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On the Hermite-Hadamard type inequalities for convex functions and convex functions on the co-ordinates in a rectangle from the plane.

1. Introduction

Let I be an interval in R, f: I [??] R [right arrow] R be a convex function and a, b [member of] I with a < b. The following double inequality

f([a + b]/2)[less than or equal to][1/[b - a]][[integral].sub.b.sup.a]f(x)dx[less than or equal to][[f(a) + f(b)]/2] (1)

is known in the literature as Hermite-Hadamard inequality for convex function (see for example (6)).

For some results which generalize, improve and extend Hermite-Hadamard inequality (1) see (1-18).

In (3), Dragomir and Buse established the following refinements of the inequality (1):

Theorem A. Let n be a natural number, [q.sub.i] [greater than or equal to] 0 (i = 1, ..., n) and [Q.sub.n] = [n.summation over (i = 1)][q.sub.i][omaga]0. If I is an interval in R, f: I [??] R [right arrow] R is convex and a, b [member of] I with a < b, then

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In (18), Yang and Wang established the following two theorems which give some refinements of the inequality (2):

Theorem B. Let 0 < [[alpha].sub.i] <] 1 (i = 1, ..., n; n [greater than or equal to] 2) with [n.summation over (i = 1)][[alpha].sub.i] = 1 and, let f

On the Hermite-hadmard Type Inequalities for Convex Functions 131

be defined as in Theorem A. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem C. Let n be a natural number, 0 [< or equal to] [[alpha].sub.i] [< or equal to] 1 (i = 1, ..., n) with [n.summation over (i = 1)][[alpha].sub.i] = 1 and let [PHI]: [0, 1] [right arrow] R be defined by

[PHI]F(t) = [1/[[(b - a)].sup.n]][b] ... [b]f(t[n.summation over (i = 1)][[alpha].sub.i][x.sub.i] + (1 - t)[a + b]/2)[n]d[x.sub.i],t[member of][0,1].

Then [PHI] is convex, increasing on [0, 1] and, for t [member of] [0, 1],

f([a + b]/2) = [PHI](0)[less than or equal to][PHI](t)[less than or equal to][PHI](1) = [b] ... [b]f([n.summation over (i = 1)][[alpha].sub.i][x.sub.i])[n]d[x.sub.i]. (4)

Let [??] [R.sup.2]. In (2), a function F: D [right arrow] R will be called convex on the co-ordinates on D if the partial mapping [F.sub.y](x):= F (x, y) is convex in x for each fixed y, and the partial mapping [F.sub.x](y):= F (x, y) is convex in y for each fixed x where (x, y) [member of] D. In (7), the author calls such a function convex separately with respect to each coodinate.

In (7), Lanina established the following two theorems:

Theorem D. Suppose

(a) F: D [right arrow] R is convex on the co-ordinates on D where [??] [R.sup.2];

(b) [DELTA] = [a, b] x [c, d] [??] D with a < b and c < d;

(c) n, qi (i = 1, ... n) and [Q.sub.n] are defined as in Theorem A.

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem E. Let D, [DELTA], n, [q.sub.i] (i = 1, ... n) and [Q.sub.n] be defined as in Theorem D. If F: D [right arrow] R is convex, then the inequality (5) also holds. In (2), Dragomir established the following two theorems:

Theorem F.Let [DELTA] = [a, b] x [c, d] [subset] [R.sup.2], F: [DELTA] [right arrow] R be convex on the co-ordinates on [DELTA] and let H: [[0, 1].sup.2] [right arrow] R be defined by

H(t, s)

= [1/[(b - a)(d - c)]][[integral].sub.a.sup.d][[integral].sub.c.sup.d]F(tx + (1 - t)[[a + b]/2],sy + (1 - s)[c + d]/2)Xdydx. (6)

Then:

(a) The function H is convex on the co-ordinates on [[0, 1].sup.2].

(b) The function H is increasing on the co-ordinates on [[0, 1].sup.2],

[sup[gamma]]H(t,s) = H(1,1) = [1/[(b - a)

(d - c)]][[integral].sub.a.sup.d][[integral].sub.c.sup.d][b][d]F(x,y)dydx

and

[inf[gamma]]H(t,s) = H(0,0) = F([a + b]/2,[c + d]/2).

Theorem G.Let [DELTA] = [a, b] x [c, d] [subset] [R.sub.2], F: [DELTA] [right arrow] R be convex on [DELTA] and let h: [0, 1] [right arrow] R be defined by h (t) = H (t, t) where H is defined as in (6).

Then:

(a) The function H is convex on [[0, 1].sup.2].

(b) The function h is convex and increasing on [0, 1],

[sup[gamma]]h(t) = h(1) = [1/[(b - a)(d - c)]][b][d]F(x,y)dydx

and

[inf[gamma]]h(t) = h(0) = F([a + b]/2,[c + d]/2).

In this paper, we shall establish some inequalities for convex functions and convex functions on the co-ordinates related to Theorems A-G.

2. Main Results

In this section, we assume that m and n are natural numbers, D [??], [R.sup.2], [DELTA] = [a, b] x [c, d] [??] D, [[DELTA].sub.x.sup.n] = [[a, b].sup.n] and [[DELTA].sub.y.sup.m] = [[c, d].sup.m] with a < b and c < d. A function H: [[0, 1].sup.2] [right arrow] R will be called increasing on the co-ordinates on [[0, 1].sup.2] if the partial mapping [H.sub.s]: [0, 1] [right arrow] R, [H.sub.s] (t):= H (t, s) is increasing on [0, 1] for each s [member of] [0, 1], and the partial mapping [H.sub.t]: [0, 1] [right arrow] R, [H.sub.t] (s):= H (t, s) is increasing on [0, 1] for each t [member of] [0, 1].

Theorem 1. Let 0 [< or equal to] [[alpha].sub.i] [< or equal to] 1 (i = 1, ..., n) and 0 [< or equal to] [[beta].sub.j] [< or equal to] 1 (j = 1, ..., m) with [n.summation over (i = 1)][[alpha].sub.i] = 1 = 1 and [m.summation over (j = 1)][[beta].sub.j] = 1.. If F: D [right arrow] R is convex on the co-ordinates on D, then

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Proof. Since F is convex on the co-ordinates on D, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using a similar argument as the proof of the inequality (8), we have the following inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[grater than equal to][[(b - a)].sup.n][[integral].sub.[?.sub.y.sup.m]]F([a + b]/2,[1/m][m.summation over (j = 1)][y.sub.j])[m]d[y.sub.j]

[grater than equal to][[(b - a)].sup.n][[(d - c)].sup.m]F([a + b]/2,[c + d]/2).

Multiplying each side of (8) and (9) by, we obtain the first inequality of (7).

To proof the second inequality and third inequalities of (7), we define [[alpha].sub.n+t] = [[alpha].sub.t] (t = 1, ..., n-1) and [[beta].sup.m+s] = [[beta].sub.s] (s = 1, ..., m-1). Then, by a simple computation, we have the following two identities

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and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where ([x.sub.i],[y.sub.j]) [member of] [DELTA](i = 1, ..., n; j = 1, ...,m). Since F is convex on the coordinates on D, using the identity (10), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly, using the convexity of F on the co-ordinates on D and the identity (11), we obtain the following inequality

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Multiplying (12) and (13) by [1/[[[(b - a)].sup.n][[(d - c)].sup.m]]],, we obtain the second and third inequalities of (7). This completes the proof.

Remark 1. Theorem 1 reduced to Theorem D if we choose m = n, [[beta].sub.i] = [[alpha].sub.i] = [[q.sub.i]/[Q.sub.n]] (i = 1, ..., n) where [q.sub.i] (i = 1, ..., n) and [Q.sub.n] are defined as in Theorem D.

Remark 2. Let f be defined as in Theorem A. If we choose F (x, y) = f (x) ((x, y) [member of] [DELTA]), then Theorem 1 reduces to Theorem A.

By convexity, it is clear that all convex functions on D are convex on the co-ordinates on D. Thus, the following corollary is a simple consequence of Theorem 1.

Corollary 1. In Theorem 1, let F: D [right arrow] R be convex. Then the inequality (7) also holds.

Remark 3. Corollary 1 reduced to Theorem E if we choose m = n, [[beta].sub.i] = [[q.sub.i]/[Q.sub.n]] (i = 1, ..., n) where [q.sub.i] (i = 1, ..., n) and [Q.sub.n] are defined as in Theorem E.

Theorem 2. Let n,m [greater than or equal to] 2, 0 < [[alpha].sub.i] < 1 (i = 1, ..., n) and 0 < [[beta].sub.j] < 1 (j = 1, ..., m) with [n.summation over (i = 1)][[alpha].sub.i] = 1 and [n.summation over (j = 1)][[beta].sub.i] = 1. If F: D [right arrow] R is convex on the co-ordinates on D, then

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Proof. The first inequality of (14) follows immediately from Theorem 1. By a simple computation, we have the following two identities

[n.summation over (i = 1)][[alpha].sub.i][x.sub.i] = [1/[n - 1]][n.summation over (i = 1)][n.summation over (s = 1,s?i)][[alpha].sub.s][x.sub.s] = [n.summation over (i = 1)][[1 - [[alpha].sub.i]]/[n - 1]][n.summation over (s = 1,s?i)][[[[alpha].sub.s][x.sub.s]]/[1 - [[alpha].sub.i]]]

and

[n.summation over (j = 1)][[beta].sub.j][y.sub.j] = [1/[m - 1]][m.summation over (j = 1)][m.summation over (t = 1,t?j)][[beta].sub.t][y.sub.t] = [m.summation over (j = 1)][[1 - [[beta].sub.j]]/[m - 1]][m.summation over (t = 1,t?j)][[[[beta].sub.t][y.sub.t]]/[1 - [[beta].sub.j]]]

where ([x.sub.i], [y.sub.j]) [member of] [DELTA] (i = 1, ..., n; j = 1, ..., m). Since 0<[[1 - [[alpha].sub.i]]/[n - 1]]<(i = 1, ...,n) and [n.summation over (i = 1)][[1 - [[alpha].sub.1]]/[n - 1]] = 1, by the convexity of F on the co-ordinates on D and the identity (15), we obtain

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Similarly, using the convexity of F on the co-ordinates on D and the identity (16), we obtain the following inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since 0<[[[alpha].sub.s]/[1 - [[alpha].sub.i]]]<1(i,s = 1, ...,n;s?i),[n.summation over (s = 1,s?i)][[[alpha].sub.s]/[1 - [[alpha].sub.i]]] = 1(i = 1, ...,n)and[n.summation over (i = 1)][[1 - [[alpha].sub.i]]/[n - 1]] = 1, by the convexity of F on the co-ordinates on D, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly, using the convexity of F on the co-ordinates on D, we obtain the following inequality

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Using the inequalities (17)-(20), we have

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Multiplying (21) by [1/[[[(b - a)].sup.n][[(d - c)].sup.m]]],, we obtain the second and third inequalities of (14). This completes the proof.

Remark 4. Let f be defined as in Theorem B. If we choose F (x, y) = f (x) ((x, y) [member of] [DELTA]), then Theorem 2 reduces to Theorem B.

Remark 5. In Theorem 2, the inequality (14) refines the inequality (7).

Theorem 3.Let F, [[alpha].sub.i] (i = 1, ..., n) and [[beta].sub.j] (j = 1, ..., m) be defined as in Theorem 1 and let G: [[0, 1].sup.2] [right arrow] R be defined by

G(t, s)

= [1/[[[(b - a)].sup.n][[(d - c)].sup.m]]][[integral].sub.[?.sub.y.sup.m]][[integral].sub.[?.sub.x.sup.n]]F(t[n.summation over (i = 1)][[alpha].sub.i][x.sub.i] + (1 - t)[[a + b]/2],s[m.summation over (j = 1)][[beta].sub.j][y.sub.j] + (1 - s)[c + d]/2)X[[PI].sub.i=1.sup.n]d[x.sub.i][[[PI].sub.j=1.sup.m]]d[y.sub.j]. (22)

Then:

(a) The function G is convex on the co-ordinates on [[0, 1].sup.2].

(b) The function G is increasing on the co-ordinates on [[0, 1].sup.2],

sup G(t, s) = G(1, 1)

(t,s)[member of][[0,1].sup.2] G(t,s) = G(1,1)

= [1/[[[(b - a)].sup.n][[(d - c)].sup.m]]][[integral].sub.[[DELTA].sub.y.sup.m]][[integral].sub.[[DELTA].sub.x.sup.n]]F([n.summation over (i = 1)][[alpha].sub.i][x.sub.i],[m.summation over (j = 1)][[beta].sub.j][y.sub.j])[n]d[x.sub.i][m]d[y.sub.j]

and

[inf.sub.(t.s)[member of][0.1].sub.2]= G(0,0)=F([a + b]/2,[c + d]/2).

Proof. (a) Fix s [member of] [0, 1]. Since F is convex on the co-ordinates on [DELTA], we have for [t.sub.1], [t.sub.2] [member of] [0, 1] and [alpha], [beta] [greater than or equal to] 0 with [alpha] + [beta] = 1 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Multiplying (23) by [1/[[[(b - a)].sup.n][[(d - c)].sup.m]]],, we obtain

G(t, [[alpha]t.sub.1] + [[beta]t.sub.2]) [< or equal to] [alpha]G([t.sub.1], s) + [beta]G([t.sub.2, s)

Similarly, if t is fixed in [0, 1], then for [s.sub.1], [s.sub.2] [member of] [0, 1] and [alpha], [beta] [greater than or equal to] 0 with [alpha] + [beta] = 1, we have

G(t, [[alpha]s.sub.1] + [[beta]s.sub.2]) [< or equal to] [alpha]G(t, [s.sub.1]) + [beta]G(t, [s.sub.2])

and the statement is proved.

(b) Since F is convex on the co-ordinates on [DELTA], we have, for all (t, s) [member of] [[0, 1].sup.2],

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Multiplying (24) by [1/[[[(b - a)].sup.n][[(d - c)].sup.m]]],, we obtain

G(t, s) [greater than or equal to] G(0, s). (25)

Similarly, we have, for all (t, s) [member of] [[0, 1].sup.2],

G(t, s) [greater than or equal to] G(t, 0). (26)

If 0 < [t.sub.1] < [t.sub.2] [< or equal to] 1 and 0 < [s.sub.1] < [s.sub.2] [< or equal to] 1, then, for all (t, s) [member of] [[0, 1].sup.2], it follows from the convexity of G on the co-ordinates on [[0, 1].sup.2], (25) and (26) that we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By (25)-(28), we obtain that G is increasing on the co-ordinates on [[0, 1].sup.2]. Hence

[sup[gamma]]G(t,s) = G(1,1)

= [1/[[[(b - a)].sup.n][[(d - c)].sup.m]]][[integral].sub.[[DELTA].sub.y.sup.m]][[integral].sub.[[DELTA].sub.x.sup.n]]F([n.summation over (i = 1)][[alpha].sub.i][x.sub.i][m.summation over (j = 1)][[beta].sub.j][y.sub.j])[n]dx[m]d[y.sub.j]

and

[inf[gamma]]G(t,s) = G(0,0) = F([a + b]/2,[c + d]/2).

This completes the proof.

Remark 6. Let f be defined as in Theorem F. If we choose m = n = 1, then Theorem 3 reduces to Theorem F.

Theorem 4. Let [[alpha].sub.i] (i = 1, ..., n) and [[beta].sub.j] (j = 1, ..., m) be defined as in Theorem 1 and let F: [DELTA] [right arrow] R be convex. Then:

(a) G is convex on [[0, 1].sup.2] where G is defined as in (22).

(b) Define g: [0, 1] [right arrow] R by g(t):= G(t, t). Then g is convex, increasing on [0, 1],

[sup[gamma]]g(t) = g(1)

and

= [1/[[[(b - a)].sup.n][[(d - c)].sup.m]]][[integral].sub.[[DELATA].sub.y.sup.m]][[integral].sub.[[DELATA].sub.x.sup.n]]F([n.summation over (i = 1)][[alpha].sub.i][x.sub.i],[m.summation over (j = 1)][[beta].sub.j][y.sub.j])[[PI].sub.i=1.sup.n]d[x.sub.i][[[PI].sub.i=1.sup.m]]d[y.sub.i] (29)

and

[inf[gamma]]g(t) = g(0) = F([a + b]/2,[c + d]/2).

Proof.(a) Since F is convex, we have for ([t.sub.1], [s.sub.1]), ([t.sub.2], [s.sub.2]) [member of] [[0, 1].sup.2] and [alpha], [beta] [greater than or equal to] 0 with [alpha] + [beta] = 1 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which shows that g is convex on [0, 1]. By Theorem 3, we have that, for 0 [< or equal to] [t.sub.1] < [t.sub.2] [< or equal to] 1,

g ([t.sub.1]) = G([t.sub.1], [t.sub.1]) [< or equal to] G([t.sub.2], [t.sub.1]) [< or equal to] G([t.sub.2], [t.sub.2]) = g([t.sub.2])

which show that g is increasing on [0, 1]. Since g is increasing on [0, 1], (29) and (30) hold. This completes the proof.

Remark 7. Let f be defined as in Theorem G. If we choose m = n = 1, then Theorem 4 reduces to Theorem G.

Shiow-Ru Hwang [dagger] China University of Science and Technology, Nankang, Taipei, Taiwan 11522 and

Kuei-Lin Tseng [double dagger] and Chung-Shin Wang Department of Mathematics, Aletheia Universty, Tamsui Taiwan 25103

Received November 14, 2009, Accepted January 13, 2010.

* 2000 Mathematics Subject Classification. Primary 26D15.

[dagger] E-mail: hsru@cc.chit.edu.tw

[double dagger] Corresponding author. E-mail: kltseng@email.au.edu.tw

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Author:Hwang, Shiow-Ru; Tseng, Kuei-Lin; Wang, Chung-Shin
Publication:Tamsui Oxford Journal of Mathematical Sciences
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Date:May 1, 2010
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