or onvex s -onvex untions on Δ =[ab [d J. Inequal. Appl. :. [ M. E. ÖZDEMIR ÇETIN YILDIZ AND A. O. AKDEMIR On some new Hadamard-Type inequalities or o-ordinated quasi-onvex untions Haet. J. Math. Stat. 5 ( (Reeived August 3 Feixiang Chen Shool o Mathematis Statistis Chongqing Three Gorges University Wanzhou Chongqing P. R. China x@6.om Journal o Mathematial Inequalities jmi@ele-math.om

arxiv: v1 [math.ca] 13 Feb 2014

arxiv: v1 [math.ca] 13 Feb 2014 arxiv:1.379v1 math.ca 13 Feb 1 SOME NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR hconvex FUNCTIONS ON THE CO-ORDINATES VIA FRACTIONAL INTEGRALS. ERHAN SET, M. ZEKI SARIKAYA, AND HATICE ÖGÜLMÜŞ Abstract.