# Hermite-Hadamard-type inequalities for $(g,{\phi}_{h})$-convex dominated functions

- Muhamet Emin Özdemir
^{1}, - Mustafa Gürbüz
^{2}Email author and - Havva Kavurmacı
^{3}

**2013**:184

https://doi.org/10.1186/1029-242X-2013-184

© Özdemir et al.; licensee Springer 2013

**Received: **29 November 2012

**Accepted: **27 March 2013

**Published: **18 April 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 Hermite-Hadamard-type inequalities for $(g,{\phi}_{h})$-convex dominated functions. Our results generalize the Hermite-Hadamard-type inequalities in Dragomir *et al.* (Tamsui Oxford Univ. J. Math. Sci. 18(2):161-173, 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 Hermite-Hadamard type, 2012).

**MSC:**26D15, 26D10, 05C38.

## Keywords

## 1 Introduction

which holds for all convex functions $f:[a,b]\to \mathbb{R}$, is known in the literature as Hermite-Hadamard’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

*g*-convex 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 *g*-convex dominated functions related to (1.1).

*g*-convex dominated mapping. Then, for all $a,b\in I$ with $a<b$,

In [1] and [2], the authors connect together some disparate threads through a Hermite-Hadamard motif. The first of these threads is the unifying concept of *g*-convex-dominated function. In [3], Hwang *et al.* established some inequalities of Fejér type for *g*-convex-dominated functions. Finally, in [4, 5] and [6], authors introduced several new different kinds of convex-dominated functions and then gave Hermite-Hadamard-type 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 non-negative functions defined on *J* and *I*, respectively.

**Definition 2**Let $h:J\to \mathbb{R}$ be a non-negative function, $h\not\equiv 0$. We say that $f:I\to \mathbb{R}$ is an

*h*-convex function, or that

*f*belongs to the class $SX(h,I)$, if

*f*is non-negative 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 *h*-concave, *i.e.* $f\in SV(h,I)$.

Youness have defined the *φ*-convex 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

*φ*-convex 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 *φ*-convexity using *h*-convexity 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 *φ*-convex, ${\phi}_{s}$-convex, *φ*-Godunova-Levin function and *φ*-*P*-function, 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 Hermite-Hadamard-type inequalities for $(g,{\phi}_{h})$-convex dominated functions. Our results generalize the Hermite-Hadamard-type 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 $(g,\phi )$-convex-dominated, $(g,{\phi}_{s})$-convex-dominated, $(g,{\phi}_{Q(I)})$-convex-dominated and $(g,{\phi}_{P(I)})$-convex-dominated 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*$g-f$*and*$g+f$*are*${\phi}_{h}$-*convex on**I*. - (3)
*There exist two*${\phi}_{h}$-*convex mappings**l*,*k**defined on**I**such that*$f=\frac{1}{2}(l-k)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}g=\frac{1}{2}(l+k).$

- 2
⟺ 3 Let we define the mappings

*f*,*g*as $f=\frac{1}{2}(l-k)$ and $g=\frac{1}{2}(l+k)$. Then if we sum and subtract*f*and*g*, respectively, we have $g+f=l$ and $g-f=k$. By the condition 2 in Lemma 1, the mappings $g-f$ 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*:

*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

If we integrate the above inequality with respect to *λ* over $[0,1]$, the inequality in (2.2) is proved.

So, the proof is completed. □

**Remark 1** If function *φ* is the identity in (2.4) and (2.5), then they reduce to Hermite-Hadamard type inequalities for convex dominated functions proved by Dragomir, Pearce and Pečarić in [2].

**Remark 2** If function *φ* is the identity in (2.6) and (2.7), then they reduce to Hermite-Hadamard type inequalities for $(g,s)$-convex dominated functions proved by Kavurmacı, Özdemir and Sarıkaya in [4].

**Remark 3** If function *φ* is the identity in (2.8), then it reduces to Hermite-Hadamard type inequality for $(g,Q(I))$-convex dominated functions proved by Özdemir, Tunç and Kavurmacı in [6].

**Remark 4** If function *φ* is the identity in (2.9) and (2.10), then they reduce to Hermite-Hadamard type inequalities for $(g,P(I))$-convex dominated functions proved by Özdemir, Tunç and Kavurmacı in [6].

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

## Authors’ Affiliations

## References

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## Copyright

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