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On some Hadamard-type inequalities for ( h 1 , h 2 ) -preinvex functions on the co-ordinates

Journal of Inequalities and Applications20132013:227

https://doi.org/10.1186/1029-242X-2013-227

Received: 5 December 2012

Accepted: 18 April 2013

Published: 7 May 2013

Abstract

We introduce the class of ( h 1 , h 2 ) -preinvex functions on the co-ordinates, and we prove some new inequalities of Hermite-Hadamard and Fejér type for such mappings.

MSC:26A15, 26A51, 52A30.

Keywords

( h 1 , h 2 ) -preinvex function on the co-ordinatesHadamard inequalitiesHermite-Hadamard-Fejér inequalities

1 Introduction

A function f : I R , I R is an interval, is said to be a convex function on I if
f ( t x + ( 1 t ) y ) t f ( x ) + ( 1 t ) f ( y )
(1.1)

holds for all x , y I and t [ 0 , 1 ] . If the reversed inequality in (1.1) holds, then f is concave.

Many important inequalities have been established for the class of convex functions, but the most famous is the Hermite-Hadamard inequality. This double inequality is stated as follows:
f ( a + b 2 ) 1 b a a b f ( x ) d x f ( a ) + f ( b ) 2 ,
(1.2)

where f : [ a , b ] R is a convex function. The above inequalities are in reversed order if f is a concave function.

In 1978, Breckner introduced an s-convex function as a generalization of a convex function [1].

Such a function is defined in the following way: a function f : [ 0 , ) R is said to be s-convex in the second sense if
f ( t x + ( 1 t ) y ) t s f ( x ) + ( 1 t ) s f ( y )
(1.3)

holds for all x , y , t [ 0 , 1 ] and for fixed s ( 0 , 1 ] .

Of course, s-convexity means just convexity when s = 1 .

In [2], Dragomir and Fitzpatrick proved the following variant of the Hermite-Hadamard inequality, which holds for s-convex functions in the second sense:
2 s 1 f ( a + b 2 ) 1 b a a b f ( x ) d x f ( a ) + f ( b ) s + 1 .
(1.4)
In the paper [3] a large class of non-negative functions, the so-called h-convex functions, is considered. This class contains several well-known classes of functions such as non-negative convex functions and s-convex in the second sense functions. This class is defined in the following way: a non-negative function f : I R , I R is an interval, is called h-convex if
f ( t x + ( 1 t ) y ) h ( t ) f ( x ) + h ( 1 t ) f ( y )
(1.5)

holds for all x , y I , t ( 0 , 1 ) , where h : J R is a non-negative function, h 0 and J is an interval, ( 0 , 1 ) J .

In the further text, functions h and f are considered without assumption of non-negativity.

In [4] Sarikaya, Saglam and Yildirim proved that for an h-convex function the following variant of the Hadamard inequality is fulfilled:
1 2 h ( 1 2 ) f ( a + b 2 ) 1 b a a b f ( x ) d x [ f ( a ) + f ( b ) ] 0 1 h ( t ) d t .
(1.6)
In [5] Bombardelli and Varošanec proved that for an h-convex function the following variant of the Hermite-Hadamard-Fejér inequality holds:
a b w ( x ) d x 2 h ( 1 2 ) f ( a + b 2 ) a b f ( x ) w ( x ) d x ( b a ) ( f ( a ) + f ( b ) ) 0 1 h ( t ) w ( t a + ( 1 t ) b ) d t ,
(1.7)

where w : [ a , b ] R , w 0 and symmetric with respect to a + b 2 .

A modification for convex functions, which is also known as co-ordinated convex functions, was introduced by Dragomir [6] as follows.

Let us consider a bidimensional Δ = [ a , b ] × [ c , d ] in R 2 with a < b and c < d . A mapping f : Δ R is said to be convex on the co-ordinates on Δ if the partial mappings f y : [ a , b ] R , f y ( u ) = f ( u , y ) and f x : [ c , d ] R , f x ( v ) = f ( x , v ) are convex for all x [ a , b ] and y [ c , d ] .

In the same article, Dragomir established the following Hadamard-type inequalities for convex functions on the co-ordinates:
f ( a + b 2 , c + d 2 ) 1 ( b a ) ( d c ) a b c d f ( x , y ) d x d y f ( a , c ) + f ( b , c ) + f ( a , d ) + f ( b , d ) 4 .
(1.8)

The concept of s-convex functions on the co-ordinates was introduced by Alomari and Darus [7]. Such a function is defined in following way: the mapping f : Δ R is s-convex in the second sense if the partial mappings f y : [ a , b ] R and f x : [ c , d ] R are s-convex in the second sense.

In the same paper, they proved the following inequality for an s-convex function:
4 s 1 f ( a + b 2 , c + d 2 ) 1 ( b a ) ( d c ) a b c d f ( x , y ) d x d y f ( a , c ) + f ( b , c ) + f ( a , d ) + f ( b , d ) ( s + 1 ) 2 .
(1.9)

For refinements and counterparts of convex and s-convex functions on the co-ordinates, see [610].

The main purpose of this paper is to introduce the class of ( h 1 , h 2 ) -preinvex functions on the co-ordinates and establish new inequalities like those given by Dragomir in [6] and Bombardelli and Varošanec in [5].

Throughout this paper, we assume that considered integrals exist.

2 Main results

Let f : X R and η : X × X R n , where X is a nonempty closed set in R n , be continuous functions. First, we recall the following well-known results and concepts; see [1116] and the references therein.

Definition 2.1 Let u X . Then the set X is said to be invex at u with respect to η if
u + t η ( v , u ) X

for all v X and t [ 0 , 1 ] .

X is said to be an invex set with respect to η if X is invex at each u X .

Definition 2.2 The function f on the invex set X is said to be preinvex with respect to η if
f ( u + t η ( v , u ) ) ( 1 t ) f ( u ) + t f ( v )

for all u , v X and t [ 0 , 1 ] .

We also need the following assumption regarding the function η which is due to Mohan and Neogy [11].

Condition C Let X R be an open invex subset with respect to η. For any x , y X and any t [ 0 , 1 ] ,
η ( y , y + t η ( x , y ) ) = t η ( x , y ) , η ( x , y + t η ( x , y ) ) = ( 1 t ) η ( x , y ) .
Note that for every x , y X and every t 1 , t 2 [ 0 , 1 ] from Condition C, we have
η ( y + t 2 η ( x , y ) , y + t 1 η ( x , y ) ) = ( t 2 t 1 ) η ( x , y ) .
In [12], Noor proved the Hermite-Hadamard inequality for preinvex functions
f ( a + 1 2 η ( b , a ) ) 1 η ( b , a ) a a + η ( b , a ) f ( x ) d x f ( a ) + f ( b ) 2 .
(2.1)
Definition 2.3 Let h : [ 0 , 1 ] R be a non-negative function, h 0 . The non-negative function f on the invex set X is said to be h-preinvex with respect to η if
f ( u + t η ( v , u ) ) h ( 1 t ) f ( u ) + h ( t ) f ( v )

for each u , v X and t [ 0 , 1 ] .

Let us note that:

− if η ( v , u ) = v u , then we get the definition of an h-convex function introduced by Varošanec in [3];

− if h ( t ) = t , then our definition reduces to the definition of a preinvex function;

− if η ( v , u ) = v u and h ( t ) = t , then we obtain the definition of a convex function.

Now let X 1 and X 2 be nonempty subsets of R n , let η 1 : X 1 × X 1 R n and η 2 : X 2 × X 2 R n .

Definition 2.4 Let ( u , v ) X 1 × X 2 . We say X 1 × X 2 is invex at ( u , v ) with respect to η 1 and η 2 if for each ( x , y ) X 1 × X 2 and t 1 , t 2 [ 0 , 1 ] ,
( u + t 1 η 1 ( x , u ) , v + t 2 η 2 ( y , v ) ) X 1 × X 2 .

X 1 × X 2 is said to be an invex set with respect to η 1 and η 2 if X 1 × X 2 is invex at each ( u , v ) X 1 × X 2 .

Definition 2.5 Let h 1 and h 2 be non-negative functions on [ 0 , 1 ] , h 1 0 , h 2 0 . The non-negative function f on the invex set X 1 × X 2 is said to be co-ordinated ( h 1 , h 2 ) -preinvex with respect to η 1 and η 2 if the partial mappings f y : X 1 R , f y ( x ) = f ( x , y ) and f x : X 2 R , f x ( y ) = f ( x , y ) are h 1 -preinvex with respect to η 1 and h 2 -preinvex with respect to η 2 , respectively, for all y X 2 and x X 1 .

If η 1 ( x , u ) = x u and η 2 ( y , v ) = y v , then the function f is called ( h 1 , h 2 ) -convex on the co-ordinates.

Remark 1 From the above definition it follows that if f is a co-ordinated ( h 1 , h 2 ) -preinvex function, then
f ( x + t 1 η 1 ( b , x ) , y + t 2 η 2 ( d , y ) ) h 1 ( 1 t 1 ) f ( x , y + t 2 η 2 ( d , y ) ) + h 1 ( t 1 ) f ( b , y + t 2 η 2 ( d , y ) ) h 1 ( 1 t 1 ) h 2 ( 1 t 2 ) f ( x , y ) + h 1 ( 1 t 1 ) h 2 ( t 2 ) f ( x , d ) + h 1 ( t 1 ) h 2 ( 1 t 2 ) f ( b , y ) + h 1 ( t 1 ) h 2 ( t 2 ) f ( b , d ) .

Remark 2 Let us note that if η 1 ( x , u ) = x u , η 2 ( y , v ) = y v , t 1 = t 2 and h 1 ( t ) = h 2 ( t ) = t , then our definition of a co-ordinated ( h 1 , h 2 ) -preinvex function reduces to the definition of a convex function on the co-ordinates proposed by Dragomir [6]. Moreover, if h 1 ( t ) = h 2 ( t ) = t s , then our definition reduces to the definition of an s-convex function on the co-ordinates proposed by Alomari and Darus [7].

Now, we will prove the Hadamard inequality for the new class functions.

Theorem 2.1 Suppose that f : [ a , a + η ( b , a ) ] R is an h-preinvex function, Condition  C for η holds and a < a + η ( b , a ) , h ( 1 2 ) > 0 . Then the following inequalities hold:
1 2 h ( 1 2 ) f ( a + 1 2 η ( b , a ) ) 1 η ( b , a ) a a + η ( b , a ) f ( x ) d x [ f ( a ) + f ( b ) ] 0 1 h ( t ) d t .
(2.2)
Proof From the definition of an h-preinvex function, we have that
f ( a + t η ( b , a ) ) h ( 1 t ) f ( a ) + h ( t ) f ( b ) .
Thus, by integrating, we obtain
0 1 f ( a + t η ( b , a ) ) d t f ( a ) 0 1 h ( 1 t ) d t + f ( b ) 0 1 h ( t ) d t = [ f ( a ) + f ( b ) ] 0 1 h ( t ) d t .
But
0 1 f ( a + t η ( b , a ) ) d t = 1 η ( b , a ) a a + η ( b , a ) f ( x ) d x .
So,
1 η ( b , a ) a a + η ( b , a ) f ( x ) d x [ f ( a ) + f ( b ) ] 0 1 h ( t ) d t .

The proof of the second inequality follows by using the definition of an h-preinvex function, Condition C for η and integrating over [ 0 , 1 ] .

That is,
f ( a + 1 2 η ( b , a ) ) = f ( a + t η ( b , a ) + 1 2 η ( a + ( 1 t ) η ( b , a ) , a + t η ( b , a ) ) h ( 1 2 ) [ f ( a + t η ( b , a ) ) + f ( a + ( 1 t ) η ( b , a ) ) ] , f ( a + 1 2 η ( b , a ) ) h ( 1 2 ) [ 0 1 f ( a + t η ( b , a ) ) d t + 0 1 f ( a + ( 1 t ) η ( b , a ) ) ] , f ( a + 1 2 η ( b , a ) ) 2 h ( 1 2 ) 1 η ( b , a ) a a + η ( b , a ) f ( x ) d x .

The proof is complete. □

Theorem 2.2 Suppose that f : [ a , a + η 1 ( b , a ) ] × [ c , c + η 2 ( d , c ) ] R is an ( h 1 , h 2 ) -preinvex function on the co-ordinates with respect to η 1 and η 2 , Condition  C for η 1 and η 2 is fulfilled, and a < a + η 1 ( b , a ) , c < c + η 2 ( d , c ) , and h 1 ( 1 2 ) > 0 , h 2 ( 1 2 ) > 0 . Then one has the following inequalities:
(2.3)
Proof Since f is ( h 1 , h 2 ) -preinvex on the co-ordinates, it follows that the mapping f x is h 2 -preinvex and the mapping f y is h 1 -preinvex. Then, by the inequality (2.2), one has
1 2 h 2 ( 1 2 ) f ( x , c + 1 2 η 2 ( d , c ) ) 1 η 2 ( d , c ) c c + η 2 ( d , c ) f ( x , y ) d y [ f ( x , c ) + f ( x , d ) ] 0 1 h 2 ( t ) d t
and
1 2 h 1 ( 1 2 ) f ( a + 1 2 η 1 ( b , a ) , y ) 1 η 1 ( b , a ) a a + η 1 ( b , a ) f ( x , y ) d x [ f ( a , y ) + f ( b , y ) ] 0 1 h 1 ( t ) d t .
Dividing the above inequalities for η 1 ( b , a ) and η 2 ( d , c ) and then integrating the resulting inequalities on [ a , a + η 1 ( b , a ) ] and [ c , c + η 2 ( d , c ) ] , respectively, we have
1 η 1 ( b , a ) 2 h 2 ( 1 2 ) a a + η 1 ( b , a ) f ( x , c + 1 2 η 2 ( d , c ) ) d x 1 η 1 ( b , a ) η 2 ( d , c ) a a + η 1 ( b , a ) c c + η 2 ( d , c ) f ( x , y ) d x d y 1 η 1 ( b , a ) 0 1 h 2 ( t ) d t [ a a + η 1 ( b , a ) f ( x , c ) d x + a a + η 1 ( b , a ) f ( x , d ) d x ]
and
1 η 2 ( b , a ) 2 h 1 ( 1 2 ) c c + η 2 ( d , c ) f ( a + 1 2 η 1 ( b , a ) , y ) d y 1 η 1 ( b , a ) η 2 ( d , c ) a a + η 1 ( b , a ) c c + η 2 ( d , c ) f ( x , y ) d x d y 1 η 2 ( d , c ) 0 1 h 1 ( t ) d t [ c c + η 2 ( c , d ) f ( a , y ) d y + c c + η 2 ( c , d ) f ( b , y ) d y ] .

Summing the above inequalities, we get the second and the third inequalities in (2.3).

By the inequality (2.2), we also have
1 2 h 2 ( 1 2 ) f ( a + 1 2 η 1 ( b , a ) , c + 1 2 η 2 ( d , c ) ) 1 η 2 ( d , c ) c c + η 2 ( d , c ) f ( a + 1 2 η 1 ( b , a ) , y ) d y
and
1 2 h 1 ( 1 2 ) f ( a + 1 2 η 1 ( b , a ) , c + 1 2 η 2 ( d , c ) ) 1 η 1 ( b , a ) a a + η 1 ( b , a ) f ( x , c + 1 2 η 2 ( d , c ) ) d x ,

which give, by addition, the first inequality in (2.3).

Finally, by the same inequality (2.2), we ca also state
1 η 2 ( d , c ) c c + η 2 ( d , c ) f ( a , y ) d y [ f ( a , c ) + f ( a , d ) ] 0 1 h 2 ( t ) d t , 1 η 2 ( d , c ) c c + η 2 ( d , c ) f ( b , y ) d y [ f ( b , c ) + f ( b , d ) ] 0 1 h 2 ( t ) d t , 1 η 1 ( b , a ) a a + η 1 ( b , a ) f ( x , c ) d x [ f ( a , c ) + f ( b , c ) ] 0 1 h 1 ( t ) d t , 1 η 1 ( b , a ) a a + η 1 ( b , a ) f ( x , d ) d x [ f ( a , d ) + f ( b , d ) ] 0 1 h 1 ( t ) d t ,

which give, by addition, the last inequality in (2.3). □

Remark 3 In particular, for η 1 ( b , a ) = b a , η 2 ( d , c ) = d c , h 1 ( t 1 ) = h 2 ( t 2 ) = t , we get the inequalities obtained by Dragomir [6] for functions convex on the co-ordinates on the rectangle from the plane R 2 .

Remark 4 If η 1 ( b , a ) = b a , η 2 ( d , c ) = d c , and h 1 ( t 1 ) = h 2 ( t 2 ) = t s , then we get the inequalities obtained by Alomari and Darus in [7] for s-convex functions on the co-ordinates on the rectangle from the plane R 2 .

Theorem 2.3 Let f , g : [ a , a + η 1 ( b , a ) ] × [ c , c + η 2 ( d , c ) ] R with a < a + η 1 ( b , a ) , c < c + η 2 ( d , c ) . If f is ( h 1 , h 2 ) -preinvex on the co-ordinates and g is ( k 1 , k 2 ) -preinvex on the co-ordinates with respect to η 1 and η 2 , then
1 η 1 ( b , a ) η 2 ( d , c ) a a + η 1 ( b , a ) c c + η 2 ( d , c ) f ( x , y ) g ( x , y ) d x d y M 1 ( a , b , c , d ) 0 1 0 1 h 1 ( t 1 ) h 2 ( t 2 ) k 1 ( t 1 ) k 2 ( t 2 ) d t 1 d t 2 + M 2 ( a , b , c , d ) 0 1 0 1 h 1 ( t 1 ) h 2 ( t 2 ) k 1 ( t 1 ) k 2 ( 1 t 2 ) d t 1 d t 2 + M 3 ( a , b , c , d ) 0 1 0 1 h 1 ( t 1 ) h 2 ( t 2 ) k 1 ( 1 t 1 ) k 2 ( t 2 ) d t 1 d t 2 + M 4 ( a , b , c , d ) 0 1 0 1 h 1 ( t 1 ) h 2 ( t 2 ) k 1 ( 1 t 1 ) k 2 ( 1 t 2 ) d t 1 d t 2 ,
where
M 1 ( a , b , c , d ) = f ( a , c ) g ( a , c ) + f ( a , d ) g ( a , d ) + f ( b , c ) g ( b , c ) + f ( b , d ) g ( b , d ) , M 2 ( a , b , c , d ) = f ( a , c ) g ( a , d ) + f ( a , d ) g ( a , c ) + f ( b , c ) g ( b , d ) + f ( b , d ) g ( b , c ) , M 3 ( a , b , c , d ) = f ( a , c ) g ( b , c ) + f ( a , d ) g ( b , d ) + f ( b , c ) g ( a , c ) + f ( b , d ) g ( a , d ) , M 4 ( a , b , c , d ) = f ( a , c ) g ( b , d ) + f ( a , d ) g ( b , c ) + f ( b , c ) g ( a , d ) + f ( b , d ) g ( a , c ) .
Proof Since f is ( h 1 , h 2 ) -preinvex on the co-ordinates and g is ( k 1 , k 2 ) -preinvex on the co-ordinates with respect to η 1 and η 2 , it follows that
f ( a + t 1 η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) h 1 ( 1 t 1 ) h 2 ( 1 t 2 ) f ( a , c ) + h 1 ( 1 t 1 ) h 2 ( t 2 ) f ( a , d ) + h 1 ( t 1 ) h 2 ( 1 t 2 ) f ( b , c ) + h 1 ( t 1 ) h 2 ( t 2 ) f ( b , d )
and
g ( a + t 1 η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) k 1 ( 1 t 1 ) k 2 ( 1 t 2 ) g ( a , c ) + k 1 ( 1 t 1 ) k 2 ( t 2 ) g ( a , d ) + k 1 ( t 1 ) k 2 ( 1 t 2 ) g ( b , c ) + k 1 ( t 1 ) k 2 ( t 2 ) g ( b , d ) .
Multiplying the above inequalities and integrating over [ 0 , 1 ] 2 and using the fact that
0 1 0 1 f ( a + t 1 η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) g ( a + t 1 η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) d t 1 d t 2 = 1 η 1 ( b , a ) η 2 ( d , c ) a a + η 1 ( b , a ) c c + η 2 ( d , c ) f ( x , y ) g ( x , y ) d x d y ,

we obtain our inequality. □

In the next two theorems, we will prove the so-called Hermite-Hadamard-Fejér inequalities for an ( h 1 , h 2 ) -preinvex function.

Theorem 2.4 Let f : [ a , a + η 1 ( b , a ) ] × [ c , c + η 2 ( d , c ) ] R be ( h 1 , h 2 ) -preinvex on the co-ordinates with respect to η 1 and η 2 , a < a + η 1 ( b , a ) , c < c + η 2 ( d , c ) , and w : [ a , a + η 1 ( b , a ) ] × [ c , c + η 2 ( d , c ) ] R , w 0 , symmetric with respect to
( a + 1 2 η 1 ( b , a ) , c + 1 2 η 2 ( d , c ) ) .
Then
1 η 1 ( b , a ) η 2 ( d , c ) a a + η 1 ( b , a ) c c + η 2 ( d , c ) f ( x , y ) w ( x , y ) d x d y [ f ( a , c ) + f ( a , d ) + f ( b , c ) + f ( b , d ) ] 0 1 0 1 h 1 ( t 1 ) h 2 ( t 2 ) w ( a + t 1 η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) d t 1 d t 2 .
(2.4)
Proof From the definition of ( h 1 , h 2 ) -preinvex on the co-ordinates with respect to η 1 and η 2 , we have
  1. (a)
    f ( a + t 1 η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) h 1 ( 1 t 1 ) h 2 ( 1 t 2 ) f ( a , c ) + h 1 ( 1 t 1 ) h 2 ( t 2 ) f ( a , d ) + h 1 ( t 1 ) h 2 ( 1 t 2 ) f ( b , c ) + h 1 ( t 1 ) h 2 ( t 2 ) f ( b , d ) ,
     
  2. (b)
    f ( a + ( 1 t 1 ) η 1 ( b , a ) , c + ( 1 t 2 ) η 2 ( d , c ) ) h 1 ( t 1 ) h 2 ( t 2 ) f ( a , c ) + h 1 ( t 1 ) h 2 ( 1 t 2 ) f ( a , d ) + h 1 ( 1 t 1 ) h 2 ( t 2 ) f ( b , c ) + h 1 ( 1 t 1 ) h 2 ( 1 t 2 ) f ( b , d ) ,
     
  3. (c)
    f ( a + t 1 η 1 ( b , a ) , c + ( 1 t 2 ) η 2 ( d , c ) ) h 1 ( 1 t 1 ) h 2 ( t 2 ) f ( a , c ) + h 1 ( 1 t 1 ) h 2 ( 1 t 2 ) f ( a , d ) + h 1 ( t 1 ) h 2 ( t 2 ) f ( b , c ) + h 1 ( t 1 ) h 2 ( 1 t 2 ) f ( b , d ) ,
     
  4. (d)
    f ( a + ( 1 t 1 ) η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) h 1 ( t 1 ) h 2 ( 1 t 2 ) f ( a , c ) + h 1 ( t 1 ) h 2 ( t 2 ) f ( a , d ) + h 1 ( 1 t 1 ) h 2 ( 1 t 2 ) f ( b , c ) + h 1 ( 1 t 1 ) h 2 ( t 2 ) f ( b , d ) .
     
Multiplying both sides of the above inequalities by w ( a + t 1 η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) , w ( a + ( 1 t 1 ) η 1 ( b , a ) , c + ( 1 t 2 ) η 2 ( d , c ) ) , w ( a + t 1 η 1 ( b , a ) , c + ( 1 t 2 ) η 2 ( d , c ) ) , w ( a + ( 1 t 1 ) η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) , respectively, adding and integrating over [ 0 , 1 ] 2 , we obtain
4 η 1 ( b , a ) η 2 ( d , c ) a a + η 1 ( b , a ) c c + η 2 ( d , c ) f ( x , y ) w ( x , y ) d x d y [ f ( a , c ) + f ( a , d ) + f ( b , c ) + f ( b , d ) ] 4 0 1 0 1 h 1 ( t 1 ) h 2 ( t 2 ) w ( a + t 1 η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) d t 1 d t 2 ,

where we use the symmetricity of the w with respect to ( a + 1 2 η 1 ( b , a ) , c + 1 2 η 2 ( d , c ) ) , which completes the proof. □

Theorem 2.5 Let f : [ a , a + η 1 ( b , a ) ] × [ c , c + η 2 ( d , c ) ] R be ( h 1 , h 2 ) -preinvex on the co-ordinates with respect to η 1 and η 2 , and a < a + η 1 ( b , a ) , c < c + η 2 ( d , c ) , w : [ a , a + η 1 ( b , a ) ] × [ c , c + η 2 ( d , c ) ] R , w 0 , symmetric with respect to ( a + 1 2 η 1 ( b , a ) , c + 1 2 η 2 ( d , c ) ) . Then, if Condition  C for η 1 and η 2 is fulfilled, we have
f ( a + 1 2 η 1 ( b , a ) , c + 1 2 η 2 ( d , c ) ) a a + η 1 ( b , a ) c c + η 2 ( d , c ) w ( x , y ) d x d y 4 h 1 ( 1 2 ) h 2 ( 1 2 ) a a + η 1 ( b , a ) c c + η 2 ( d , c ) f ( x , y ) w ( x , y ) d x d y .
(2.5)
Proof Using the definition of an ( h 1 , h 2 ) -preinvex function on the co-ordinates and Condition C for η 1 and η 2 , we obtain
f ( a + 1 2 η 1 ( b , a ) , c + 1 2 η 2 ( d , c ) ) h 1 ( 1 2 ) h 2 ( 1 2 ) [ f ( a + t 1 η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) + f ( a + t 1 η 1 ( b , a ) , c + ( 1 t 2 ) η 2 ( d , c ) ) + f ( a + ( 1 t 1 ) η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) + f ( a + ( 1 t 1 ) η 1 ( b , a ) , c + ( 1 t 2 ) η 2 ( d , c ) ) ] .
Now, we multiply it by w ( a + t 1 η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) = w ( a + t 1 η 1 ( b , c ) , c + ( 1 t 2 ) η 2 ( d , c ) ) = w ( a + ( 1 t 1 ) η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) = w ( a + ( 1 t 1 ) η 1 ( b , a ) , c + ( 1 t 2 ) η 2 ( d , c ) ) and integrate over [ 0 , 1 ] 2 to obtain the inequality
f ( a + 1 2 η 1 ( b , a ) , c + 1 2 η 2 ( d , c ) ) 0 1 0 1 w ( a + t 1 η 1 ( b , a ) , c + t 2 η 2 ( d , c ) ) d t 1 d t 2 = f ( a + 1 2 η 1 ( b , a ) , c + 1 2 η 2 ( d , c ) ) 1 η 1 ( b , a ) η 2 ( d , c ) a a + η 1 ( b , a ) c c + η 2 ( d , c ) w ( x , y ) d x d y 4 h 1 ( 1 2 ) h 2 ( 1 2 ) 1 η 1 ( b , a ) η 2 ( d , c ) a a + η 1 ( b , a ) c c + η 2 ( d , c ) f ( x , y ) w ( x , y ) d x d y ,

which completes the proof. □

Now, for a mapping f : [ a , b ] × [ c , d ] R , let us define a mapping H : [ 0 , 1 ] 2 R in the following way:
H ( t , r ) = 1 ( b a ) ( d c ) a b c d f ( t x + ( 1 t ) a + b 2 , r y + ( 1 r ) c + d 2 ) d x d y .
(2.6)

Some properties of this mapping for a convex on the co-ordinates function and an s-convex on the co-ordinates function are given in [6, 7], respectively. Here we investigate which of these properties can be generalized for ( h 1 , h 2 ) -convex on the co-ordinates functions.

Theorem 2.6 Suppose that f : [ a , b ] × [ c , d ] is ( h 1 , h 2 ) -convex on the co-ordinates. Then:
  1. (i)

    The mapping H is ( h 1 , h 2 ) -convex on the co-ordinates on [ 0 , 1 ] 2 ,

     
  2. (ii)

    4 h 1 ( 1 2 ) h 2 ( 1 2 ) H ( t , r ) H ( 0 , 0 ) for any ( t , r ) [ 0 , 1 ] 2 .

     
Proof (i) The ( h 1 , h 2 ) -convexity on the co-ordinates of the mapping H is a consequence of the ( h 1 , h 2 ) -convexity on the co-ordinates of the function f. Namely, for r [ 0 , 1 ] and for all α , β 0 with α + β = 1 and t 1 , t 2 [ 0 , 1 ] , we have:
H ( α t 1 + β t 2 , r ) = 1 ( b a ) ( d c ) a b c d f ( ( α t 1 + β t 2 , r ) x + ( 1 ( α t 1 + β t 2 ) ) a + b 2 , r y + ( 1 r ) c + d 2 ) d x d y = 1 ( b a ) ( d c ) a b c d f ( α ( t 1 x + ( 1 t 1 ) a + b 2 ) + β ( t 2 x + ( 1 t 2 ) a + b 2 ) , r y + ( 1 r ) c + d 2 ) d x d y h 1 ( α ) 1 ( b a ) ( d c ) a b c d f ( t 1 x + ( 1 t 1 ) a + b 2 , r y + ( 1 r ) c + d 2 ) d x d y + h 1 ( β ) 1 ( b a ) ( d c ) a b c d f ( t 2 x + ( 1 t 2 ) a + b 2 , r y + ( 1 r ) c + d 2 ) d x d y = h 1 ( α ) H ( t 1 , r ) + h 1 ( β ) H ( t 2 , r ) .
Similarly, if t [ 0 , 1 ] is fixed, then for all r 1 , r 2 [ 0 , 1 ] and α , β 0 with α + β = 1 , we also have
H ( t , α r 1 + β r 2 ) h 2 ( α ) H ( t , r 1 ) + h 2 ( β ) H ( t , r 2 ) ,
which means that H is ( h 1 , h 2 ) -convex on the co-ordinates.
  1. (ii)
    After changing the variables u = t x + ( 1 t ) a + b 2 and v = r y + ( 1 r ) c + d 2 , we have
    H ( t , r ) = 1 ( b a ) ( d c ) a b c d f ( t x + ( 1 t ) a + b 2 , r y + ( 1 r ) c + d 2 ) d x d y = 1 ( b a ) ( d c ) u L u U v L v U f ( u , v ) b a u U u L d c v U v L d u d v = 1 ( u U u L ) ( v U v L ) u L u U v L v U f ( u , v ) d u d v 1 4 h 1 ( 1 2 ) h 2 ( 1 2 ) f ( a + b 2 , c + d 2 ) ,
     

where u L = t a + ( 1 t ) a + b 2 , u U = t b + ( 1 t ) a + b 2 , v L = r c + ( 1 r ) c + d 2 and v U = r d + ( 1 r ) c + d 2 , which completes the proof. □

Remark 5 If f is convex on the co-ordinates, then we get H ( t , r ) H ( 0 , 0 ) . If f is s-convex on the co-ordinates in the second sense, then we have the inequality H ( t , r ) 4 s 1 H ( 0 , 0 ) .

Declarations

Authors’ Affiliations

(1)
Department of Applied Mathematics, Poznań University of Economics

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© Matłoka; licensee Springer 2013

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