On some Hermite-Hadamard type inequalities for (s, QC)-convex functions.

In the paper, the authors introduce a new notion "[Formula: see text]-convex function on the co-ordinates" and establish some Hermite-Hadamard type integral inequalities for [Formula: see text]-convex functions on the co-ordinates.

Background

Let be a convex function and with . The double inequalityis known in the literature as Hermite–Hadamard’s inequality for convex functions.

Definition 1

(Dragomir and Pearce 1998; Pečarić et al. 1992) A function is said to be quasi-convex (QC), ifholds for all and .

Definition 2

(Dragomir and Pearce 1998) The function is Jensen- or J-quasi-convex (JQC) ifholds for all .

Definition 3

(Hudzik and Maligranda 1994) Let . A function is said to be s-convex (in the second sense) ifholds for all and

Definition 4

(Xi and Qi 2015a) For some , a function is said to be extended s-convex ifis valid for all and .

Definition 5

(Dragomir 2001; Dragomir and Pearce 2000) A function is said to be convex on co-ordinates on if the partial functionsare convex for all and .

Definition 6

A function is said to be convex on co-ordinates on if the inequalityholds for all and .

Definition 7

(Alomari and Darus 2008) A function is s-convex on for some fixed ifholds for all and .

Definition 8

(Özdemir et al. 2012a, Definition 7) A function is called a Jensen- or J-quasi-convex function on the co-ordinates on ifholds for all .

Definition 9

(Özdemir et al. 2012a, Definition 5) A function is called a quasi-convex function on the co-ordinates on ifholds for all and .

Theorem 1

(Dragomir 2001; Dragomir and Pearce 2000 Theorem 2.2) Letbe convex on the co-ordinates onwithand. Then

Theorem 2

(Özdemir et al. 2012a, Lemma 8) Every J-quasi-convex mappingis J-quasi-convex on the co-ordinates.

Theorem 3

(Özdemir et al. 2012a, Lemma 6) Every quasi-convex mappingis quasi-convex on the coordinates.

For more information on this topic, please refer to Bai et al. (2016), Hwang et al. (2007), Özdemir et al. (2011, 2012a, b, c, 2014), Qi and Xi (2013), Roberts and Varberg (1973), Sarikaya et al. (2012), Wu et al. (2016), Xi et al. (2012, 2015), Xi and Qi (2012, 2013, 2015a, b, c) and related references therein.

In this paper, we introduce a new concept “-convex functions on the co-ordinates on the rectangle of ” and establish some new integral inequalities of Hermite–Hadamard type for -convex functions on the co-ordinates.

Definitions and Lemmas

We now introduce three new definitions

Definition 10

For , a function is said to be -convex on the co-ordinates on with and , ifholds for all and .

Remark 1

By Definitions 8 and 10 and Lemma 1, we see that, for and ,

If is a J-quasi-convex function on the co-ordinates on , then f is a -convex function on the co-ordinates on ;

Every J-quasi-convex function is a -convex function on the co-ordinates on .

Definition 11

A function is called -convex on the co-ordinates on with and , ifholds for all , , and some .

Definition 12

For some , a function is called -convex on the co-ordinates on with and , ifis valid for all , , and .

Remark 2

For and ,

If taking and in (13), then ;

If is a s-convex function on , then f is an -convex function on the co-ordinates on .

Remark 3

Considering Definitions 9 and 12 and Lemma 1, for and ,

If is a quasi-convex function on the co-ordinates on , then it is an -convex function on the co-ordinates on ;

Every quasi-convex function is an -convex function on the co-ordinates on .

Lemma 1

(Latif and Dragomir 2012) Ifhas partial derivatives andwithand, thenwhere

Lemma 2

Letand. Thenandwhereis defined by (14).

Proof

This follows from a straightforward computation.

Some integral inequalities of Hermite–Hadamard type

In this section, we will establish Hermite–Hadamard type integral inequalities for -convex functions on the co-ordinates on rectangle from the plane .

Theorem 4

Lethave partial derivatives and. Ifis an (s, QC)-convex function on the co-ordinates onwithandfor someand, thenwhere

When,

When,

By Lemma 1 and Hölder’s integral inequality, we haveWhen , using the co-ordinated -convexity of and by Lemma 2, we obtainSimilarly, we also haveApplying inequalities (21) to (24) into the inequality (20) yieldsWhen , similar to the proof of inequalities (21) to (24), we can writeSubstituting inequalities (25) to (28) into (20) leads to the inequality (18). Theorem 4 is thus proved.

Corollary 1

Under the conditions of Theorem 4,

Ifand, then

Ifand, then

Corollary 2

Under the conditions of Theorem 4,

If, then

If, then

Theorem 5

Lethave partial derivatives and. Ifis an (s, QC)-convex function on the co-ordinates onwithandfor some, , and, thenwhereis defined by (19).

When ,

When,

If , similar to the proof of the inequality (17), we can acquireIf , similarly one can see thatThe proof of Theorem 5 is complete.

Corollary 3

Under the conditions of Theorem 5, when,

If, then

if, then

Corollary 4

Under the conditions of Theorem 5, when,

If, then

If, then

Theorem 6

Lethave partial derivatives and. Ifis an (s, QC)-convex function on the co-ordinates onwithandfor someand, then

From Lemma 1, Hölder’s integral inequality, the co-ordinated -convexity of , and Lemma 2, it follows thatTheorem 6 is thus proved.

Conclusions

Our main results in this paper are Definitions  11 to 12 and those integral inequalities of Hermite–Hadamard type in Theorems 4 to 6.