On the Hadamard's type inequalities for co-ordinated convex functions via fractional integrals

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ORIGINAL ARTICLE

On the Hadamard’s type inequalities for

co-ordinated convex functions via fractional

integrals

Abdullah Akkurt

a,*

_{, Mehmet Zeki Sar}

ıkaya

b

_{, Hu¨seyin Budak}

b

_{, Hu¨seyin Y}

ıldırım

a

a_{Department of Mathematics, Faculty of Science and Arts, University of Kahramanmaras}_{ß Su¨tc¸u¨ _Imam, 46100}

Kahramanmarasß, Turkey

b

Department of Mathematics, Faculty of Science and Arts, Du¨zce University, Du¨zce, Turkey Received 30 March 2016; accepted 18 June 2016

KEYWORDS Riemann–Liouville fractional integrals; Hadamard’s type Inequalities; Co-ordinated convex functions; Ho¨lder’s inequality

Abstract In this paper, we establish two identities for functions of two variables and apply them to give new Hermite–Hadamard type fractional integral inequalities for double fractional integrals involving functions whose derivatives are bounded or co-ordinates convex function on D :¼ ½a; b ½c; d in R2

with a< b; c < d.

Ó 2016 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Let f: I # R ! R be a convex mapping defined on the interval Iof real numbers and a; b 2 I with a < b. The following double inequality: f aþ b 2 6 1 b a Z b a fðxÞdx 6fðaÞ þ fðbÞ 2 : ð1Þ

is known in the literature as Hermite–Hadamard inequality for convex mappings. Note that some of the classical inequalities for means can be derived from(1)for appropriate particular selections of the mapping f. Both inequalities hold in the reversed direction if f is concave.

It is well known that the Hermite–Hadamard’s inequality plays an important role in nonlinear analysis. Over the last decade, this classical inequality has been improved and gener-alized in a number of ways; there have been a large number of studies on Hermite–Hadamard’s inequality reporting its role in nonlinear analysis (Alomari et al., 2009; Azpeitia, 1994; Bakula and Pecˇaric´, 2004; Dragomir and Pearce, 2000), later, this classical inequality has been improved (Kırmacı and Dikici, 2013; Set et al., 2011; Latif and Dragomir, 2012; Ozdemir et al., 2010) and is generalized in a number of ways (Hussain et al., 2009; Sarikaya and Aktan, 2011; Sarikaya et al., 2014a).

Let us now consider a bidemensional interval

D ¼: a; b½ c; d½ in R2 _{with a}_{< b and c < d. A mapping}

f: D ! R is said to be convex on D if the following inequality: fðtx þ 1 tð Þz; ty þ 1 tð ÞwÞ 6 tf x; yð Þ þ 1 tð Þf z; wð Þ ð2Þ holds, for all ðx; yÞ; z; wð Þ 2 D and t 2 0; 1½ . A function f: D ! R is said to be convex on the co-ordinates on D if the partial mappings fy: a; b½ ! R, fyð Þ ¼ f u; yu ð Þ and * Corresponding author.

E-mail address:abdullahmat@gmail.com(A. Akkurt). Peer review under responsibility of King Saud University.

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f_x: c; d½ ! R, f_xð Þ ¼ f x; vv ð Þ are convex where defined for all x2 a; b½ and y 2 c; d½ (Dragomir and Pearce, 2000).

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

Definition 1. A function f: D ! R will be called co-ordinated convex on D, for all t; s 2 ½0; 1 and ðx; yÞ; ðu; wÞ 2 D, if the following inequality holds:

fðtx þ 1 tð Þy; su þ 1 sð ÞwÞ 6 tsfðx; uÞ þ sð1 tÞfðy; uÞ

þ tð1 sÞfðx; wÞ þ ð1 tÞð1 sÞfðy; wÞ: ð3Þ

Clearly, every convex function is co-ordinated convex. Furthermore, there exist co-ordinated convex function which is not convex (Dragomir, 2001). Several recent studies have expressed concerns on Hermite–Hadamard’s inequality for some convex function on the co-ordinates on a rectangle from the planeR2₍_{Sarikaya and Yaldiz, 2013; Ozdemir et al., 2011;} Sarikaya et al., 2012; Sarikaya et al. (2014c)). More details, one can consult Sarikaya (2014), Sarikaya et al. (2014b) and Sarikaya (2015).

Earlier,Dragomir (2001)establish the following inequality of Hermite–Hadamard type for co-ordinated convex mapping on a rectangle from the planeR2. Later, another proof of a special version of the following theorem, using the definition of the co-ordinated convex function was reported (Sarikaya and Yaldiz, 2013).

Theorem 2. Suppose that f: D ! R is co-ordinated convex on D. Then one has the inequalities:

f aþ b 2 ; cþ d 2 61 2 1 b a Zb a f x;cþ d 2 dxþ 1 d c Z d c f aþ b 2 ; y dy 6 1 b a ð Þ d cð Þ Zb a Zd c fðx; yÞdydx 61 4 1 b a Zb a fðx; cÞdx þ 1 b a Zb a fðx; dÞdx þ 1 d c Zd c fða; yÞdy þ 1 d c Zd c fðb; yÞdy 6fða; cÞ þ fða; dÞ þ fðb; cÞ þ fðb; dÞ 4 ð4Þ

The above inequalities are sharp.

In the following section, some relevant definitions and mathematical preliminaries of fractional calculus theory are presented. For more details, one can consult Gorenflo and Mainardi (1997), Kilbas et al. (2006), Samko et al. (1993), Miller and Ross (1993).

Definition 3. Let f2 L1½a; b. The Riemann–Liouville integrals

Ja_aþfand Ja_bfof ordera > 0 with a P 0 are defined by

Ja_aþfðxÞ ¼ 1 C að Þ Z x a x t ð Þa1_f_{ðtÞdt; x > a;} _ð5Þ Ja_bfðxÞ ¼ 1 C að Þ Z b x t x ð Þa1fðtÞdt; x < b ð6Þ

respectively. Here,C að Þ is the Gamma function.

It is remarkable thatSarikaya et al. (2012)first give the fol-lowing interesting integral inequalities of Hermite–Hadamard type involving Riemann–Liouville fractional integrals. Theorem 4. Let f: a; b½ ! R be a positive function with 06 a < b and f 2 L1½a; b. If f is a convex function on ½a; b,

then the following inequalities for fractional integrals hold:

f aþ b 2 6C a þ 1ð Þ 2 bð aÞa J a aþfðbÞ þ J a bfðaÞ 6fðaÞ þ fðbÞ 2 ð7Þ witha > 0.

Meanwhile, Sarikaya et al. (2012)presented the following important integral identity including the first-order derivative of f to establish many interesting Hermite–Hadamard type inequalities for convexity functions via Riemann–Liouville fractional integrals of the ordera > 0.

Lemma 5. Let f: a; b½ ! R be a differentiable mapping on a; b

ð Þ with a < b. If f0_{2 L}

1½a; b, then the following equality for

fractional integrals holds: fðaÞ þ fðbÞ 2 C a þ 1ð Þ 2 bð aÞa J a aþfðbÞ þ JabfðaÞ ð8Þ ¼b a 2 Z 1 0 1 t ð Þa ta ½ f0_{ðta þ ð1 tÞbÞdt:} _ð9Þ

Definition 6. Let f2 L1ð½a; b c; d½ Þ. The Riemann–Liouville

integrals Ja;b_aþ;cþ; Ja;baþ;d; Ja;bb;cþ and Ja;bb;d of ordera; b > 0 with

a; c P 0 are defined by Ja;b_aþ;cþfðx; yÞ ¼ 1 C að ÞC bð Þ Z x a Z y c x t ð Þa1_ð_y_s_Þb1_f_{ðt; sÞdsdt;} ð10Þ Ja;b_aþ;dfðx; yÞ ¼ 1 C að ÞC bð Þ Z x a Z d y x t ð Þa1_ð_s_y_Þb1_f_{ðt; sÞdsdt;} ð11Þ Ja;b_b_;cþfðx; yÞ ¼ 1 C að ÞC bð Þ Z b x Z y c t x ð Þa1ðy sÞb1fðt; sÞdsdt; ð12Þ and Ja;b_b_;dfðx; yÞ ¼ 1 C að ÞC bð Þ Z b x Z d y t x ð Þa1_ð_s_y_Þb1_f_{ðt; sÞdsdt;} ð13Þ respectively. Similar to Definitions 3and6we introduce the following fractional integrals:

Ja_aþf x; cþ d 2 ¼ 1 C að Þ Z x a x t ð Þa1_{f t;}cþ d 2 dt; ð14Þ Ja_bf x; cþ d 2 ¼_{C a}_{ð Þ}1 Z b x t x ð Þa1_{f t}_;cþ d 2 dt; ð15Þ Jb_cþf aþ b 2 ; y ¼_{C b}_{ð Þ}1 Z y c y s ð Þb1_f aþ b 2 ; s ds; ð16Þ

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Jb_df aþ b 2 ; y ¼_{C b}_{ð Þ}1 Z d y s y ð Þb1_f aþ b 2 ; s ds: ð17Þ

Objective of the present study is to state and prove the Hermite–Hadamard type inequality for co-ordinated convex mapping on a rectangle from the planeR2. In order to achieve our goal, we first give two important identities and then by using these identities we prove some integral inequalities. We have obtained some results which are a simpler proof of the results presented bySarikaya (2012).

2. Main results

To establish our main results, we need the following first identity:

Lemma 7. Let f: D # R2! R be a partial differentiable mapping on D :¼ ½a; b ½c; d in R2 _{with a}_{< b and c < d and}

f_rs2 LðDÞ. Then the following equality holds: 4C a þ 1ð ÞC b þ 1ð Þ

b a

ð Þa

d c

ð Þb Ja;baþ;cþfðb;dÞ þ Jaa;bþ;dfðb;cÞ þ Ja;bb;cþfða;dÞ

h þJa;b b;dfða;cÞ i 2C a þ 1ð Þ b a ð Þa Jaaþfðb;cÞ þ J a aþfðb;dÞ þJa bfða;cÞ þ J a bfða;dÞ 2C b þ 1ð Þ d c ð Þb Jbcþfða;dÞ þ J b cþfðb;dÞ þJb dfða;cÞ þ J b dfðb;cÞ þ F ¼ ab b a ð Þaðd cÞb Z b a Z d c b x ð Þa1_ð_{d y}_Þb1 h þ b xð Þa1 y c ð Þb1_{þ x a}_ð _Þa1_ð_{y c}_Þb1 þ x að Þa1_ð_d_y_Þb1i_Iðx;yÞdydxo _ð18Þ where Iðx; yÞ ¼ Z x a Z y c f_rsðr; sÞdsdr þ Z x a Z y d f_rsðr; sÞdsdr þ Z x b Z y c f_rsðr; sÞdsdr þ Z x b Z y d f_rsðr; sÞdsdr; ð19Þ and F¼ fða; cÞ þ fða; dÞ þ fðb; cÞ þ fðb; dÞ: ð20Þ

Proof. For any x; t 2 a; b½ and s; y 2 c; d½ ; x – t; s – y, we have Z x t Z y s f_rsðr; sÞdsdr ¼ Z x t f_rðr; yÞ f_rðr; sÞ ½ dr ¼ f r; y½ð Þ f r; sð Þjx t ¼ fðx; yÞ fðx; sÞ fðt; yÞ þ fðt; sÞ: ð21Þ Choose t¼ a; s ¼ c; t ¼ a; s ¼ d; t ¼ b; s ¼ c; t ¼ b; s ¼ d in(21), respectively, we get I1¼ Z x a Z y c

f_rsðr; sÞdsdr ¼ fðx; yÞ fðx; cÞ fða; yÞ þ fða; cÞ; ð22Þ I2¼ Z x a Z y d

f_rsðr; sÞdsdr ¼ fðx; yÞ fðx; dÞ fða; yÞ þ fða; dÞ; ð23Þ I3¼ Z x b Z y c f_rsðr; sÞdsdr ¼ fðx; yÞ fðx; cÞ fðb; yÞ þ fðb; cÞ; ð24Þ and I4¼ Z x b Z y d f_rsðr; sÞdsdr ¼ fðx; yÞ fðx; dÞ fðb; yÞ þ fðb; dÞ: ð25Þ Adding these four integrals side by side, we obtain

Iðx; yÞ ¼ I1þ I2þ I3þ I4

¼ 4fðx; yÞ 2 fðx; cÞ þ fðx; dÞ½ 2 fða; yÞ þ fðb; yÞ½ þ fða; cÞ þ fða; dÞ þ fðb; cÞ þ fðb; dÞ: ð26Þ Multiplying(26)byðbx_{4C a}Þa1_{ð ÞC b}ðdy_{ð Þ}Þb1and integrating the resulting equality with respect toðx; yÞ on ½a; b ½c; d, we have

1 4C að ÞC bð Þ Z b a Z d c bx ð Þa1_ð_d_y_Þb1_I_ðx;yÞdydx ¼_{C a}_{ð ÞC b}1_{ð Þ} Z b a Z d c bx ð Þa1_ð_d_y_Þb1_f_ðx;yÞdydx 1 2C að ÞC bð Þ Z b a Z d c bx ð Þa1ðdyÞb1½fðx;cÞþfðx;dÞdx 1 2C að ÞC bð Þ Z b a Z d c bx

ð Þa1_ð_d_y_Þb1_½_f_{ða;yÞþfðb;yÞ}_dydx

þ F 4C að ÞC bð Þ Z b a Z d c bx

ð Þa1_ð_dy_Þb1_dydx:

ð27Þ Thus, in(27)by means of simple calculations, we have

Ja;b_aþ;cþðb; dÞ d c ð Þb 2C b þ 1ð Þ J a aþfðb; cÞ þ Jaaþfðb; dÞ ðb aÞa 2C a þ 1ð Þ J b cþfða; dÞ þ Jbcþfðb; dÞ þ_{C a þ 1}ð_ðb aÞa_{ÞC b þ 1}ðd_ð cÞb_ÞF ¼ 1 4C að ÞC bð Þ Z b a Z d c b x ð Þa1_ð_d_y_Þb1_I_{ðx; yÞdydx: ð28Þ}

Multiplying(26) byðbx_{4C a}Þa1_{ð ÞC b}ðyc_{ð Þ}Þb1 and integrating the resulting equality with respect toðx; yÞ on ½a; b ½c; d, and by similar calculations, we have Ja;b_aþ_;dfðb; cÞ d c ð Þb 2C b þ 1ð Þ Jaaþfðb; cÞ þ J a aþfðb; dÞ _{2C a þ 1}ðb_ð aÞa_Þ Jb_dfða; cÞ þ Jbdfðb; cÞ þ_{4C a þ 1}ðb_ð aÞa_{ÞC b þ 1}ðd_ð cÞb_ÞF ¼_{4C a}_{ð ÞC b}1 _{ð Þ} Z b a Z d c b x ð Þa1_ð_y_c_Þb1_I_{ðx; yÞdydx: ð29Þ}

Multiplying(26) byðxa_{4C a}Þa1_{ð ÞC b}ðyc_{ð Þ}Þb1 and integrating the resulting equality with respect toðx; yÞ on ½a; b ½c; d, we have

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Ja;b_b_;cþfða; dÞ d c ð Þb 2C b þ 1ð Þ J a bfða; cÞ þ Jabfða; dÞ ðb aÞ a 2C a þ 1ð Þ J b cþfða; dÞ þ Jbcþfðb; dÞ þ ðb aÞ a d c ð Þb 4C a þ 1ð ÞC b þ 1ð ÞF ¼ 1 4C að ÞC bð Þ Z b a Z d c x a ð Þa1_ð_y_c_Þb1_I_{ðx; yÞdydx: ð30Þ}

Multiplying(26)byðxa_{4C a}Þa1_{ð ÞC b}ðdy_{ð Þ}Þb1and integrating the resulting equality with respect toðx; yÞ on ½a; b ½c; d, we have Ja;b_b_;dfða; cÞ ðd cÞ b 2C b þ 1ð Þ J a bfða; cÞ þ Jabfða; dÞ ðb aÞ a 2C a þ 1ð Þ J b dfða; cÞ þ J b dfðb; cÞ þ ðb aÞ a d c ð Þb 4C a þ 1ð ÞC b þ 1ð ÞF ¼ 1 4C að ÞC bð Þ Z b a Z d c x a ð Þa1_ð_d_y_Þb1_I_{ðx; yÞdydx: ð31Þ}

Adding these(28)–(31)side by side, which completes the proof.

Corollary 8. If we takea ¼ b ¼ 1 inLemma7, we get 4 b a ð Þ d cð Þ Z b a Z d c fðx;yÞdydx 2 b a ð Þ Z b a f xð ;cÞ þ f x;dð Þ ½ dx _ð_{d c}2 _Þ Z d c f að ;yÞ þ f b;yð Þ ½ dy þ F ¼_ð_{b a}1 Þ d cð Þ Z b a Z d c Iðx;yÞdydx: ð32Þ

Theorem 9. Let f: D # R2! R be a partial differentiable mapping onD :¼ ½a; b ½c; d in R2 with a< b and c < d and f_rs2 LðDÞ. If f_rs2 L1ðDÞ, i.e fjj jrsj1¼ sup r;s ð Þ2 a;bð Þ c;dð Þ @2_fðr;sÞ @r@s < 1, then one has the inequality:

4C a þ 1ð ÞC b þ 1ð Þ b a ð Þa d c ð Þb Ja;baþ;cþfðb; dÞ þ J a;b aþ;dfðb; cÞ þ J a;b b;cþfða; dÞ h þJa;b b;dfða; cÞ i 2C a þ 1ð Þ b a ð Þa Jaaþfðb; cÞ þ J a aþfðb; dÞ þJa bfða; cÞ þ J a bfða; dÞ 2C b þ 1ð Þ d c ð Þb Jbcþfða; dÞ þ J b cþfðb; dÞ þJb dfða; cÞ þ J b dfðb; cÞ _{þ F 6} 4 fjj j_rsj₁ðb aÞ d cð Þ: ð33Þ

Proof. FromLemma 7, taking the modulus, it follows that

J

j j ¼ 4C a þ 1ð ÞC b þ 1ð Þ b a

ð Þa

d c

ð Þb Ja;baþ;cþfðb;dÞ þ Jaa;bþ;dfðb;cÞ þ Ja;b_b_;cþfða;dÞ

h þJa;b b;dfða;cÞ i 2C a þ 1ð Þ b a ð Þa Jaaþfðb;cÞ þ J a aþfðb;dÞ þ J a bfða;cÞ þJa bfða;dÞ 2C b þ 1ð Þ d c ð Þb Jbcþfða;dÞ þ J b cþfðb;dÞ þ J b dfða;cÞ þJb dfðb;cÞ þ F ð34Þ 6 ab b a ð Þa d c ð Þb Z b a Z d c b x ð Þa1_ð_d_y_Þb1 h þ b xð Þa1_ð_y_c_Þb1_{þ x a}_ð _Þa1_ð_y_c_Þb1 þ x að Þa1 d y ð Þb1i Z x a Z y c f_rsðr; sÞ j jdsdr þ Z x a Z d y f_rsðr; sÞ j jdsdrþ Z b x Z y c f_rsðr; sÞ j jdsdr þ Z b x Z d y f_rsðr; sÞ j jdsdr dydx : ð35Þ Since f_rs2 L1ðDÞ, we get J j j 6 ab fjj jrsj1 b a ð Þa d c ð Þb Zb a Zd c b x ð Þa1_ð_d_y_Þb1 Zb a Z d c dsdr dydx þ Zb a Zd c b x ð Þa1_ð_y_c_Þb1 Zb a Zd c dsdr dydx ð36Þ þ Z b a Z d c x a ð Þa1_ð_y_c_Þb1 Z b a Z d c dsdr dydx þ Z b a Z d c x a ð Þa1_ð_d_y_Þb1 Z b a Z d c dsdr dydx ¼ ab fjj jrsj1 b a ð Þa d c ð Þb 4 bð aÞaþ1 a d c ð Þbþ1 b ¼ 4 fjj jrsj1ðb aÞ d cð Þ: ð37Þ

This completes the proof.

Corollary 10. If we takea ¼ b ¼ 1 inTheorem9, we get 4 ba ð Þ dcð Þ Z b a Z d c fðx;yÞdydx 2 ba ð Þ Z b a f xð ;cÞþf x;dð Þ ½ dx 2 dc ð Þ Z d c f að ;yÞþf b;yð Þ ½ dyþF 6 4 fjj jrsj1ðbaÞ dcð Þ: ð38Þ Theorem 11. Let f: D # R2! R be a partial differentiable mapping on D :¼ ½a; b ½c; d in R2 with a< b and c < d and f_rs2 LðDÞ. If fj j is a convex function on the co-ordinates on_rs D, then the following inequality holds:

4C a þ 1ð ÞC b þ 1ð Þ b a

ð Þa_ð_d_c_Þb Ja;baþ;cþfðb; dÞ þ Ja;baþ;dfðb; cÞ þ Ja;b_b_;cþfða; dÞ

h þJa;b b;dfða; cÞ i 2C a þ 1ð Þ b a ð Þa Jaaþfðb; cÞ þ Jaaþfðb; dÞ þJa bfða; cÞ þ Jabfða; dÞ 2C b þ 1ð Þ d c ð Þb J b cþfða; dÞ þ J b cþfðb; dÞ þJb dfða; cÞ þ J b dfðb; cÞ _{þ F 6} b a ð Þ d cð Þ f½jrsða; cÞþj fjrsða; dÞj þ fjrsðb; cÞj þ fjrsðb; dÞj ð39Þ

Proof. Since fj_rsðr; sÞj is co-ordinates on D, we know that x2 a; b½ ; y 2 c; d½

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f_rsðr;sÞ j j ¼ frs b r b aaþ r a b ab; d s d ccþ s c d cd 6b r b a d s d cjfrsða;cÞj þ b r b a s c d cjfrsða;dÞj þr a b a d s d cjfrsðb;cÞj þ r a b a s c d cjfrsðb;dÞj: ð40Þ FromLemma 7, we have

J j j 6 ab b a ð Þa_ð_d_c_Þb Z b a Z d c b x ð Þa1 d y ð Þb1 h þ b xð Þa1 y c ð Þb1_{þ x a}_ð _Þa1 y c ð Þb1 þ x að Þa1_ð_d_y_Þb1i Z x a Z y c f_rsðr; sÞ j jdsdr þ Z x a Z d y f_rsðr; sÞ j jdsdrþ Z b x Z y c f_rsðr; sÞ j jdsdr þ Z b x Z d y f_rsðr; sÞ j jdsdr dydx ð41Þ By using co-ordinated convexity of fj j, we get_rs

J j j 6 ab b a ð Þa_ð_d_c_Þb Z b a Z d c b x ð Þa1 d y ð Þb1 h þ b xð Þa1 y c ð Þb1_{þ x a}_ð _Þa1 y c ð Þb1 þ x að Þa1_ð_d_y_Þb1i Z x a Z y c b r b a d s d cjfrsða;cÞj þb r b a s c d cjfrsða;dÞj þ r a b a d s d cjfrsðb;cÞj þr a b a s c d cjfrsðb;dÞj i dsdr þ Z x a Z d y b r b a d s d cjfrsða;cÞj þb r b a s c d cjfrsða;dÞj þ r a b a d s d cjfrsðb;cÞj þr a b a s c d cjfrsðb;dÞj i dsdr ð42Þ þ Z b x Z y c b r b a d s d cjfrsða;cÞj þ b r b a s c d cjfrsða;dÞj þr a b a d s d cjfrsðb;cÞj þ r a b a s c d cjfrsðb;dÞj dsdr þ Z b x Z d y b r b a d s d cjfrsða;cÞj þ b r b a s c d cjfrsða;dÞj þr a b a d s d cjfrsðb;cÞjþ r a b a s c d cjfrsðb;dÞj dsdr dydx ð43Þ ¼ ab b a ð Þaðd cÞb Z b a Z d c b x ð Þa1_ð_{d y}_Þb1 h þ b xð Þa1_ð_y_c_Þb1_{þ x a}_ð _Þa1_ð_y_c_Þb1 þ x að Þa1 d y ð Þb1i Z b a Z d c b r b a d s d cjfrsða;cÞj þb r b a s c d cjfrsða;dÞj þr a_{b a} d s d cjfrsðb;cÞj þr a b a s c d cjfrsðb;dÞj i dsdr o dydx¼ A1þ A2þ A3þ A4: ð44Þ

With a simple calculation, we have

A1¼ ab ba ð Þaþ1_ð_d_c_Þbþ1 Zb a Zd c bx ð Þa1_ð_d_y_Þb1 (Z b a Zd c b r ð Þ d sð Þ fjrsða;cÞj þ b rð Þ s cð Þ fjrsða;dÞj ½ þ r að Þ d sð Þ fjrsðb;cÞjþ r að Þ s cð Þ fjrsðb;dÞjdsdr ) dydx ð45Þ ¼ ab b a ð Þaþ1ðd cÞbþ1 ðb aÞaþ2 2a d c ð Þbþ2 2b ½jfrsða; cÞþj fjrsða; dÞj ( þ fjrsðb; cÞj þ fjrsðb; dÞj ¼ b a ð Þ d cð Þ 4 ½jfrsða; cÞþj fjrsða; dÞj þ fjrsðb; cÞj þ fjrsðb; dÞj: ð46Þ

Similarly, we also have the following equalities

A2¼ ab b a ð Þaþ1_ð_d_c_Þbþ1 Z b a Z d c b x ð Þa1_ð_y_c_Þb1 Z b a Z d c b r ð Þ d sð Þ fjrsða;cÞj þ b rð Þ s cð Þ fjrsða;dÞj ½ þ r að Þ d sð Þ fjrsðb;cÞj þ r að Þ s cð Þ fjrsðb;dÞjdsdr dydx ¼ðb aÞ d cð Þ 4 ½jfrsða;cÞþj fjrsða;dÞj þ fjrsðb;cÞj þ fjrsðb;dÞj; ð47Þ A3¼ ab b a ð Þaþ1_ð_d_c_Þbþ1 Z b a Z d c x a ð Þa1_ð_y_c_Þb1 Z b a Z d c b r ð Þ d sð Þ fjrsða;cÞj þ b rð Þ s cð Þ fjrsða;dÞj ½ þ r að Þ d sð Þ fjrsðb;cÞj þ r að Þ s cð Þ fjrsðb;dÞjdsdr dydx ¼ðb aÞ d c₄ð Þ½jfrsða;cÞþj fjrsða;dÞj þ fjrsðb;cÞj þ fjrsðb;dÞj ð48Þ and A4¼ ab b a ð Þaþ1_ð_d_c_Þbþ1 Z b a Z d c x a ð Þa1_ð_d_y_Þb1 Z b a Z d c b r ð Þ d sð Þ fjrsða;cÞj þ b rð Þ s cð Þ fjrsða;dÞj ½ þ r að Þ d sð Þ fjrsðb;cÞj þ r að Þ s cð Þ fjrsðb;dÞjdsdr dydx ¼ðb aÞ d c₄ð Þ½jfrsða;cÞþj fjrsða;dÞj þ fjrsðb;cÞj þ fjrsðb;dÞj: ð49Þ

Adding these(46)–(49)side by side, if we put in(44), we obtain

(39). This completes the proof of the theorem.

Corollary 12. If we takea ¼ b ¼ 1 inTheorem11, we get 4 ba ð Þ dcð Þ Z b a Z d c fðx;yÞdydx 2 ba ð Þ Z b a f xð ;cÞþf x;dð Þ ½ dx _ð_dc2 _Þ Z d c f að ;yÞþf b;yð Þ ½ dyþF 6 bað Þ dcð Þ fj½rsð Þa;c þj fjrsða;dÞjþ fjrsð Þb;cjþ fjrsðb;dÞj: ð50Þ

Lemma 13. Let f: D # R2! R be a partial differentiable map-ping on D :¼ ½a; b ½c; d in R2 with a< b; c < d and f_rs2 LðDÞ. Then the following equality holds:

(6)

f aþ b 2 ; cþ d 2 C b þ 1ð Þ 2 dð cÞb J b cþf aþ b 2 ;d þ Jb df aþ b 2 ;c C a þ 1ð Þ 2 bð aÞa J a bf a; cþ d 2 þ Ja aþf b; cþ d 2 þC a þ 1ð ÞC b þ 1ð Þ 4 b að Þa_ð_{d c}_Þb Ja;baþ;cþfðb;dÞ þ J a;b aþ;dfðb;cÞ þ J a;b b;cþfða;dÞ h þJa;b b;dfða;cÞ i ¼ ab 4 bð aÞaðd cÞb Z b a Z d c b t ð Þa1 h n þ t að Þa1i ðd sÞb1þ s cð Þb1 h i Z aþb 2 t Z cþd 2 s f_rsðr;sÞdsdr !) dsdt: ð51Þ

Proof. Choose x¼aþb₂ and y¼cþd₂ in(21), we have Z aþb 2 t Z cþd 2 s f_rsðr; sÞdsdr ¼ f aþ b 2 ; cþ d 2 f aþ b 2 ; s f t;cþ d 2 þ fðt; sÞ: ð52Þ

Multiplying(52)byðbt_{C a}Þa1_{ð ÞC b}ðds_{ð Þ}Þb1 and integrating the resulting equality with respect toðs; tÞ on ½a; b ½c; d, we get

1 C að ÞC bð Þ Z b a Z d c b t ð Þa1ðd sÞb1 Z aþb 2 t Z cþd 2 s f_rsðr;sÞdsdr ( ) dsdt ¼f aþb2 ;cþd2 C að ÞC bð Þ Z b a Z d c b t ð Þa1ðd sÞb1dsdt _{C a}_{ð ÞC b}1_{ð Þ} Z b a Z d c b t ð Þa1_ð_d_s_Þb1_f aþ b 2 ;s dsdt _{C a}_{ð ÞC b}1_{ð Þ} Z b a Z d c b t ð Þa1_ð_d_s_Þb1_{f t}_;cþ d 2 dsdt þ_{C a}_{ð ÞC b}1_{ð Þ} Z b a Z d c b t ð Þa1_ð_d_s_Þb1_f_ðt;sÞdsdt: ð53Þ By simple calculations, we have

b a ð Þa_ð_d_c_Þb C a þ 1ð ÞC b þ 1ð Þf aþ b 2 ; cþ d 2 ðb aÞa C a þ 1ð ÞJ b cþf aþ b 2 ;d ðd cÞb C b þ 1ð ÞJaaþf b; cþ d 2 þ Ja;b aþ;cþfðb;dÞ ¼ 1 C að ÞC bð Þ Z b a Z d c b t ð Þa1 d s ð Þb1 Z aþb 2 t Z cþd 2 s f_rsðr;sÞdsdr ( ) dsdt: ð54Þ

Multiplying (52) by ðbt_{C a}Þa1_{ð ÞC b}ðsc_{ð Þ}Þb1, integrating the resulting equality with respect toðs; tÞ on ½a; b ½c; d, and by similar methods above we have

b a ð Þa_ð_d_c_Þb C a þ 1ð ÞC b þ 1ð Þf aþ b 2 ; cþ d 2 _{C a þ 1}ðb_ð aÞa_ÞJb_df aþ b 2 ;c ðd cÞb C b þ 1ð ÞJaaþf b; cþ d 2 þ Ja;b aþ;dfðb;cÞ ¼ 1 C að ÞC bð Þ Z b a Z d c b t ð Þa1 d s ð Þb1 Z aþb 2 t Z cþd 2 s f_rsðr;sÞdsdr ( ) dsdt: ð55Þ

Multiplying (52) by ðta_{C a}Þa1_{ð ÞC b}ðds_{ð Þ}Þb1 integrating the resulting equality with respect toðs; tÞ on ½a; b ½c; d, and by similar methods above we have

b a ð Þa_ð_{d c}_Þb C a þ 1ð ÞC b þ 1ð Þf aþ b 2 ; cþ d 2 _{C a þ 1}ðb a_ð Þa_ÞJbcþf aþ b 2 ;d _{C b þ 1}ðd c_ð Þb_ÞJabf a; cþ d 2 þ Ja;b b;cþfða;dÞ ¼_{C a}_{ð ÞC b}1_{ð Þ}Z b a Z d c b t ð Þa1_ð_{d s}_Þb1 Z aþb 2 t Z cþd 2 s f_rsðr;sÞdsdr ( ) dsdt: ð56Þ Multiplying (52) by ðta_{C a}Þa1_{ð ÞC b}ðsc_{ð Þ}Þb1 integrating the resulting equality with respect toðs; tÞ on ½a; b ½c; d, and by similar methods above we have

b a ð Þa_ð_{d c}_Þb C a þ 1ð ÞC b þ 1ð Þf aþ b 2 ; cþ d 2 _{C a þ 1}ðb a_ð Þa_ÞJbdf aþ b 2 ;c _{C b þ 1}ðd c_ð Þb_ÞJabf a; cþ d 2 þ Ja;b b;dfða;cÞ ¼_{C a}_{ð ÞC b}1_{ð Þ} Z b a Z d c b t ð Þa1_ð_d_s_Þb1 Z aþb 2 t Z cþd 2 s f_rsðr;sÞdsdr ( ) dsdt: ð57Þ

Adding these(54)–(57)side by side and multiplying both sides by_{4 ba}C aþ1_ðð _ÞÞC bþ1a_ðð_dc_ÞbÞ, we get the desired equality(51).

Corollary 14. If we takea ¼ b ¼ 1 inLemma13, we get

f aþ b 2 ; cþ d 2 1 b a ð Þ Z b a f x;cþ d 2 dx _ð_d_c1 _Þ Z d c f aþ b 2 ; y dyþ 1 b a ð Þ d cð Þ Z b a Z d c fðx; yÞdydx ¼ 1 16 bð aÞ d cð Þ Z b a Z d c Z aþb 2 t Z cþd 2 s f_rsðr; sÞdsdr ( ) dsdt: ð58Þ Theorem 15. Let f: D # R2_{! R be a partial differentiable}

mapping on D :¼ ½a; b ½c; d in R2 _{with a}_{< b and c < d. If}

f_rs2 L₁ðDÞ, then the following equality holds:

f aþ b 2 ; cþ d 2 C b þ 1ð Þ 2 dð cÞb J b cþf aþ b 2 ;d þ Jb df aþ b 2 ;c C a þ 1ð Þ 2 bð aÞa J a bf a; cþ d 2 þ Ja aþf b; cþ d 2 þC a þ1ð ÞC b þ 1ð Þ 4 bð aÞaðd cÞb J a;b aþ;cþfðb;dÞ þ J a;b aþ;dfðb;cÞ þ J a;b b;cþfða;dÞ h þJa;b b;dfða;cÞ i ₆jj jf_rsj₁ðb aÞ d cð Þ 4 21aþ a 1ð Þ a þ 1 21bþ b 1ð Þ b þ 1 " # : ð59Þ

(7)

Proof. InLemma 13, taking the modulus, it follows that f aþ b 2 ; cþ d 2 C b þ 1ð Þ 2 dð cÞb J b cþf aþ b 2 ;d þ Jb df aþ b 2 ;c C a þ 1ð Þ 2 bð aÞa J a bf a; cþ d 2 þJa aþf b; cþ d 2 þC a þ 1ð ÞC b þ 1ð Þ 4 bð aÞaðd cÞb J a;b aþ;cþfðb;dÞ þ Ja;baþ;dfðb;cÞ h þJa;b

b;cþfða;dÞ þ Ja;bb;dfða;cÞi ð60Þ

6 ab fjj jrsj1 4 bð aÞaðd cÞb Z b a Z d c b t ð Þa1_{þ t a}_ð _Þa1 h i n d shð Þb1_{þ s c}_ð _Þb1i aþ b 2 t cþ d₂ s dsdt ¼jj jfrsj1ðb aÞ d cð Þ 4 21aþ a 1ð Þ a þ 1 21bþ b 1ð Þ b þ 1 " # ð61Þ for f_rs2 L₁ðDÞ.

Remark 16. If we takea ¼ b ¼ 1 inTheorem 15, we get

f aþ b 2 ; cþ d 2 1 b a ð Þ Zb a f x;cþ d 2 dx 1 d c ð Þ Zd c f aþ b 2 ; y dy þ 1 b a ð Þ d cð Þ Zb a Z d c fðx; yÞdydx ₆jj jfrsj1 16 ðb aÞ d cð Þ: ð62Þ

which is proved by Sarikaya inSarikaya (2012).

Theorem 17. Let f: D # R2_{! R be a partial differentiable}

mapping on D :¼ ½a; b ½c; d in R2 _{with a}_{< b and c < d. If}

f_rs

j j is a convex function on the co-ordinates on D, then the following equality holds:

f aþ b 2 ; cþ d 2 C b þ 1ð Þ 2 dð cÞb J b cþf aþ b 2 ;d þ Jb df aþ b 2 ;c C a þ 1ð Þ 2 bð aÞa J a bf a; cþ d 2 þ Ja aþf b; cþ d 2 þC a þ 1ð ÞC b þ 1ð Þ 4 bð aÞaðd cÞb J a;b aþ;cþfðb;dÞ þ Ja;baþ;dfðb;cÞ h þJa;b b;cþfða;dÞ þ J a;b b;dfða;cÞi ð63Þ 6 b að Þ d cð Þa2a a þ 1ð Þ2a1þ 1 2aða þ 1Þ b2b_{b þ 1}_ð _Þ2b1_{þ 1} 2bðb þ 1Þ jfrsða; cÞj þ fjrsða; dÞj þ fjrsðb; cÞj þ fjrsðb; dÞj 4 : ð64Þ

Proof. Since fj_rsðr; sÞj is co-ordinates on D, we know that t2 a; b½ ; s 2 c; d½ f_rsðr;sÞ j j ¼ frs b r b aaþ r a b ab; d s d ccþ s c d cd 6b r b a d s d cjfrsða;cÞj þ b r b a s c d cjfrsða;dÞj þr a b a d s d cjfrsðb;cÞj þ r a b a s c d cjfrsðb;dÞj: ð65Þ From Lemma 13, using co-ordinated convexity of fj j, we_rs have f aþ b 2 ; cþ d 2 C b þ 1ð Þ 2 dð cÞb J b cþf aþ b 2 ;d þ Jb df aþ b 2 ;c C a þ 1ð Þ 2 bð aÞa J a bf a; cþ d 2 þ Ja aþf b; cþ d 2 þC a þ 1ð ÞC b þ 1ð Þ 4 bð aÞaðd cÞb J a;b aþ;cþfðb;dÞ þ Ja;baþ;dfðb;cÞ h þJa;b

b;cþfða;dÞ þ Ja;bb;dfða;cÞi ð66Þ

6 ab 4 b að Þa_ð_d_c_Þb Z b a Z d c b t ð Þa1_{þ t a}_ð _Þa1 h i n d shð Þb1_{þ s c}_ð _Þb1iZ aþb 2 t Z cþd 2 s frsðr;sÞ j jdsdr ) dsdt ð67Þ 6 ab 4 bð aÞaþ1ðd cÞbþ1 Z b a Z d c b t ð Þa1þ t að Þa1 h i n d shð Þb1_{þ s c}_ð _Þb1iZ aþb 2 t Z cþd 2 s b r ð Þ d sð Þ fjrsða;cÞj ½ þ b rð Þ s cð Þ fjrsða;dÞjþ r að Þ d sð Þ fjrsðb;cÞj þ _ðr aÞ s cð Þ fjrsðb;dÞj i dsdr o dsdt¼ K1þ K2þ K3þ K4: ð68Þ

With a simple calculation, we have K1¼ ab 4 bð aÞaþ1ðd cÞbþ1 Z b a Z d c b t ð Þa1_{þ t a}_ð _Þa1 h i d shð Þb1þ s cð Þb1i fjrsða;cÞj Z aþb 2 t Z cþd 2 s b r ð Þ d sð Þdsdr dsdt¼ ab fjrsða;cÞj 4 bð aÞaþ1ðd cÞbþ1 Z b a b t ð Þa1þ t að Þa1 h i Z aþb 2 t b r ð Þdr dt " # Z d c d s ð Þb1_{þ s c}_ð _Þb1 h i Z cþd 2 s d s ð Þds ds " # ð69Þ ¼ ab fjrsða;cÞj 4 bð aÞaþ1_ð_d_c_Þbþ1 Z aþb 2 a bt ð Þa1_{þ t a}_ð _Þa1 h iZ aþb 2 t br ð Þdrdt " þ Zb aþb 2 b t ð Þa1_{þ t a}_ð _Þa1 h iZt aþb 2 b r ð Þdrdt # Zcþd 2 c d s ð Þb1_{þ s c}_ð _Þb1 h iZ cþd 2 s d s ð Þdsds " þ Zd cþd 2 d s ð Þb1_{þ s c}_ð _Þb1 h iZ s cþd 2 d s ð Þdsds # ¼jfrsða;cÞj 4 a2a_{a þ 1}_ð _Þ2a1_{þ 1} 2aða þ1Þ b2b_{b þ 1}_ð _Þ2b1_{þ 1} 2bðb þ1Þ ðb aÞ d cð Þ: ð70Þ

(8)

Similarly, we also have the following equalities K2¼ ab 4 bð aÞaþ1ðd cÞbþ1 Zb a Zd c b t ð Þa1_{þ t a}_ð _Þa1 h i d shð Þb1_{þ s c}_ð _Þb1i_f rsða;dÞ j j Zaþb 2 t Zcþd 2 s b r ð Þ s cð Þdsdr dsdt ¼jfrsða;dÞj 4 a2a_{a þ 1}_ð _Þ2a1_{þ 1} 2aða þ 1Þ b2b_{b þ 1}_ð _Þ2b1_{þ 1} 2bðb þ 1Þ ðb aÞ d cð Þ; ð71Þ K3¼ ab 4 bð aÞaþ1ðd cÞbþ1 Zb a Zd c b t ð Þa1_{þ t a}_ð _Þa1 h i d s ð Þb1_{þ s c}_ð _Þb1 h i fjrsðb;cÞj Zaþb 2 t Zcþd 2 s r a ð Þ d sð Þdsdr dsdt ¼jfrsðb;cÞj 4 a2a_{a þ 1}_ð _Þ2a1_{þ 1} 2aða þ 1Þ b2b_{b þ 1}_ð _Þ2b1_{þ 1} 2bðb þ 1Þ ðb aÞ d cð Þ ð72Þ and K4¼ ab 4 bð aÞaþ1ðd cÞbþ1 Zb a Zd c b t ð Þa1_{þ t a}_ð _Þa1 h i d shð Þb1_{þ s c}_ð _Þb1i_f rsðb;dÞ j jZ aþb 2 t Zcþd 2 s r a ð Þ s cð Þdsdr dsdt ¼jfrsðb;dÞj 4 a2a_{a þ 1}_ð _Þ2a1_{þ 1} 2aða þ 1Þ b2b_{b þ 1}_ð _Þ2b1_{þ 1} 2bðb þ 1Þ ðb aÞ d cð Þ: ð73Þ

Thus, if we put the last four equalities in(68), we obtain(64). This completes the proof of the theorem.

Corollary 18. If we takea ¼ b ¼ 1 inTheorem17, we get f aþ b 2 ; cþ d 2 _ð_b_a1 _Þ Z b a f x;cþ d 2 dx _ð_d_c1 _Þ Zd c f aþ b 2 ;y dy þ 1 b a ð Þ d cð Þ Z b a Zd c fðx;yÞdydx 6ðb aÞ d cð Þ 16 jfrsða;cÞj þ fjrsða;dÞj þ fjrsðb;cÞj þ fjrsðb;dÞj 4 : ð74Þ 3. Conclusion

In this work we give two identities for functions of two variables and apply them to give new Hermite–Hadamard type Fractional integral inequalities for double Fractional integrals involving functions whose derivatives are bounded or co-ordinates convex function onD :¼ ½a; b ½c; d in R2 _with

a< b; c < d. Acknowledgement

The authors would like to express their appreciation to the referees for their valuable suggestions which helped to better presentation of this paper.

References

Alomari, M., Darus, M., Dragomir, S.S., 2009. New inequalities of Hermite–Hadamard type for functions whose second derivates absolute values are quasi-convex, RGMIA Res. Rep. Coll., 12 Supplement, Article 14.

Azpeitia, A.G., 1994. Convex functions and the Hadamard inequality. Rev. Colomb. Math. 28, 7–12.

Bakula, M.K., Pecˇaric´, J., 2004. Note on some Hadamard-type inequalities. J. Inequal. Pure Appl. Math. 5 (3), article 74.

Dragomir, S.S., 2001. On Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane. Taiwanese J. Math. 4, 775–788.

Dragomir, S.S., Pearce, C.E.M., 2000. Selected Topics on Hermite– Hadamard Inequalities and Applications. RGMIA Monographs, Victoria University.

Gorenflo, R., Mainardi, F., 1997. Fractional Calculus: Integral and Differential Equations of Fractional Order. Springer Verlag, Wien, 223–276.

Hussain, S., Bhatti, M.I., Iqbal, M., 2009. Hadamard-type inequalities for s-convex functions I. Punjab Univ. J. Math. 41, 51–60.

Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., 2006. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Sci. B.V., Amsterdam.

Kırmacı, U.S., Dikici, R., 2013. On some Hermite–Hadamard type inequalities for twice differentiable mappings and applications. Tamkang J. Math. 44 (1), 41–51.

Latif, M.A., Dragomir, S.S., 2012. On some new inequalities for differentiable co-ordinated convex functions. J. Inequal. Appl. 1 (December), 1–13.

Miller, S., Ross, B., 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, USA, p. 2.

Ozdemir, M.E., Avc, M., Set, E., 2010. On some inequalities of Hermite–Hadamard type via m-convexity. Appl. Math. Lett. 23, 1065–1070.

Ozdemir, M.E., Set, E., Sarikaya, M.Z., 2011. Some new Hadamard’s type inequalities for co-ordinated m-convex and ða; mÞ-convex functions. Hacettepe J. Math. Stat. 40, 219–229.

Samko, S.G., Kilbas, A.A., Marichev, O.I., 1993. Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Yverdon et alibi.

Sarikaya, M.Z., 2012. On the Ostrowski type integral inequality for double integrals. Demon. Math. 45, 533–540.

Sarikaya, M.Z., 2014. On the Hermite–Hadamard-type inequalities for co-ordinated convex function via fractional integrals. Integr. Transf. Spec. Funct. 25, 134–147.

Sarikaya, M.Z., 2015. Some inequalities for differentiable co-ordinated convex mappings. Asian-Eur. J. Math. 8 (21 pages).

Sarikaya, M.Z., Aktan, N., 2011. On the generalization of some integral inequalities and their applications. Math. Comput. Model. 54, 2175–2182.

Sarikaya, M.Z., Yaldiz, H., 2013. On the Hadamard’s type inequalities for L-Lipschitzian mapping. Konuralp J. Math. 1, 33–40.

Sarikaya, M.Z., Set, E., Ozdemir, M.E., Dragomir, S.S., 2012. New some Hadamard’s type inequalities for co-ordinated convex func-tions. Tamsui Oxford J. Inf. Math. Sci. 28, 137–152.

Sarikaya, M.Z., Set, E., Yaldiz, H., Basak, N., 2012. Hermite– Hadamard’s inequalities for fractional integrals and related frac-tional inequalities. Math. Comput. Model. 57, 2403–2407.

Sarikaya, M.Z., Set, E., Ozdemir, M.E., 2014a. On some integral inequalities for twice differantiable mappings. Stud. Univ. Babes-Bolyai Math. 59, 11–24.

Sarikaya, M.Z., Budak, H., Yaldiz, H., 2014b. Cebysev type inequal-ities for co-ordinated convex functions. Pure Appl. Math. Lett. 2, 36–40.

Sarikaya, M.Z., Budak, H., Yaldiz, H., 2014c. Some new Ostrowski type inequalities for co-ordinated convex functions. Turkish J. Anal. Numb. Theory 2, 176–182.

Set, E., Sarikaya, M.Z., Ahmad, A., 2011. A generalization of Chebychev type inequalities for first differentiable mappings. Miskolc Math. Notes 12, 245–253.