 Research
 Open Access
 Published:
HermiteHadamardtype inequalities for (g,{\phi}_{h})convex dominated functions
Journal of Inequalities and Applications volume 2013, Article number: 184 (2013)
Abstract
In this paper, we introduce the notion of (g,{\phi}_{h})convex dominated function and present some properties of them. Finally, we present a version of HermiteHadamardtype inequalities for (g,{\phi}_{h})convex dominated functions. Our results generalize the HermiteHadamardtype inequalities in Dragomir et al. (Tamsui Oxford Univ. J. Math. Sci. 18(2):161173, 2002), Kavurmacı et al. (New Definitions and Theorems via Different Kinds of Convex Dominated Functions, 2012) and Özdemir et al. (Two new different kinds of convex dominated functions and inequalities via HermiteHadamard type, 2012).
MSC:26D15, 26D10, 05C38.
1 Introduction
The inequality
which holds for all convex functions f:[a,b]\to \mathbb{R}, is known in the literature as HermiteHadamard’s inequality.
In [1], Dragomir and Ionescu introduced the following class of functions.
Definition 1 Let g:I\to \mathbb{R} be a convex function on the interval I. The function f:I\to \mathbb{R} is called gconvex dominated on I if the following condition is satisfied:
for all x,y\in I and \lambda \in [0,1].
In [2], Dragomir et al. proved the following theorem for gconvex dominated functions related to (1.1).
Let g:I\to \mathbb{R} be a convex function and f:I\to \mathbb{R} be a gconvex dominated mapping. Then, for all a,b\in I with a<b,
and
In [1] and [2], the authors connect together some disparate threads through a HermiteHadamard motif. The first of these threads is the unifying concept of gconvexdominated function. In [3], Hwang et al. established some inequalities of Fejér type for gconvexdominated functions. Finally, in [4, 5] and [6], authors introduced several new different kinds of convexdominated functions and then gave HermiteHadamardtype inequalities for these classes of functions.
In [7], Varošanec introduced the following class of functions.
I and J are intervals in ℝ, (0,1)\subseteq J and functions h and f are real nonnegative functions defined on J and I, respectively.
Definition 2 Let h:J\to \mathbb{R} be a nonnegative function, h\not\equiv 0. We say that f:I\to \mathbb{R} is an hconvex function, or that f belongs to the class SX(h,I), if f is nonnegative and for all x,y\in I, \alpha \in (0,1], we have
If the inequality (1.2) is reversed, then f is said to be hconcave, i.e. f\in SV(h,I).
Youness have defined the φconvex functions in [8]. A function \phi :[a,b]\to [c,d] where [a,b]\subset \mathbb{R}:
Definition 3 A function f:[a,b]\to \mathbb{R} is said to be φconvex on [a,b] if for every two points x\in [a,b], y\in [a,b] and t\in [0,1] the following inequality holds:
In [9], Sarıkaya defined a new kind of φconvexity using hconvexity as following:
Definition 4 Let I be an interval in ℝ and h:(0,1)\to (0,\mathrm{\infty}) be a given function. We say that a function f:I\to [0,\mathrm{\infty}) is {\phi}_{h}convex if
for all x,y\in I and t\in (0,1).
If inequality (1.3) is reversed, then f is said to be {\phi}_{h}concave. In particular, if f satisfies (1.3) with h(t)=t, h(t)={t}^{s} (s\in (0,1)), h(t)=\frac{1}{t}, and h(t)=1, then f is said to be φconvex, {\phi}_{s}convex, φGodunovaLevin function and φPfunction, respectively.
In the following sections, our main results are given: we introduce the notion of (g,{\phi}_{h})convex dominated function and present some properties of them. Finally, we present a version of HermiteHadamardtype inequalities for (g,{\phi}_{h})convex dominated functions. Our results generalize the HermiteHadamardtype inequalities in [2, 4] and [6].
2 (g,{\phi}_{h})convex dominated functions
Definition 5 Let h:(0,1)\to (0,\mathrm{\infty}) be a given function, g:I\to [0,\mathrm{\infty}) be a given {\phi}_{h}convex function. The real function f:I\to [0,\mathrm{\infty}) is called (g,{\phi}_{h})convex dominated on I if the following condition is satisfied:
for all x,y\in I and t\in (0,1).
In particular, if f satisfies (2.1) with h(t)=t, h(t)={t}^{s} (s\in (0,1)), h(t)=\frac{1}{t} and h(t)=1, then f is said to be (g,\phi )convexdominated, (g,{\phi}_{s})convexdominated, (g,{\phi}_{Q(I)})convexdominated and (g,{\phi}_{P(I)})convexdominated functions, respectively.
The next simple characterization of (g,{\phi}_{h})convex dominated functions holds.
Lemma 1 Let h:(0,1)\to (0,\mathrm{\infty}) be a given function, g:I\to [0,\mathrm{\infty}) be a given {\phi}_{h}convex function and f:I\to [0,\mathrm{\infty}) be a real function. The following statements are equivalent:

(1)
f is (g,{\phi}_{h})convex dominated on I.

(2)
The mappings gf and g+f are {\phi}_{h}convex on I.

(3)
There exist two {\phi}_{h}convex mappings l, k defined on I such that
f=\frac{1}{2}(lk)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}g=\frac{1}{2}(l+k).
Proof 1 ⟺ 2 The condition (2.1) is equivalent to
for all x,y\in I and t\in [0,1]. The two inequalities may be rearranged as
and
which are equivalent to the {\phi}_{h}convexity of g+f and gf, respectively.

2
⟺ 3 Let we define the mappings f, g as f=\frac{1}{2}(lk) and g=\frac{1}{2}(l+k). Then if we sum and subtract f and g, respectively, we have g+f=l and gf=k. By the condition 2 in Lemma 1, the mappings gf and g+f are {\phi}_{h}convex on I, so l, k are {\phi}_{h}convex mappings on I, also. □
Theorem 1 Let h:(0,1)\to (0,\mathrm{\infty}) be a given function, g:I\to [0,\mathrm{\infty}) be a given {\phi}_{h}convex function. If f:I\to [0,\mathrm{\infty}) is Lebesgue integrable and (g,{\phi}_{h})convex dominated on I for linear continuous function \phi :[a,b]\to [a,b], then the following inequalities hold:
and
for all x,y\in I and t\in [0,1].
Proof By the Definition 5 with t=\frac{1}{2}, x=\lambda a+(1\lambda )b, y=(1\lambda )a+\lambda b and \lambda \in [0,1], as the mapping f is (g,{\phi}_{h})convex dominated function, we have that
Then using the linearity of φfunction, we have
If we integrate the above inequality with respect to λ over [0,1], the inequality in (2.2) is proved.
To prove the inequality in (2.3), firstly we use the Definition 5 for x=a and y=b, we have
Then we integrate the above inequality with respect to t over [0,1], we get
If we substitute x=t\phi (a)+(1t)\phi (b) and use the fact that {\int}_{0}^{1}h(t)\phantom{\rule{0.2em}{0ex}}dt={\int}_{0}^{1}h(1t)\phantom{\rule{0.2em}{0ex}}dt, we get
So, the proof is completed. □
Corollary 1 Under the assumptions of Theorem 1 with h(t)=t, t\in (0,1), we have
and
Remark 1 If function φ is the identity in (2.4) and (2.5), then they reduce to HermiteHadamard type inequalities for convex dominated functions proved by Dragomir, Pearce and Pečarić in [2].
Corollary 2 Under the assumptions of Theorem 1 with h(t)={t}^{s}, t,s\in (0,1), we have
and
Remark 2 If function φ is the identity in (2.6) and (2.7), then they reduce to HermiteHadamard type inequalities for (g,s)convex dominated functions proved by Kavurmacı, Özdemir and Sarıkaya in [4].
Corollary 3 Under the assumptions of Theorem 1 with h(t)=\frac{1}{t}, t\in (0,1), we have
Remark 3 If function φ is the identity in (2.8), then it reduces to HermiteHadamard type inequality for (g,Q(I))convex dominated functions proved by Özdemir, Tunç and Kavurmacı in [6].
Corollary 4 Under the assumptions of Theorem 1 with h(t)=1, t\in (0,1), we have
and
Remark 4 If function φ is the identity in (2.9) and (2.10), then they reduce to HermiteHadamard type inequalities for (g,P(I))convex dominated functions proved by Özdemir, Tunç and Kavurmacı in [6].
References
Dragomir SS, Ionescu NM: On some inequalities for convexdominated functions. Anal. Numér. Théor. Approx. 1990, 19: 21–28.
Dragomir SS, Pearce CEM, Pečarić JE: Means, g convex dominated & Hadamardtype inequalities. Tamsui Oxford Univ. J. Math. Sci. 2002, 18(2):161–173.
Hwang SR, Ho MI, Wang CS: Inequalities of Fejér type for G convex dominated functions. Tamsui Oxford Univ. J. Math. Sci. 2009, 25(1):55–69.
Kavurmacı H, Özdemir ME, Sarıkaya MZ: New definitions and theorems via different kinds of convex dominated functions. RGMIA Research Report Collection (Online) 2012., 15: Article ID 9
Özdemir, ME, Kavurmacı, H, Tunç, M: HermiteHadamardtype inequalities for new different kinds of convex dominated functions. arXiv:1202.2054v1 [math.CA] 9 Feb 2012
Özdemir, ME, Tunç, M, Kavurmacı, H: Two new different kinds of convex dominated functions and inequalities via HermiteHadamard type. arXiv:1202.2054v1 [math.CA] 9 Feb 2012
Varošanec S: On h convexity. J. Math. Anal. Appl. 2007, 326: 303–311. 10.1016/j.jmaa.2006.02.086
Youness EA: E convex sets, E convex functions and E convex programming. J. Optim. Theory Appl. 1999, 102(2):439–450. 10.1023/A:1021792726715
Sarıkaya MZ:On HermiteHadamardtype inequalities for {\phi}_{h}convex functions. RGMIA Research Report Collection (Online) 2012., 15: Article ID 37
Acknowledgements
Dedicated to Professor Hari M Srivastava.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MG and HK carried out the design of the study and performed the analysis. MEÖ participated in its design and coordination. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Özdemir, M.E., Gürbüz, M. & Kavurmacı, H. HermiteHadamardtype inequalities for (g,{\phi}_{h})convex dominated functions. J Inequal Appl 2013, 184 (2013). https://doi.org/10.1186/1029242X2013184
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2013184
Keywords
 convex dominated functions
 HermiteHadamard inequality
 {\phi}_{h}convex functions
 (g,s)convex dominated functions