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

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.

1 Introduction

The inequality

$$f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}{\int }_a^{b}f(x)dx\le \frac{f(a)+f(b)}{2}$$

(1.1)

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).

Let $g:I\to \mathbb{R}$ be a convex function and $f:I\to \mathbb{R}$ be a g-convex dominated mapping. Then, for all $a,b\in I$ with $a<b$,

$$|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)dx|\le \frac{1}{b-a}{\int }_a^{b}g(x)dx-g\left(\frac{a+b}{2}\right)$$

and

$$|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)dx|\le \frac{g(a)+g(b)}{2}-\frac{1}{b-a}{\int }_a^{b}g(x)dx.$$

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

$$f(\alpha x+(1-\alpha )y)\le h(\alpha )f(x)+h(1-\alpha )f(y).$$

(1.2)

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:

$$f(t\phi (x)+(1-t)\phi (y))\le tf(\phi (x))+(1-t)f(\phi (y)).$$

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

$$f(t\phi (x)+(1-t)\phi (y))\le h(t)f(\phi (x))+h(1-t)f(\phi (y))$$

(1.3)

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:

(2.1)

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

  $$f=\frac{1}{2}(l-k)\mathit{\text{and}}g=\frac{1}{2}(l+k).$$

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

$$(g+f)(t\phi (x)+(1-t)\phi (y))\le h(t)(g+f)(\phi (x))+h(1-t)(g+f)(\phi (y))$$

and

$$(g-f)(t\phi (x)+(1-t)\phi (y))\le h(t)(g-f)(\phi (x))+h(1-t)(g-f)(\phi (y))$$

which are equivalent to the ${\phi }_h$-convexity of $g+f$ and $g-f$, respectively.

• 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:

(2.2)

and

(2.3)

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 φ-function, 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 $x=t\phi (a)+(1-t)\phi (b)$ and use the fact that ${\int }_0^{1}h(t)dt={\int }_0^{1}h(1-t)dt$, 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

(2.4)

and

(2.5)

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].

Corollary 2 Under the assumptions of Theorem 1 with $h(t)=t^s$, $t,s\in (0,1)$, we have

(2.6)

and

(2.7)

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].

Corollary 3 Under the assumptions of Theorem 1 with $h(t)=\frac{1}{t}$, $t\in (0,1)$, we have

(2.8)

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].

Corollary 4 Under the assumptions of Theorem 1 with $h(t)=1$, $t\in (0,1)$, we have

(2.9)

and

(2.10)

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].