Gas-phase advanced oxidation as an integrated air pollution control technique

Getachew A. Adnew; Carl Meusinger; Nicolai Bork; Michael Gallus; Mildrid Kyte; Vitalijs Rodins; Thomas Rosenørn; Matthew S. Johnson; Getachew A. Adnew; Carl Meusinger; Nicolai Bork; Michael Gallus; Mildrid Kyte; Vitalijs Rodins; Thomas Rosenørn; Matthew S. Johnson

doi:10.3934/environsci.2016.1.141

AIMS Environmental Science

2016, Volume 3, Issue 1: 141-158. doi: 10.3934/environsci.2016.1.141

Previous Article Next Article

Research article Special Issues

Gas-phase advanced oxidation as an integrated air pollution control technique

1.
Department of Chemistry, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark
2.
Infuser ApS, Ole Maaløes vej 5, DK-2200 Copenhagen N, Denmark
3.
Rīga Stradiņš University, 16 Dzirciema Street, Rīga, LV-1007, Latvia
4.
Airlabs, Ole Maaløes vej 3, DK-2200 Copenhagen N, Denmark

Received: 21 January 2016 Accepted: 21 March 2016 Published: 28 March 2016

Gas-phase advanced oxidation (GPAO) is an emerging air cleaning technology based on the natural self-cleaning processes that occur in the Earth’s atmosphere. The technology uses ozone, UV-C lamps and water vapor to generate gas-phase hydroxyl radicals that initiate oxidation of a wide range of pollutants. In this study four types of GPAO systems are presented: a laboratory scale prototype, a shipping container prototype, a modular prototype, and commercial scale GPAO installations. The GPAO systems treat volatile organic compounds, reduced sulfur compounds, amines, ozone, nitrogen oxides, particles and odor. While the method covers a wide range of pollutants, effective treatment becomes difficult when temperature is outside the range of 0 to 80 °C, for anoxic gas streams and for pollution loads exceeding ca. 1000 ppm. Air residence time in the system and the rate of reaction of a given pollutant with hydroxyl radicals determine the removal efficiency of GPAO. For gas phase compounds and odors including VOCs (e.g. C₆H₆ and C₃H₈) and reduced sulfur compounds (e.g. H₂S and CH₃SH), removal efficiencies exceed 80%. The method is energy efficient relative to many established technologies and is applicable to pollutants emitted from diverse sources including food processing, foundries, water treatment, biofuel generation, and petrochemical industries.

Keywords:

Citation: Getachew A. Adnew, Carl Meusinger, Nicolai Bork, Michael Gallus, Mildrid Kyte, Vitalijs Rodins, Thomas Rosenørn, Matthew S. Johnson. Gas-phase advanced oxidation as an integrated air pollution control technique[J]. AIMS Environmental Science, 2016, 3(1): 141-158. doi: 10.3934/environsci.2016.1.141

Related Papers:

[1]	Muhammad Uzair Awan, Nousheen Akhtar, Artion Kashuri, Muhammad Aslam Noor, Yu-Ming Chu . 2D approximately reciprocal ρ-convex functions and associated integral inequalities. AIMS Mathematics, 2020, 5(5): 4662-4680. doi: 10.3934/math.2020299
[2]	Yousaf Khurshid, Muhammad Adil Khan, Yu-Ming Chu . Conformable fractional integral inequalities for GG- and GA-convex functions. AIMS Mathematics, 2020, 5(5): 5012-5030. doi: 10.3934/math.2020322
[3]	Hengxiao Qi, Muhammad Yussouf, Sajid Mehmood, Yu-Ming Chu, Ghulam Farid . Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity. AIMS Mathematics, 2020, 5(6): 6030-6042. doi: 10.3934/math.2020386
[4]	Mehmet Eyüp Kiriş, Miguel Vivas-Cortez, Gözde Bayrak, Tuğba Çınar, Hüseyin Budak . On Hermite-Hadamard type inequalities for co-ordinated convex function via conformable fractional integrals. AIMS Mathematics, 2024, 9(4): 10267-10288. doi: 10.3934/math.2024502
[5]	Ghulam Farid, Saira Bano Akbar, Shafiq Ur Rehman, Josip Pečarić . Boundedness of fractional integral operators containing Mittag-Leffler functions via (s,m)-convexity. AIMS Mathematics, 2020, 5(2): 966-978. doi: 10.3934/math.2020067
[6]	Yousaf Khurshid, Muhammad Adil Khan, Yu-Ming Chu . Conformable integral version of Hermite-Hadamard-Fejér inequalities via η-convex functions. AIMS Mathematics, 2020, 5(5): 5106-5120. doi: 10.3934/math.2020328
[7]	Yue Wang, Ghulam Farid, Babar Khan Bangash, Weiwei Wang . Generalized inequalities for integral operators via several kinds of convex functions. AIMS Mathematics, 2020, 5(5): 4624-4643. doi: 10.3934/math.2020297
[8]	M. Emin Özdemir, Saad I. Butt, Bahtiyar Bayraktar, Jamshed Nasir . Several integral inequalities for (α, s,m)-convex functions. AIMS Mathematics, 2020, 5(4): 3906-3921. doi: 10.3934/math.2020253
[9]	Muhammad Zakria Javed, Muhammad Uzair Awan, Loredana Ciurdariu, Omar Mutab Alsalami . Pseudo-ordering and $ \delta^{1} $-level mappings: A study in fuzzy interval convex analysis. AIMS Mathematics, 2025, 10(3): 7154-7190. doi: 10.3934/math.2025327
[10]	Xiuzhi Yang, G. Farid, Waqas Nazeer, Muhammad Yussouf, Yu-Ming Chu, Chunfa Dong . Fractional generalized Hadamard and Fejér-Hadamard inequalities for m-convex functions. AIMS Mathematics, 2020, 5(6): 6325-6340. doi: 10.3934/math.2020407

Abstract

1. Introduction

The inequalities discovered by C. Hermite and J. Hadamard for convex functions are considerable significant in the literature (see, e.g., ^[9], ^[18], [27,p.137]). These inequalities state that if $ f:I\rightarrow \mathbb{R} $ is a convex function on the interval $ I $ of real numbers and $ a, b\in I $ with $ a < b $, then

$\begin{equation} f\left( \frac{a+b}{2}\right) \leq \frac{1}{b-a}\int \limits_{a}^{b}f(x)dx\leq \frac{f\left( a\right) +f\left( b\right) }{2}. \end{equation}$ $

(1.1)

Both inequalities hold in the reversed direction if $ f $ is concave.

The Hermite-Hadamard inequality, which is the first fundamental result for convex mappings with a natural geometrical interpretation and many applications, has drawn attention much interest in elementary mathematics. A number of mathematicians have devoted their efforts.

The most well-known inequalities related to the integral mean of a convex function are the Hermite Hadamard inequalities or its weighted versions, the so-called Hermite-Hadamard-Fejér inequalities. In ^[17], Fejer gave a weighted generalization of the inequalities (1.1) as the following:

Theorem 1. $ f:[a, b]\rightarrow \mathbb{R} $, be a convex function, then the inequality

$\begin{equation} f\left( \frac{a+b}{2}\right) \int \limits_{a}^{b}g(x)dx\leq \int \limits_{a}^{b}f(x)g(x)dx\leq \frac{f(a)+f(b)}{2}\int \limits_{a}^{b}g(x)dx \end{equation}$ $

(1.2)

holds, where $ g:[a, b]\rightarrow \mathbb{R} $ is nonnegative, integrable, and symmetric about $ x = \frac{a+b}{2} $ (i.e. $ g(x) = g(a+b-x) $)$. $

In this paper we will establish some new Fejér type inequalities for the new concept of co-ordinated hyperbolic $ \rho $-convex functions.

The overall structure of the paper takes the form of four sections including introduction. The paper is organized as follows: we first give the definition of co-ordinated convex functions, the definition of fractional integrals and related Hermite-Hadamard inequality in Section 1. We also recall the concept of hyperbolic $ \rho $-convex functions and co-ordinated hyperbolic $ \rho $-convex functions introduced by Özçelik et. al in ^[23]. Moreover, we give a lemma and a theorem which will be frequently used in the next section. Some Hermite-Hadamard-Fejer type inequalities for co-ordinated hyperbolic $ \rho $-convex functions are obtained and some special cases of the results are also given in Section 2. Then, we also apply the inequalities obtained in Section 2 to establish some fractional Fejer type inequalities in Section 3. Finally, in Section 4, some conclusions and further directions of research are discussed.

A formal definition for co-ordinated convex function may be stated as follows:

Definition 1. A function $ f:\Delta : = [a, b]\times \lbrack c, d]\rightarrow \mathbb{R} $ is called co-ordinated convex on $ \Delta, $ for all $ (x, u), (y, v)\in \Delta $ and $ t, s\in \lbrack 0, 1] $, if it satisfies the following inequality:

$\begin{eqnarray} &&f(tx+(1-t)\text{ }y,su+(1-s)\text{ }v) \\ && \\ &\leq &ts\text{ }f(x,u)+t(1-s)f(x,v)+s(1-t)f(y,u)+(1-t)(1-s)f(y,v). \end{eqnarray}$ $

(1.3)

The mapping $ f $ is a co-ordinated concave on $ \Delta $ if the inequality (1.3) holds in reversed direction for all $ t, s\in \lbrack 0, 1] $ and $ (x, u), (y, v)\in \Delta $.

In ^[11], Dragomir proved the following inequalities which is Hermite-Hadamard type inequalities for co-ordinated convex functions on the rectangle from the plane $ \mathbb{R} ^{2}. $

Theorem 2. Suppose that $ f:\Delta : = \left[ a, b\right] \times \left[ c, d\right] \rightarrow \mathbb{R} $ is co-ordinated convex, then we have the following inequalities:

$\begin{eqnarray} f\left( \frac{a+b}{2},\frac{c+d}{2}\right) &\leq &\frac{1}{2}\left[ \frac{1}{ b-a}\int \limits_{a}^{b}f\left( x,\frac{c+d}{2}\right) dx+\frac{1}{d-c}\int \limits_{c}^{d}f\left( \frac{a+b}{2},y\right) dy\right] \\ &\leq &\frac{1}{(b-a)(d-c)}\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)dydx \\ &\leq &\frac{1}{4}\left[ \frac{1}{b-a}\int \limits_{a}^{b}f(x,c)dx+\frac{1}{ b-a}\int \limits_{a}^{b}f(x,d)dx\right. \\ &+&\left. \frac{1}{d-c}\int \limits_{c}^{d}f(a,y)dy+\frac{1}{d-c}\int \limits_{c}^{d}f(b,y)dy\right] \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4}. \end{eqnarray}$ $

(1.4)

The above inequalities are sharp. The inequalities in (1.4) hold in reverse direction if the mapping $ f $ is a co-ordinated concave mapping.

Over the years, the numerous studies have focused on to establish generalization of the inequality (1.1) and (1.4). For some of them, please see (^{[1,2,3,4,5,6,7,8]}, ^{[19,20,21,22,23,24,25,26]}, ^{[28,29,30,31,32,33,34,35,36]}).

Definition 2. ^[29] Let $ f\in L_{1}\left(\Delta \right). $The Riemann-Lioville integrals $ J_{a+, c+}^{\alpha, \beta }, J_{a+, d-}^{\alpha, \beta }, +J_{b-, c+}^{\alpha, \beta } $ and $ J_{b-, d-}^{\alpha, \beta } $of order $ \alpha, \beta > 0 $ with $ a, c\geq 0 $ are defined by

$\begin{eqnarray*} J_{a+,c+}^{\alpha ,\beta }f(x,y) & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\int \limits_{a}^{x}\int \limits_{c}^{y}\left( x-t\right) ^{\alpha -1}\left( y-s\right) ^{\beta -1}f\left( t,s\right) dsdt,\ \ x \gt a,\ y \gt c, \\ J_{a+,d-}^{\alpha ,\beta }f(x,y) & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\int \limits_{a}^{x}\int \limits_{y}^{d}\left( x-t\right) ^{\alpha -1}\left( s-y\right) ^{\beta -1}f\left( t,s\right) dsdt,\ \ x \gt a,\ y \gt d, \\ J_{b-,c+}^{\alpha ,\beta }f(x,y) & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\int \limits_{x}^{b}\int \limits_{c}^{y}\left( t-x\right) ^{\alpha -1}\left( y-s\right) ^{\beta -1}f\left( t,s\right) dsdt,\ \ x \lt b,\ y \gt c, \\ J_{b-,d-}^{\alpha ,\beta }f(x,y) & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\int \limits_{x}^{b}\int \limits_{y}^{d}\left( t-x\right) ^{\alpha -1}\left( s-y\right) ^{\beta -1}f\left( t,s\right) dsdt,\ \ x \lt b,\ y \lt d, \end{eqnarray*}$ $

respectively. Here, $ \Gamma $ is the Gamma funtion,

$\begin{equation*} J_{a+,c+}^{0,0}f(x,y) = J_{a+,d-}^{0,0}f(x,y) = J_{b-,c+}^{0,0}f(x,y) = J_{b-,d-}^{0,0}f(x,y) \end{equation*}$ $

and

$\begin{equation*} J_{a+,c+}^{1,1}f(x,y) = \int \limits_{a}^{x}\int \limits_{c}^{y}f\left( t,s\right) dsdt. \end{equation*}$ $

First, we give the definition of hyperbolic $ \rho $-convex functions and some related inequalities. Then we define the co-ordinated hyperbolic $ \rho $ -convex functions.

Definition 3. ^[10] A function $ f:I\rightarrow \mathbb{R} $ is said to be hyperbolic $ \rho $-convex, if for any arbitrary closed subinterval $ \left[ a, b\right] $ of $ I $ such that we have

$\begin{equation} f(x)\leq \frac{\sinh \left[ \rho \left( b-x\right) \right] }{\sinh \left[ \rho \left( b-a\right) \right] }f(a)+\frac{\sinh \left[ \rho \left( x-a\right) \right] }{\sinh \left[ \rho \left( b-a\right) \right] }f(b) \end{equation}$ $

(1.5)

for all $ x\in \left[ a, b\right]. $ If we take $ x = (1-t)a+tb, $ $ t\in \left[ 0, 1 \right] $ in (1.5), then the condition (1.5) becomes

$\begin{equation} f\left( \left( 1-t\right) a+tb\right) \leq \frac{\sinh \left[ \rho \left( 1-t\right) \left( b-a\right) \right] }{\sinh \left[ \rho \left( b-a\right) \right] }f(a)+\frac{\sinh \left[ \rho t\left( b-a\right) \right] }{\sinh \left[ \rho \left( b-a\right) \right] }f(b). \end{equation}$ $

(1.6)

If the inequality (1.5) holds with "$ \geq $", then the function will be called hyperbolic $ \rho $-concave on $ I. $

The following Hermite-Hadamard inequality for hyperbolic $ \rho $-convex function is proved by Dragomir in ^[10].

Theorem 3. Suppose that $ f:I\rightarrow \mathbb{R} $ is hyperbolic $ \rho $-convex on $ I $. Then for any $ a, b\in I $, we have

$\begin{equation} \frac{2}{\rho }f\left( \frac{a+b}{2}\right) \sinh \left[ \frac{\rho \left( b-a\right) }{2}\right] \leq \int \limits_{a}^{b}f(x)dx\leq \frac{f(a)+f(b)}{ \rho }\tanh \left[ \frac{\rho \left( b-a\right) }{2}\right] . \end{equation}$ $

(1.7)

Moreover in ^[12], Dragomir prove the following Hermite Hadamard-Fejer type inequalities for hyperbolic $ \rho $-convex functions.

Theorem 4. Assume that the function $ f:I\rightarrow \mathbb{R} $ is hyperbolic $ \rho $-convex on $ I $ and $ a, b\in I. $ Assume also that $ p:[a, b]\longrightarrow \mathbb{R} $ is a positive, symmetric and integrable function on $ [a, b], $ then we have

$\begin{eqnarray} &&f\left( \frac{a+b}{2}\right) \int \limits_{a}^{b}\cosh \left[ \rho \left( x-\frac{a+b}{2}\right) \right] p(x)dx \\ &\leq &\int \limits_{a}^{b}f(x)p(x)dx \\ &\leq &\frac{f(a)+f(b)}{2}\sec h\left[ \frac{\rho \left( b-a\right) }{2} \right] \int \limits_{a}^{b}\cosh \left[ \rho \left( x-\frac{a+b}{2}\right) \right] p(x)dx. \end{eqnarray}$ $

(1.8)

For the other inequalities for hyperbolic $ \rho $-convex functions, please refer to (^{[12,13,14,15]}).

Now we give the definition of co-ordinated hyperbolic $ \rho $-convex functions.

Definition 4. ^[23] A function $ f:\Delta \rightarrow \mathbb{R} $ is said to co-ordinated hyperbolic $ \rho $-convex on $ \Delta $, if the inequality

$\begin{eqnarray} f(x,y) &\leq &\frac{\sinh \left[ \rho _{1}\left( b-x\right) \right] }{\sinh \left[ \rho _{1}\left( b-a\right) \right] }\frac{\sinh \left[ \rho _{2}\left( d-y\right) \right] }{\sinh \left[ \rho _{2}\left( d-c\right) \right] }f(a,c)+\frac{\sinh \left[ \rho _{1}\left( b-x\right) \right] }{ \sinh \left[ \rho _{1}\left( b-a\right) \right] }\frac{\sinh \left[ \rho _{2}\left( y-c\right) \right] }{\sinh \left[ \rho _{2}\left( d-c\right) \right] }f(a,d) \\ && \\ &&+\frac{\sinh \left[ \rho _{1}\left( x-a\right) \right] }{\sinh \left[ \rho _{1}\left( b-a\right) \right] }\frac{\sinh \left[ \rho _{2}\left( d-y\right) \right] }{\sinh \left[ \rho _{2}\left( d-c\right) \right] }f(b,c)+\frac{ \sinh \left[ \rho _{1}\left( x-a\right) \right] }{\sinh \left[ \rho _{1}\left( b-a\right) \right] }\frac{\sinh \left[ \rho _{2}\left( y-c\right) \right] }{\sinh \left[ \rho _{2}\left( d-c\right) \right] }f(b,d). \end{eqnarray}$ $

(1.9)

holds.

If the inequality (1.9) holds with "$ \geq $", then the function will be called co-ordinated hyperbolic $ \rho $-concave on $ \Delta. $

If we take $ x = (1-t)a+tb $ and $ y = (1-s)c+sd $ for $ t, s, \in \left[ 0, 1\right], $ then the inequality (1.9) can be written as

$\begin{eqnarray} &&f(\left( 1-t\right) a+tb,\left( 1-s\right) c+sd) \\ && \\ &\leq &\frac{\sinh \left[ \rho _{1}\left( 1-t\right) \left( b-a\right) \right] }{\sinh \left[ \rho _{1}\left( b-a\right) \right] }\frac{\sinh \left[ \rho _{2}\left( 1-s\right) \left( d-y\right) \right] }{\sinh \left[ \rho _{2}\left( d-c\right) \right] }f(a,c) \\ &&+\frac{\sinh \left[ \rho _{1}\left( 1-t\right) \left( b-a\right) \right] }{ \sinh \left[ \rho _{1}\left( b-a\right) \right] }\frac{\sinh \left[ \rho _{2}s\left( d-y\right) \right] }{\sinh \left[ \rho _{2}\left( d-c\right) \right] }f(a,d) \\ && \\ &&+\frac{\sinh \left[ \rho _{1}t\left( b-a\right) \right] }{\sinh \left[ \rho _{1}\left( b-a\right) \right] }\frac{\sinh \left[ \rho _{2}\left( 1-s\right) \left( d-y\right) \right] }{\sinh \left[ \rho _{2}\left( d-c\right) \right] }f(b,c) \\ &&+\frac{\sinh \left[ \rho _{1}\left( b-a\right) \right] }{\sinh \left[ \rho _{1}\left( b-a\right) \right] }\frac{\sinh \left[ \rho _{2}s\left( d-y\right) \right] }{\sinh \left[ \rho _{2}\left( d-c\right) \right] }f(b,d). \end{eqnarray}$ $

(1.10)

Now we give the following useful lemma:

Lemma 1. ^[23] If $ f:\Delta = \left[ a, b\right] \times \left[ c, d \right] \rightarrow \mathbb{R} $ is co-ordinated $ \rho $-convex function on $ \Delta $, then we have the following inequality

$\begin{eqnarray} &&\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] \cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] f\left( \frac{a+b}{2},\frac{ c+d}{2}\right) \\ && \\ &\leq &\frac{1}{4}\left[ f(x,y)+f(x,c+d-y)+f(a+b-x,y)+f(a+b-x,c+d-y)\right] \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4}\frac{\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] }{\cosh \left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] }\frac{\cosh \left[ \rho _{2}\left( y- \frac{c+d}{2}\right) \right] }{\cosh \left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] } \end{eqnarray}$ $

(1.11)

for all $ (x, y)\in \Delta. $

2. Fejer type inequalities for co-ordinated hyperbolic $ \rho $-convex functions

Theorem 5. Let $ p:\Delta \rightarrow \mathbb{R} $ be a positive, integrable and symmetric about $ \frac{a+b}{2} $ and $ \frac{ c+d}{2}. $ Let, $ f:\Delta \rightarrow \mathbb{R} $ be a co-ordinated hyperbolic $ \rho $-convex functions on $ \Delta $. We have the following Hermite-Hadamard-Fejer type inequalities:

(2.1)

Proof. Multiplying the inequality (1.1) by $ p(x, y) > 0 $ and then integrating with respect to $ \left(x, y\right) $ on $ \Delta, $ we obtain

(2.2)

Since $ p $ is symmetric about $ \frac{a+b}{2} $ and $ \frac{c+d}{2}, $ one can show that

$\begin{eqnarray*} \int \limits_{a}^{b}\int \limits_{c}^{d}f(x,c+d-y)p(x,y)dydx & = &\int \limits_{a}^{b}\int \limits_{c}^{d}f(a+b-x,y)p(x,y)dydx \\ & = &\int \limits_{a}^{b}\int \limits_{c}^{d}f(a+b-x,c+d-y)p(x,y)dydx \\ & = &\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)p(x,y)dydx. \end{eqnarray*}$ $

This completes the proof.

Remark 1. If we choose $ p(x, y) = 1 $ in Theorem 5, then we have the following the inequality

$\begin{eqnarray*} &&\frac{4}{\rho _{1}\rho _{2}}\sinh \left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \sinh \left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \\ && \\ &\leq &\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)dydx \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{\rho _{1}\rho _{2}}\tanh \left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \tanh \left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \end{eqnarray*}$ $

which is proved by Özçelik et. al in ^[23].

Corollary 1. Suppose that all assumptions of Theorem 5 are satisfied. Then we have the following inequality,

$\begin{eqnarray} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \int \limits_{a}^{b}\int \limits_{c}^{d}w\left( x,y\right) dydx \\ && \\ &\leq &\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)w(x,y)\sec h\left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] \sec h\left[ \rho _{2}\left( y- \frac{c+d}{2}\right) \right] dydx \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \int \limits_{a}^{b}\int \limits_{c}^{d}w\left( x,y\right) dydx. \end{eqnarray}$ $

(2.3)

Proof. Let us define the function $ p(x, y) $ by

$\begin{equation*} w(x,y) = \frac{p(x,y)}{\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] \cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] }. \end{equation*}$ $

Clearly, $ w(x.y) $ is a a positive, integrable and symmetric about $ \frac{a+b }{2} $ and $ \frac{c+d}{2}. $ If we apply Theorem 5 for the function $ w(x, y) $ then we establish the desired inequality (2.3).

Remark 2. If we choose $ w(x, y) = 1 $ for all $ \left(x, y\right) \epsilon \Delta $ in Corollary 1, then we have the following the inequality

$\begin{eqnarray} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \\ && \\ &\leq &\frac{1}{(b-a)(d-c)}\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)\sec h\left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] \sec h\left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] dydx \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] . \end{eqnarray}$ $

(2.4)

which is proved by Özçelik et. al in ^[23].

Theorem 6. Let $ p:\Delta \rightarrow \mathbb{R} $ be a positive, integrable and symmetric about $ \frac{a+b}{2} $ and $ \frac{ c+d}{2}. $ Let $ f:\Delta \rightarrow \mathbb{R} $ be a co-ordinated hyperbolic $ \rho $-convex on $ \Delta $, then we have the following Hermite-Hadamard-Fejer type inequalities

$\begin{eqnarray} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \int \limits_{a}^{b}\int \limits_{c}^{d}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] \cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] p(x,y)dydx \\ &\leq &\frac{1}{2}\left[ \int \limits_{a}^{b}\int \limits_{c}^{d}f\left( x, \frac{c+d}{2}\right) \cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] p(x,y)dydx\right. \\ &&+\left. \int \limits_{a}^{b}\int \limits_{c}^{d}f\left( \frac{a+b}{2} ,y\right) \cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] p(x,y)dydx\right] \\ &\leq &\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)p(x,y)dydx \\ &\leq &\frac{1}{4}\left[ \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2} \right] \int \limits_{a}^{b}\int \limits_{c}^{d}\left[ f(x,c)+f(x,d)\right] \cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] p(x,y)dydx\right. \\ &&+\left. \sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \int \limits_{a}^{b}\int \limits_{c}^{d}\left[ f(a,y)+f(b,y)\right] \cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] p(x,y)dydx\right] \\ &\leq &\frac{f(a,c)+f(b,c)+f(a,d)+f(b,d)}{4}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \\ &&\times \int \limits_{a}^{b}\int \limits_{c}^{d}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] \cosh \left[ \rho _{2}\left( y- \frac{c+d}{2}\right) \right] p(x,y)dydx. \end{eqnarray}$ $

(2.5)

Proof. Since $ f $ is co-ordinated hyperbolic $ \rho $-convex on $ \Delta $, if we define the mappings $ f_{x}:\left[ c, d\right] \rightarrow \mathbb{R}, $ $ f_{x}(y) = f(x, y) $ and $ p_{x}:\left[ c, d\right] \rightarrow \mathbb{R}, $ $ p_{x}(y) = p(x, y), $ then $ f_{x}(y) $ is hyperbolic $ \rho $-convex on $ \left[ c, d\right] $ and $ p_{x}(y) $ is positive, integrable and symmetric about $ \frac{c+d}{2} $ for all $ x\in \left[ a, b\right]. $ If we apply the inequality (1.8) for the hyperbolic $ \rho $-convex function $ f_{x}(y), $ then we have

$\begin{eqnarray} &&f_{x}\left( \frac{c+d}{2}\right) \int \limits_{c}^{d}\cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] p_{x}(y)dy \\ &\leq &\int \limits_{c}^{d}f_{x}(y)p_{x}(y)dy \\ &\leq &\frac{f_{x}(c)+f_{x}(d)}{2}\sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \int \limits_{c}^{d}\cosh \left[ \rho _{2}\left( y- \frac{c+d}{2}\right) \right] p_{x}(y)dy. \end{eqnarray}$ $

(2.6)

That is,

$\begin{eqnarray} &&f\left( x,\frac{c+d}{2}\right) \int \limits_{c}^{d}\cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] p(x,y)dy \\ &\leq &\int \limits_{c}^{d}f(x,y)p(x,y)dy \\ &\leq &\frac{f(x,c)+f(x,d)}{2}\sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \int \limits_{c}^{d}\cosh \left[ \rho _{2}\left( y-\frac{c+d}{2} \right) \right] p(x,y)dy. \end{eqnarray}$ $

(2.7)

Integrating the inequality (2.7) with respect to $ x $ from $ a $ to $ b, $ we obtain

$\begin{eqnarray} &&\int \limits_{a}^{b}\int \limits_{c}^{d}f\left( x,\frac{c+d}{2}\right) \cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] p(x,y)dydx \\ &\leq &\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)p(x,y)dydx \\ &\leq &\frac{1}{2}\int \limits_{a}^{b}\int \limits_{c}^{d}\left[ f(x,c)+f(x,d)\right] \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2} \right] \cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] p(x,y)dydx. \end{eqnarray}$ $

(2.8)

Similarly, as $ f $ is co-ordinated hyperbolic $ \rho $-convex on $ \Delta $, if we define the mappings $ f_{y}:\left[ a, b\right] \rightarrow \mathbb{R}, $ $ f_{y}(x) = f(x, y) $ and $ p_{y}:\left[ a, b\right] \rightarrow \mathbb{R}, $ $ p_{y}(x) = p(x, y), $ then $ f_{y}(x) $ is hyperbolic $ \rho $-convex on $ \left[ a, b\right] $ and $ p_{y}(x) $ is positive, integrable and symmetric about $ \frac{a+b}{2} $ for all $ y\in \left[ c, d\right]. $ Utilizing the inequality (1.8) for the hyperbolic $ \rho $-convex function $ f_{y}(x), $ then we obtain the inequality

$\begin{eqnarray} &&f_{y}\left( \frac{a+b}{2}\right) \int \limits_{a}^{b}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] p_{y}(x)dx \\ &\leq &\int \limits_{a}^{b}f_{y}(x)p_{y}(x)dx \\ &\leq &\frac{f_{y}(a)+f_{y}(b)}{2}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \int \limits_{a}^{b}\cosh \left[ \rho _{1}\left( x- \frac{a+b}{2}\right) \right] p_{y}(x)dx \end{eqnarray}$ $

(2.9)

i.e.

$\begin{eqnarray} &&f\left( \frac{a+b}{2},y\right) \int \limits_{a}^{b}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] p(x,y)dx \\ &\leq &\int \limits_{a}^{b}f(x,y)p(x,y)dx \\ &\leq &\frac{f(a,y)+f(b,y)}{2}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \int \limits_{a}^{b}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2} \right) \right] p(x,y)dx. \end{eqnarray}$ $

(2.10)

Integrating the inequality (2.10) with respect to $ y $ on $ \left[ c, d \right], $ we get

$\begin{eqnarray} &&\int \limits_{a}^{b}\int \limits_{c}^{d}f\left( \frac{a+b}{2},y\right) \cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] p(x,y)dydx \\ &\leq &\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)p(x,y)dydx \\ &\leq &\frac{1}{2}\int \limits_{a}^{b}\int \limits_{c}^{d}\left[ f(a,y)+f(b,y)\right] \sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2} \right] \cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] p(x,y)dydx. \end{eqnarray}$ $

(2.11)

Summing the inequalities (2.8) and (2.11), we obtain the second and third inequalities in (2.5).

Since $ f\left(\frac{a+b}{2}, y\right) $ is hyperbolic $ \rho $-convex on $ \left[ c, d\right] $ and $ p_{x}(y) $ is positive, integrable and symmetric about $ \frac{c+d}{2}, $ using the first inequality in (1.8), we have

$\begin{eqnarray} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \int \limits_{c}^{d}\cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] p(x,y)dy \\ &\leq &\int \limits_{c}^{d}f\left( \frac{a+b}{2},y\right) p(x,y)dy. \end{eqnarray}$ $

(2.12)

Multiplying the inequality (2.12) by $ \cosh \left[ \rho _{1}\left(x- \frac{a+b}{2}\right) \right] $ and integrating resulting inequality with respect to $ x $ on $ \left[ a, b\right], $ we get

$\begin{eqnarray} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \int \limits_{a}^{b}\int \limits_{c}^{d}\cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] \cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] p(x,y)dydx \\ &\leq &\int \limits_{a}^{b}\int \limits_{c}^{d}f\left( \frac{a+b}{2} ,y\right) \cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] p(x,y)dydx. \end{eqnarray}$ $

(2.13)

Since $ f\left(x, \frac{c+d}{2}\right) $ is hyperbolic $ \rho $-convex on $ \left[ a, b\right] $ and $ p_{y}(x) $ is positive, integrable and symmetric about $ \frac{a+b}{2}, $ utilizing the first inequality in (1.8), we have the following inequality

$\begin{eqnarray} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \int \limits_{a}^{b}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] p(x,y)dx \\ &\leq &\int \limits_{a}^{b}f\left( x,\frac{c+d}{2}\right) p(x,y)dx. \end{eqnarray}$ $

(2.14)

Multiplying the inequality (2.14) by $ \cosh \left[ \rho _{2}\left(y- \frac{c+d}{2}\right) \right] $ and integrating resulting inequality with respect to $ y $ on $ \left[ c, d\right], $ we get

(2.15)

From the inequalities (2.13) and (2.15), we obtain the first inequality in (2.5).

For the proof of last inequality in (2.5), using the second inequality in (1.8) for the hyperbolic $ \rho $-convex functions $ f(x, c) $ and $ f(x, d) $ on $ \left[ a, b\right] $ and for the symmetric function $ p_{y}(x) $, we obtain the inequalities

$\begin{eqnarray} &&\int \limits_{a}^{b}f(x,c)p(x,y)dx \\ &\leq &\frac{f(a,c)+f(b,c)}{2}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \int \limits_{a}^{b}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2} \right) \right] p(x,y)dx \end{eqnarray}$ $

(2.16)

and

$\begin{eqnarray} &&\int \limits_{a}^{b}f(x,d)p(x,y)dx \\ &\leq &\frac{f(a,d)+f(b,d)}{2}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \int \limits_{a}^{b}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2} \right) \right] p(x,y)dx. \end{eqnarray}$ $

(2.17)

If we multiply the inequalities (2.16) and (2.17) by $ \sec h \left[ \frac{\rho _{2}\left(d-c\right) }{2}\right] \cosh \left[ \rho _{2}\left(y-\frac{c+d}{2}\right) \right] $ and integrating the resulting inequalities on $ \left[ c, d\right], $ then we have

$\begin{eqnarray} &&\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,c)\sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \cosh \left[ \rho _{2}\left( y-\frac{c+d}{2 }\right) \right] p(x,y)dydx \\ &\leq &\frac{f(a,c)+f(b,c)}{2}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \\ &&\times \int \limits_{a}^{b}\int \limits_{c}^{d}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] \cosh \left[ \rho _{2}\left( y- \frac{c+d}{2}\right) \right] p(x,y)dydx \end{eqnarray}$ $

(2.18)

and

$\begin{eqnarray} &&\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,d)\sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \cosh \left[ \rho _{2}\left( y-\frac{c+d}{2 }\right) \right] p(x,y)dydx \\ &\leq &\frac{f(a,d)+f(b,d)}{2}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \\ &&\times \int \limits_{a}^{b}\int \limits_{c}^{d}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] \cosh \left[ \rho _{2}\left( y- \frac{c+d}{2}\right) \right] p(x,y)dydx. \end{eqnarray}$ $

(2.19)

Similarly, applying the second inequality in (1.8) for the hyperbolic $ \rho $-convex functions $ f(a, y) $ and $ f(b, y) $ on $ \left[ c, d\right] $ and for the symmetric function $ p_{x}(y), $ we have

$\begin{eqnarray} &&\int \limits_{c}^{d}f(a,y)p(x,y)dy \\ &\leq &\frac{f(a,c)+f(a,d)}{2}\sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \int \limits_{c}^{d}\cosh \left[ \rho _{2}\left( y-\frac{c+d}{2} \right) \right] p(x,y)dy \end{eqnarray}$ $

(2.20)

and

$\begin{eqnarray} &&\int \limits_{c}^{d}f(b,y)p(x,y)dy \\ &\leq &\frac{f(b,c)+f(b,d)}{2}\sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \int \limits_{c}^{d}\cosh \left[ \rho _{2}\left( y-\frac{c+d}{2} \right) \right] p(x,y)dy. \end{eqnarray}$ $

(2.21)

Multiplying the inequalities (2.20) and (2.21) by $ \sec h\left[ \frac{\rho _{1}\left(b-a\right) }{2}\right] \cosh \left[ \rho _{1}\left(x- \frac{a+b}{2}\right) \right] $ and integrating the resulting inequalities on $ \left[ a, b\right], $ then we have

$\begin{eqnarray} &&\int \limits_{a}^{b}\int \limits_{c}^{d}f(a,y)\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \cosh \left[ \rho _{1}\left( x-\frac{a+b}{2 }\right) \right] p(x,y)dydx \\ &\leq &\frac{f(a,c)+f(a,d)}{2}\sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \\ &&\times \int \limits_{a}^{b}\int \limits_{c}^{d}\cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] \cosh \left[ \rho _{1}\left( x- \frac{a+b}{2}\right) \right] p(x,y)dydx \end{eqnarray}$ $

(2.22)

and

$\begin{eqnarray} &&\int \limits_{a}^{b}\int \limits_{c}^{d}f(b,y)\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \cosh \left[ \rho _{1}\left( x-\frac{a+b}{2 }\right) \right] p(x,y)dydx \\ &\leq &\frac{f(b,c)+f(b,d)}{2}\sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \\ &&\times \int \limits_{a}^{b}\int \limits_{c}^{d}\cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] \cosh \left[ \rho _{1}\left( x- \frac{a+b}{2}\right) \right] p(x,y)dydx. \end{eqnarray}$ $

(2.23)

Summing the inequalities (2.18), (2.19), (2.22) and (2.23), we establish the last inequality in (2.5). This completes the proof.

Remark 3. If we choose $ p(x, y) = 1 $ in Theorem 6, then we have

$\begin{eqnarray} &&\frac{4}{\rho _{1}\rho _{2}}\sinh \left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \sinh \left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \\ && \\ &\leq &\frac{1}{\rho _{1}}\sinh \left[ \frac{\rho _{1}\left( b-a\right) }{2} \right] \int \limits_{c}^{d}f\left( \frac{a+b}{2},y\right) dy+\frac{1}{\rho _{2}}\sinh \left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \int \limits_{a}^{b}f\left( x,\frac{c+d}{2}\right) dx \\ && \\ &\leq &\int \limits_{a}^{b}\int \limits_{c}^{d}f\left( x,y\right) dydx \\ && \\ &\leq &\frac{1}{2}\left[ \frac{1}{\rho _{2}}\tanh \left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \int \limits_{a}^{b}\left[ f\left( x,c\right) +f\left( x,d\right) \right] dx\right. \\ &&\left. +\frac{1}{\rho _{1}}\tanh \left[ \frac{\rho _{1}\left( b-a\right) }{ 2}\right] \int \limits_{c}^{d}\left[ f\left( a,y\right) +f\left( b,y\right) \right] dy\right] \\ && \\ &\leq &\tanh \left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \tanh \left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \frac{f\left( a,c\right) +f\left( a,d\right) +f\left( b,c\right) +f\left( b,d\right) }{\rho _{1}\rho _{2}} \end{eqnarray}$ $

(2.24)

which is proved by Özçelik et. al in ^[23].

Remark 4. Choosing $ \rho _{1} = \rho _{2} = 0 $ in Theorem 6, we obtain

$\begin{eqnarray*} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \int \limits_{a}^{b}\int \limits_{c}^{d}p(x,y)dydx \\ &\leq &\frac{1}{2}\int \limits_{a}^{b}\int \limits_{c}^{d}\left[ f\left( x, \frac{c+d}{2}\right) +f\left( \frac{a+b}{2},y\right) \right] p(x,y)dydx \\ && \\ &\leq &\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)p(x,y)dydx \\ && \\ &\leq &\frac{1}{4}\int \limits_{a}^{b}\int \limits_{c}^{d}\left[ f(x,c)+f(x,d)+f(a,y)+f(b,y)\right] p(x,y)dydx \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4}\int \limits_{a}^{b}\int \limits_{c}^{d}p(x,y)dydx. \end{eqnarray*}$ $

which is proved by Budak and Sarikaya in ^[5].

Corollary 2. Let $ g_{1}:\left[ a, b\right] \rightarrow \mathbb{R} $ and $ g_{1}:\left[ c, d\right] \rightarrow \mathbb{R} $ be two positive, integrable and symmetric about $ \frac{a+b}{2} $ and $ \frac{ c+d}{2}, $ respectively. If we choose $ p(x, y) = \frac{g_{1}(x)g_{2}(y)}{ G_{1}G_{2}} $ for all $ \left(x, y\right) \in \Delta $ in Theorem 6, then we have

$\begin{eqnarray} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \\ &\leq &\frac{1}{2}\left[ \frac{1}{G_{1}}\int \limits_{a}^{b}f\left( x,\frac{ c+d}{2}\right) g_{1}(x)dx+\frac{1}{G_{2}}\int \limits_{c}^{d}f\left( \frac{ a+b}{2},y\right) g_{2}(y)dy\right] \\ &\leq &\frac{1}{G_{1}G_{2}}\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)g_{1}(x)g_{2}(y)dydx \\ &\leq &\frac{1}{4}\left[ \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2} \right] \frac{1}{G_{1}}\int \limits_{a}^{b}\left[ f(x,c)+f(x,d)\right] g_{1}(x)dx\right. \\ &&+\left. \sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \frac{1}{ G_{2}}\int \limits_{c}^{d}\left[ f(a,y)+f(b,y)\right] g_{2}(y)dy\right] \\ &\leq &\frac{f(a,c)+f(b,c)+f(a,d)+f(b,d)}{4}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \end{eqnarray}$ $

(2.25)

where

$\begin{equation*} G_{1} = \int \limits_{a}^{b}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2} \right) \right] g_{1}(x)dx\mathit{\text{and}}G_{2} = \int \limits_{c}^{d}\cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] g_{2}(y)dy. \end{equation*}$ $

Remark 5. If we choose $ \rho _{1} = \rho _{2} = 0 $ in Corollary 2, then we have

$\begin{eqnarray*} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \\ && \\ &\leq &\frac{1}{2}\left[ \frac{1}{G_{1}}\int \limits_{a}^{b}f\left( x,\frac{ c+d}{2}\right) g_{1}(x)dx+\frac{1}{G_{2}}\int \limits_{c}^{d}f\left( \frac{ a+b}{2},y\right) g_{2}(y)dy\right] \\ && \\ &\leq &\frac{1}{G_{1}G_{2}}\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)g_{1}(x)g_{2}(y)dydx \\ && \\ &\leq &\frac{1}{4}\left[ \frac{1}{G_{1}}\int \limits_{a}^{b}\left[ f(x,c)+f(x,d)\right] g_{1}(x)dx+\frac{1}{G_{2}}\int \limits_{c}^{d}\left[ f(a,y)+f(b,y)\right] g_{2}(y)dy\right] \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4} \end{eqnarray*}$ $

which is proved by Farid et al. in ^[16].

3. Fractional inequalities for co-ordinated hyperbolic $ \rho $-convex functions

In this section we obtain some fractional Hermite-Hadamard an Fejer type inequalities for co-ordinated hyperbolic $ \rho $-convex functions.

Theorem 7. If $ f:\Delta \rightarrow \mathbb{R} $ is a co-ordinated hyperbolic $ \rho $-convex functions on $ \Delta $, then we have the following Hermite-Hadamard and Fejer type inequalities,

$\begin{eqnarray*} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) H(\alpha ,\beta ) \\ && \\ &\leq &\left[ J_{a+,c+}^{\alpha ,\beta }f(b,d)+J_{a+,d-}^{\alpha ,\beta }f(b,c)+J_{b-,c+}^{\alpha ,\beta }f(a,d)+J_{b-,d-}^{\alpha ,\beta }f(a,c) \right] \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] H(\alpha ,\beta ) \end{eqnarray*}$ $

where

$\begin{eqnarray*} H(\alpha ,\beta ) & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\int \limits_{a}^{b}\int \limits_{c}^{d}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] \cosh \left[ \rho _{2}\left( y- \frac{c+d}{2}\right) \right] \\ &&\times \left[ (b-x)^{\alpha -1}(d-y)^{\beta -1}+(b-x)^{\alpha -1}(y-c)^{\beta -1}\right. \\ &&+\left. (x-a)^{\alpha -1}(d-y)^{\beta -1}+(x-a)^{\alpha -1}(y-c)^{\beta -1} \right] dydx. \end{eqnarray*}$ $

Proof. If we apply Theorem 5 for the symmetric function

$\begin{eqnarray*} p(x,y) & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \left[ (b-x)^{\alpha -1}(d-y)^{\beta -1}+(b-x)^{\alpha -1}(y-c)^{\beta -1}\right. \\ &&+\left. (x-a)^{\alpha -1}(d-y)^{\beta -1}+(x-a)^{\alpha -1}(y-c)^{\beta -1} \right] , \end{eqnarray*}$ $

then we get the following inequality

$\begin{eqnarray*} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) H(\alpha ,\beta ) \\ && \\ &\leq &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)\left[ (b-x)^{\alpha -1}(d-y)^{\beta -1}+(b-x)^{\alpha -1}(y-c)^{\beta -1}\right. \\ &&+\left. (x-a)^{\alpha -1}(d-y)^{\beta -1}+(x-a)^{\alpha -1}(y-c)^{\beta -1} \right] dydx \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] H(\alpha ,\beta ). \end{eqnarray*}$ $

From the definition of the double fractional integrals we have

$\begin{eqnarray*} &&\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)\left[ (b-x)^{\alpha -1}(d-y)^{\beta -1}+(b-x)^{\alpha -1}(y-c)^{\beta -1}\right. \\ &&+\left. (x-a)^{\alpha -1}(d-y)^{\beta -1}+(x-a)^{\alpha -1}(y-c)^{\beta -1} \right] dydx \\ & = &\left[ J_{a+,c+}^{\alpha ,\beta }f(b,d)+J_{a+,d-}^{\alpha ,\beta }f(b,c)+J_{b-,c+}^{\alpha ,\beta }f(a,d)+J_{b-,d-}^{\alpha ,\beta }f(a,c) \right] \end{eqnarray*}$ $

which completes the proof.

Remark 6. If we choose $ \rho _{1} = \rho _{2} = 0 $ in Theorem 7, then we have the following fractional Hermite-Hadamard inequality,

$\begin{eqnarray*} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \\ && \\ &\leq &\frac{\Gamma \left( \alpha +1\right) \Gamma \left( \beta +1\right) }{ 4(b-a)^{\alpha }(d-c)^{\beta }}\left[ J_{a+,c+}^{\alpha ,\beta }f(b,d)+J_{a+,d-}^{\alpha ,\beta }f(b,c)+J_{b-,c+}^{\alpha ,\beta }f(a,d)+J_{b-,d-}^{\alpha ,\beta }f(a,c)\right] \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4} \end{eqnarray*}$ $

which was proved by Sarikaya in [29,Theorem 4].

Remark 7. If we choose $ \alpha $ $ = \beta = 1 $ in Theorem 7, then we have

$\begin{equation*} H(1,1) = \frac{16}{\rho _{1}\rho _{2}}\sinh \left( \frac{\rho _{1}(b-a)}{2} \right) \sinh \left( \frac{\rho _{2}(d-c)}{2}\right) . \end{equation*}$ $

Thus, we get the following Hermite-Hadamard inequality,

$\begin{eqnarray*} &&\frac{4}{\rho _{1}\rho _{2}}f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \sinh \left( \frac{\rho _{1}(b-a)}{2}\right) \sinh \left( \frac{\rho _{2}(d-c)}{2}\right) \\ && \\ &\leq &\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)dydx \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{\rho _{1}\rho _{2}}\tanh \left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \tanh \left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] \end{eqnarray*}$ $

which is proved by Özçelik et al. in ^[23].

Theorem 8. Let $ p:\Delta \rightarrow \mathbb{R} $ be a positive, integrable and symmetric about $ \frac{a+b}{2} $ and $ \frac{ c+d}{2}. $ If $ f:\Delta \rightarrow \mathbb{R} $ is a co-ordinated hyperbolic $ \rho $-convex functions on $ \Delta $, then we have the following Hermite-Hadamard-Fejer type inequalities,

$\begin{eqnarray*} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) H_{p}(\alpha ,\beta ) \\ && \\ &\leq &\left[ J_{a+,c+}^{\alpha ,\beta }\left( fp\right) (b,d)+J_{a+,d-}^{\alpha ,\beta }\left( fp\right) (b,c)+J_{b-,c+}^{\alpha ,\beta }\left( fp\right) (a,d)+J_{b-,d-}^{\alpha ,\beta }\left( fp\right) (a,c)\right] \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] H_{p}(\alpha ,\beta ) \end{eqnarray*}$ $

where

$\begin{eqnarray*} H_{p}(\alpha ,\beta ) & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\int \limits_{a}^{b}\int \limits_{c}^{d}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] \cosh \left[ \rho _{2}\left( y- \frac{c+d}{2}\right) \right] \\ &&\times \left[ (b-x)^{\alpha -1}(d-y)^{\beta -1}+(b-x)^{\alpha -1}(y-c)^{\beta -1}\right. \\ &&+\left. (x-a)^{\alpha -1}(d-y)^{\beta -1}+(x-a)^{\alpha -1}(y-c)^{\beta -1} \right] p(x,y)dydx. \end{eqnarray*}$ $

Proof. Let us define the function $ k(x, y) $ by

$\begin{eqnarray*} k(x,y) & = &\frac{p(x,y)}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\left[ (b-x)^{\alpha -1}(d-y)^{\beta -1}+(b-x)^{\alpha -1}(y-c)^{\beta -1}\right. \\ &&+\left. (x-a)^{\alpha -1}(d-y)^{\beta -1}+(x-a)^{\alpha -1}(y-c)^{\beta -1} \right] , \end{eqnarray*}$ $

Clearly, $ k(x.y) $ is a a positive, integrable and symmetric about $ \frac{a+b }{2} $ and $ \frac{c+d}{2}. $ If we apply Theorem 5 for the function $ k(x, y) $ then we obtain,

$\begin{eqnarray*} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) H_{p}(\alpha ,\beta ) \\ && \\ &\leq &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)p(x,y)\left[ (b-x)^{\alpha -1}(d-y)^{\beta -1}+(b-x)^{\alpha -1}(y-c)^{\beta -1}\right. \\ &&+\left. (x-a)^{\alpha -1}(d-y)^{\beta -1}+(x-a)^{\alpha -1}(y-c)^{\beta -1} \right] dydx \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4\cosh \left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \cosh \left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] }H_{p}(\alpha ,\beta ). \end{eqnarray*}$ $

From the definition of the double fractional integrals we have

$\begin{eqnarray*} &&\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)\left[ (b-x)^{\alpha -1}(d-y)^{\beta -1}+(b-x)^{\alpha -1}(y-c)^{\beta -1}\right. \\ &&+\left. (x-a)^{\alpha -1}(d-y)^{\beta -1}+(x-a)^{\alpha -1}(y-c)^{\beta -1} \right] p(x,y)dydx \\ && \\ & = &\left[ J_{a+,c+}^{\alpha ,\beta }\left( fp\right) (b,d)+J_{a+,d-}^{\alpha ,\beta }\left( fp\right) (b,c)+J_{b-,c+}^{\alpha ,\beta }\left( fp\right) (a,d)+J_{b-,d-}^{\alpha ,\beta }\left( fp\right) (a,c)\right] . \end{eqnarray*}$ $

This completes the proof.

Remark 8. If we choose $ \rho _{1} = \rho _{2} = 0 $ in Theorem 3, then we have the following fractional Hermite-Hadamard inequality,

$\begin{eqnarray*} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \left[ J_{a+,c+}^{\alpha ,\beta }p(b,d)+J_{a+,d-}^{\alpha ,\beta }p(b,c)+J_{b-,c+}^{\alpha ,\beta }p(a,d)+J_{b-,d-}^{\alpha ,\beta }p(a,c)\right] \\ && \\ &\leq &\left[ J_{a+,c+}^{\alpha ,\beta }\left( fp\right) (b,d)+J_{a+,d-}^{\alpha ,\beta }\left( fp\right) (b,c)+J_{b-,c+}^{\alpha ,\beta }\left( fp\right) (a,d)+J_{b-,d-}^{\alpha ,\beta }\left( fp\right) (a,c)\right] \\ && \\ &\leq &\frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4}\left[ J_{a+,c+}^{\alpha ,\beta }p(b,d)+J_{a+,d-}^{\alpha ,\beta }p(b,c)+J_{b-,c+}^{\alpha ,\beta }p(a,d)+J_{b-,d-}^{\alpha ,\beta }p(a,c)\right] \end{eqnarray*}$ $

which is proved by Yaldız et all in ^[34].

Remark 9. If we choose $ \alpha $ $ = \beta = 1 $ in Theorem 3, then we have Theorem 1.3 reduces to Theorem 5.

Theorem 9. If $ f:\Delta \rightarrow \mathbb{R} $ is a co-ordinated hyperbolic $ \rho $-convex functions on $ \Delta $. Then we have the following Hermite-Hadamard type inequalities for fractional integrals,

$\begin{eqnarray} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) H_{1}(\alpha ,\beta ) \\ && \\ &\leq &\frac{1}{2}\left[ \left( J_{a+}^{\alpha }f\left( b,\frac{c+d}{2} \right) +J_{b-}^{\alpha }f\left( a,\frac{c+d}{2}\right) \right) H_{2}(\beta )\right. \\ && \\ &&+\left. J_{c+}^{\beta }f\left( d,\frac{a+b}{2}\right) +J_{d-}^{\beta }f\left( c,\frac{a+b}{2}\right) H_{3}(\alpha )\right] \\ && \\ &\leq &\left[ J_{a+,c+}^{\alpha ,\beta }f(b,d)+J_{a+,d-}^{\alpha ,\beta }f(b,c)+J_{b-,c+}^{\alpha ,\beta }f(a,d)+J_{b-,d-}^{\alpha ,\beta }f(a,c) \right] \\ && \\ &\leq &\frac{1}{4}\left[ \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2} \right] \left( J_{a+}^{\alpha }f(b,c)+J_{a+}^{\alpha }f(b,d)+J_{b-}^{\alpha }f(a,c)+J_{b-}^{\alpha }f(a,d)\right) H_{2}(\beta )\right. \\ && \\ &&+\left. \sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \left( J_{c+}^{\beta }f(a,d)+J_{c+}^{\beta }f(b,d)+J_{d-}^{\beta }f(a,c)+J_{d-}^{\beta }f(b,c)\right) H_{3}(\alpha )\right] \\ && \\ &\leq &\frac{f(a,c)+f(b,c)+f(a,d)+f(b,d)}{4}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] H_{1}(\alpha ,\beta ) \end{eqnarray}$ $

(3.1)

where

$\begin{eqnarray*} H_{1}(\alpha ,\beta ) & = &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) }\int \limits_{a}^{b}\int \limits_{c}^{d}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] \cosh \left[ \rho _{2}\left( y- \frac{c+d}{2}\right) \right] \\ &&\times \left[ (b-x)^{\alpha -1}(d-y)^{\beta -1}+(b-x)^{\alpha -1}(y-c)^{\beta -1}\right. \\ &&+\left. (x-a)^{\alpha -1}(d-y)^{\beta -1}+(x-a)^{\alpha -1}(y-c)^{\beta -1} \right] dydx, \end{eqnarray*}$ $

$\begin{equation*} H_{2}(\beta ) = \frac{1}{\Gamma \left( \beta \right) }\int \limits_{c}^{d}\cosh \left[ \rho _{2}\left( y-\frac{c+d}{2}\right) \right] \left[ (d-y)^{\beta -1}+(y-c)^{\beta -1}\right] dy \end{equation*}$ $

and

$\begin{equation*} H_{3}(\alpha ,\beta ) = \frac{1}{\Gamma \left( \alpha \right) }\int \limits_{a}^{b}\cosh \left[ \rho _{1}\left( x-\frac{a+b}{2}\right) \right] \left[ (b-x)^{\alpha -1}+(x-a)^{\alpha -1}\right] dx. \end{equation*}$ $

Proof. If we apply Theorem 6 for the symmetric function

then we get the following inequality

$\begin{eqnarray*} &&f\left( \frac{a+b}{2},\frac{c+d}{2}\right) H_{1}(\alpha ,\beta ) \\ &\leq &\frac{1}{2}\left[ \left( \frac{1}{\Gamma \left( \alpha \right) }\int \limits_{a}^{b}f\left( x,\frac{c+d}{2}\right) \left[ (b-x)^{\alpha -1}+(x-a)^{\alpha -1}\right] dx\right) H_{2}(\beta )\right. \\ &&+\left. \left( \frac{1}{\Gamma \left( \beta \right) }\int \limits_{c}^{d}f\left( \frac{a+b}{2},y\right) \left[ (d-y)^{\beta -1}+(y-c)^{\beta -1}\right] dy\right) H_{3}(\alpha )\right] \\ &\leq &\frac{1}{\Gamma \left( \alpha \right) \Gamma \left( \beta \right) } \int \limits_{a}^{b}\int \limits_{c}^{d}f(x,y)\left[ (b-x)^{\alpha -1}(d-y)^{\beta -1}+(b-x)^{\alpha -1}(y-c)^{\beta -1}\right. \\ &&+\left. (x-a)^{\alpha -1}(d-y)^{\beta -1}+(x-a)^{\alpha -1}(y-c)^{\beta -1} \right] dydx \\ &\leq &\frac{1}{4}\left[ \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2} \right] \left( \frac{1}{\Gamma \left( \alpha \right) }\int \limits_{a}^{b} \left[ f(x,c)+f(x,d)\right] \left[ (b-x)^{\alpha -1}+(x-a)^{\alpha -1}\right] dx\right) H_{2}(\beta )\right. \\ &&+\left. \sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \left( \frac{1}{\Gamma \left( \beta \right) }\int \limits_{a}^{b}\left[ f(a,y)+f(b,y)\right] \left[ (d-y)^{\beta -1}+(y-c)^{\beta -1}\right] dx\right) H_{3}(\alpha )\right] \\ &\leq &\frac{f(a,c)+f(b,c)+f(a,d)+f(b,d)}{4}\sec h\left[ \frac{\rho _{1}\left( b-a\right) }{2}\right] \sec h\left[ \frac{\rho _{2}\left( d-c\right) }{2}\right] H_{1}(\alpha ,\beta ). \end{eqnarray*}$ $

This completes the proof.

Remark 10. Under assumptions of Theorem 9 with $ \alpha = \beta = 1, $ the inequalities (3.1) reduce to inequalities (2.5) proved by Özçelik et. al in ^[23].

Remark 11. Under assumptions of Theorem 9 with $ \rho _{1} = \rho _{2} = 0, $ the inequalities (3.1) reduce to inequalities proved by Sarikaya in [29,Theorem 4]

4. Conclusions

In this paper, we establish some Fejer type inequalities for co-ordinated hyperbolic $ \rho $-convex functions. By using these inequalities we present some inequalities for Riemann-Liouville fractional integrals. In the future works, authors can prove similar inequalities for other fractional integrals.

Conflict of interest

All authors declare no conflicts of interest.

References

[1]	Ramanathan V, Feng Y (2009) Air pollution, greenhouse gases and climate change: Global and regional perspectives. Atmos Environ 43: 37-50.
[2]	Brunekreef B, Holgate ST (2002) Air pollution and health. Lancet 360: 1233-1242. doi: 10.1016/S0140-6736(02)11274-8
[3]	Burney J, Ramanathan V (2014) Recent climate and air pollution impacts on Indian agriculture. Proc Natl Acad Sci 111: 16319-16324. doi: 10.1073/pnas.1317275111
[4]	Kampa M, Castanas E (2008) Human health effects of air pollution. Environ Pollut 151: 362-367. doi: 10.1016/j.envpol.2007.06.012
[5]	Bernstein JA, Alexis N, Barnes C, et al. (2004) Health effects of air pollution. J Allergy Clin Immunol 114: 1116-1123. doi: 10.1016/j.jaci.2004.08.030
[6]	Harnung SE, Johnson MS (2012) Chemistry and the Environment: Cambridge University Press; 448 p.
[7]	WHO (2014) Burden of disease from the joint effects of Household and Ambient Air Pollution for 2012. Public Health, Social and Environmental Determinants of Health Department, World Health Organization, Geneva Switzerland.
[8]	Zhang Y, Mo J, Li Y, et al. (2011) Can commonly-used fan-driven air cleaning technologies improve indoor air quality? A literature review. Atmos Environ 45: 4329-4343.
[9]	Johnson MS, Nilsson EJ, Svensson EA, et al. (2014) Gas-phase advanced oxidation for effective, efficient in situ control of pollution. Environ Sci Technol 48: 8768-8776. doi: 10.1021/es5012687
[10]	Rohde RA, Muller RA (2015) Air pollution in China: Mapping of concentrations and sources. PloS One 10: e0135749.
[11]	Hyttinen M, Pasanen P, Björkroth M, et al. (2007) Odors and volatile organic compounds released from ventilation filters. Atmos Environ 41: 4029-4039.
[12]	Clausen G (2004) Ventilation filters and indoor air quality: a review of research from the International Centre for Indoor Environment and Energy. Indoor Air 14: 202-207. doi: 10.1111/j.1600-0668.2004.00289.x
[13]	Nøjgaard JK, Christensen KB, Wolkoff P (2005) The effect on human eye blink frequency of exposure to limonene oxidation products and methacrolein. Toxicol Lett 156: 241-251. doi: 10.1016/j.toxlet.2004.11.013
[14]	Klenø J, Wolkoff P (2004) Changes in eye blink frequency as a measure of trigeminal stimulation by exposure to limonene oxidation products, isoprene oxidation products and nitrate radicals. Int Arch Occup Environ Health 77: 235-243. doi: 10.1007/s00420-003-0502-1
[15]	Weschler CJ (2000) Ozone in indoor environments: concentration and chemistry. Indoor Air 10: 269-288.
[16]	Waring MS, Siegel JA, Corsi RL (2008) Ultrafine particle removal and generation by portable air cleaners. Atmos Environ 42: 5003-5014.
[17]	Muzenda E (2012) Pre-treatment methods in the abatement of volatile organic compounds: a discussion. Int Conf Nanotechnol Chem Eng.
[18]	Yu B, Hu Z, Liu M, et al. (2009) Review of research on air-conditioning systems and indoor air quality control for human health. Int J Refrig 32: 3-20. doi: 10.1016/j.ijrefrig.2008.05.004
[19]	Guieysse B, Hort C, Platel V, et al. (2008) Biological treatment of indoor air for VOC removal: Potential and challenges. Biotechnol Adv 26: 398-410.
[20]	Zhao J, Yang X (2003) Photocatalytic oxidation for indoor air purification: a literature review. Build Environ 38: 645-654.
[21]	Ardkapan SR, Johnson MS, Yazdi S, et al. (2014) Filtration efficiency of an electrostatic fibrous filter: Studying filtration dependency on ultrafine particle exposure and composition. J Aerosol Sci 72: 14-20. doi: 10.1016/j.jaerosci.2014.02.002
[22]	Hyttinen M, Pasanen P, Salo J, et al. (2003) Reactions of ozone on ventilation filters. Indoor Built Environ 12: 151-158. doi: 10.1177/1420326X03012003002
[23]	Sekine Y, Nishimura A (2001) Removal of formaldehyde from indoor air by passive type air-cleaning materials. Atmos Environ 35:2001-2007. doi: 10.1016/S1352-2310(00)00465-9
[24]	Khan FI, Ghoshal AK (2000) Removal of volatile organic compounds from polluted air. J Loss Prev Process Ind 13: 527-545. doi: 10.1016/S0950-4230(00)00007-3
[25]	Borhan MS, Mukhtar S, Capareda S, et al. (2012) Greenhouse gas emissions from housing and manure management systems at confined livestock operations. INTECH Open Access Publisher.
[26]	Nicolai R, Clanton C, Janni K, et al. (2006) Ammonia removal during biofiltration as affected by inlet air temperature and media moisture content. Trans ASABE 49: 1125-1138. doi: 10.13031/2013.21730
[27]	Harrop O (2001) Air quality assessment and management: A practical guide: CRC Press.
[28]	Moretti EC (2002) Reduce VOC and HAP emissions. Chem Eng Progress 98: 30-40.
[29]	Mills B (1998) Abatement of VOCs. Surf Coat Int 81: 223-229. doi: 10.1007/BF02693863
[30]	Andreozzi R, Caprio V, Insola A, et al. (1999) Advanced oxidation processes (AOP) for water purification and recovery. Catal Today 53: 51-59. doi: 10.1016/S0920-5861(99)00102-9
[31]	Wu G, Conti B, Leroux A, et al. (1999) A high performance biofilter for VOC emission control. J Air Waste Manag Assoc 49: 185-192. doi: 10.1080/10473289.1999.10463793
[32]	Leson G, Winer AM (1991) Biofiltration: an innovative air pollution control technology for VOC emissions. J Air Waste Manag Assoc 41: 1045-1054.
[33]	Ergas SJ, Schroeder ED, Chang DP, et al. (1995) Control of volatile organic compound emissions using a compost biofilter. Water Environ Res 67: 816-821. doi: 10.2175/106143095X131736
[34]	Wani AH, Branion RM, Lau AK (1997) Biofiltration: A promising and cost‐effective control technology for Odors, VOCs and air toxics. J Environ Sci Health 32: 2027-2055.
[35]	Chang J-S (2001) Recent development of plasma pollution control technology: a critical review. Sci Technol Adv Mater 2: 571-576. doi: 10.1016/S1468-6996(01)00139-5
[36]	Magureanu M, Mandache NB, Eloy P, et al. (2005) Plasma-assisted catalysis for volatile organic compounds abatement. Appl Catal B 61: 12-20.
[37]	Subrahmanyam C, Renken A, Kiwi-Minsker L (2007) Novel catalytic non-thermal plasma reactor for the abatement of VOCs. Chem Eng J 134: 78-83. doi: 10.1016/j.cej.2007.03.063
[38]	Johnson MS, Arlemark J (2012) Method and device for cleaning air. European Patent EP 2119974, 2009; International Patent Cooperation Treaty PCT/EP2009/055849, 2009; U.S. Patent 8,318,084 B2, 2011.
[39]	Li R, Palm BB, Ortega AM, et al. (2015) Modeling the Radical Chemistry in an Oxidation Flow Reactor: Radical Formation and Recycling, Sensitivities, and the OH Exposure Estimation Equation. J Phys Chem A 119: 4418-4432. doi: 10.1021/jp509534k
[40]	Jimenez J, Canagaratna M, Donahue N, et al. (2009) Evolution of organic aerosols in the atmosphere. Science 326: 1525-1529. doi: 10.1126/science.1180353
[41]	Donahue NM, Epstein S, Pandis SN, et al. (2011) A two-dimensional volatility basis set: 1. organic-aerosol mixing thermodynamics. Atmos Chem Phys 11: 3303-3318. doi: 10.5194/acp-11-3303-2011
[42]	Kroll JH, Donahue NM, Jimenez JL, et al. (2011) Carbon oxidation state as a metric for describing the chemistry of atmospheric organic aerosol. Nat Chem 3: 133-139. doi: 10.1038/nchem.948
[43]	Alfassi ZB (1997) The chemistry of free radicals: peroxyl radicals: Wiley; 546 p.
[44]	Hanst PL, Spence JW, Edney EO (1980) Carbon monoxide production in photooxidation of organic molecules in the air. Atmos Environ (1967) 14: 1077-1088. doi: 10.1016/0004-6981(80)90038-4
[45]	Jacob D (1999) Introduction to Atmospheric Chemistry: Princeton University Press; 267 p.
[46]	Atkinson R, Baulch D, Cox R, et al. (2006) Evaluated Kinetic and Photochemical Data for Atmospheric Chemistry: Volume II–Gas Phase Reactions of Organic Species. Atmos Chem Phys 6: 3625-4055. doi: 10.5194/acp-6-3625-2006
[47]	Sharkey TD, Yeh S (2001) Isoprene emission from plants. Ann Rev Plant Biol 52: 407-436. doi: 10.1146/annurev.arplant.52.1.407
[48]	Nilsson EJK, Eskebjerg C, Johnson MS (2009) A photochemical reactor for studies of atmospheric chemistry. Atmos Environ 43: 3029-3033.
[49]	Atkinson R (1986) Kinetics and mechanisms of the gas-phase reactions of the hydroxyl radical with organic compounds under atmospheric conditions. Chem Rev 86: 69-201. doi: 10.1021/cr00071a004
[50]	Khamaganov VG, Hites RA (2001) Rate constants for the gas-phase reactions of ozone with isoprene, α-and β-pinene, and limonene as a function of temperature. J Phys Chem A 105: 815-822. doi: 10.1021/jp002730z
[51]	DeMore WB, Sander SP, Golden D, et al. (1997) Chemical kinetics and photochemical data for use in stratospheric modeling. Evaluation number 12; NASA panel for data evaluation. JPL Publ 97-412.
[52]	Toby S, Van de Burgt L, Toby F (1985) Kinetics and chemiluminescence of ozone-aromatic reactions in the gas phase. J Phys Chem 89: 1982-1986. doi: 10.1021/j100256a034
[53]	Atkinson R (2003) Kinetics of the gas-phase reactions of OH radicals with alkanes and cycloalkanes. Atmos Chem Phys 3: 2233-2307. doi: 10.5194/acp-3-2233-2003
[54]	Atkinson R, Baulch D, Cox R, et al. (2004) Evaluated kinetic and photochemical data for atmospheric chemistry: Volume I-gas phase reactions of Ox, HOx, NOx and SOx species. Atmos Chem Phys 4: 1461-1738. doi: 10.5194/acp-4-1461-2004
[55]	Król S, Namieśnik J, Zabiegała B (2014) α-Pinene, 3-carene and d-limonene in indoor air of Polish apartments: the impact on air quality and human exposure. Sci Total Environ 468: 985-995.
[56]	Yu Y, Ezell MJ, Zelenyuk A, et al. (2008) Photooxidation of α-pinene at high relative humidity in the presence of increasing concentrations of NOx. Atmos Environ 42: 5044-5060. doi: 10.1016/j.atmosenv.2008.02.026
[57]	Jenkin ME, Shallcross DE, Harvey JN (2000) Development and application of a possible mechanism for the generation of cis-pinic acid from the ozonolysis of α-and β-pinene. Atmos Environ 34: 2837-2850.
[58]	Camredon M, Hamilton J, Alam M, et al. (2010) Distribution of gaseous and particulate organic composition during dark α-pinene ozonolysis. Atmos Chem Phys 10: 2893-2917. doi: 10.5194/acp-10-2893-2010
[59]	Capouet M, Müller JF, Ceulemans K, et al. (2008) Modeling aerosol formation in alpha‐pinene photo‐oxidation experiments. J Geophys Res 113.
[60]	Capouet M, Peeters J, Noziere B, et al. (2004) Alpha-pinene oxidation by OH: simulations of laboratory experiments. Atmos Chem Phys 4: 2285-2311. doi: 10.5194/acp-4-2285-2004
[61]	WHO (2002) IARC monographs of the evaluation carcinogenic risk to humans. Vol.3, some traditional herbal medicines, some mycotoxins, naphthalene and styrene. Lyon (France): IARC Press; 590 p.
[62]	Rodins V. Photochemical air purification, MSc Thesis, Department of Chemistry, University of Copenhagen; 2013. 80 p.
[63]	Staples E, Zeiger K (2007) On-site Measurement of VOCs and Odors from Metal Casting Operations Using an Ultra-Fast Gas Chromatograph. Electronic Sensor Technology, Inc, USA.
[64]	Fatta D, Marneri M, Papadopoulos A, et al. (2004) Industrial pollution and control measures for a foundry in Cyprus. J Clean Prod 12: 29-36. doi: 10.1016/S0959-6526(02)00180-4
[65]	Kim KY, Ko HJ, Kim HT, et al. (2008) Quantification of ammonia and hydrogen sulfide emitted from pig buildings in Korea. J Environ Manag 88: 195-202. doi: 10.1016/j.jenvman.2007.02.003
[66]	Hobbs P, Misselbrook T, Cumby T (1999) Production and emission of odours and gases from ageing pig waste. J Agric Eng Res 72: 291-298. doi: 10.1006/jaer.1998.0372
[67]	Aladedunye FA, Przybylski R (2009) Degradation and nutritional quality changes of oil during frying. J Am Oil Chem Soc 86: 149-156. doi: 10.1007/s11746-008-1328-5
[68]	Ghoshal A, Manjare S (2002) Selection of appropriate adsorption technique for recovery of VOCs: an analysis. J Loss Prev Process Ind 15: 413-421. doi: 10.1016/S0950-4230(02)00042-6

This article has been cited by:

1.	Dumitru Baleanu, Artion Kashuri, Pshtiwan Othman Mohammed, Badreddine Meftah, General Raina fractional integral inequalities on coordinates of convex functions, 2021, 2021, 1687-1847, 10.1186/s13662-021-03241-y
2.	Han Li, Muhammad Shoaib Saleem, Imran Ahmed, Kiran Naseem Aslam, Hermite–Hadamard and Fejér-type inequalities for strongly reciprocally (p, h)-convex functions of higher order, 2023, 2023, 1029-242X, 10.1186/s13660-023-02960-y
3.	Silvestru Sever Dragomir, Mohamed Jleli, Bessem Samet, Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension, 2024, 22, 2391-5455, 10.1515/math-2024-0028

Reader Comments

Your name:*

Email:*
© 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)