849–861 DOI: 10.18514/MMN.2019.2796 NEW BOUNDS FOR HERMITE-HADAMARD’S TRAPEZOID AND MID-POINT TYPE INEQUALITIES VIA FRACTIONAL INTEGRALS M

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Vol. 20 (2019), No. 2, pp. 849–861 DOI: 10.18514/MMN.2019.2796

NEW BOUNDS FOR HERMITE-HADAMARD’S TRAPEZOID AND MID-POINT TYPE INEQUALITIES VIA FRACTIONAL

INTEGRALS

M. ROSTAMIAN DELAVAR Received 18 December, 2018

Abstract. Some trapezoid and mid-point type inequalities with new bounds for Hermite-Hadamard inequality related to Riemann-Liouville integrals of order˛ > 0are obtained. Also a refinement of Hermite-Hadamard inequality for nonnegative monotone convex functions is presented. Fur- thermore some applications in connection with special means are given.

2010Mathematics Subject Classification: 26A15; 26A51; 26D15

Keywords: Hermite-Hadamard inequality, trapezoid and mid-point type inequalities, fractional integrals

1. INTRODUCTION

The following inequality is known in literature as Hermite-Hadamard inequality:

faCb 2

1

b a Z b

a

f .x/dx f .a/Cf .b/

2 ; (1.1)

wheref WŒaIb!Ris convex onŒa; b. For historical information about inequality (1.1), see [6].

In inequality (1.1), we may deal with two issues:

(i) Estimation of the difference between left and middle terms which we call it trapezoid type estimation

(ii) Estimation of the difference between right and middle terms which we call it mid-point (rectangle) type estimation.

At first S. S. Dragomir et al, in [3], obtained the trapezoid type inequality related to (1.1) as well:

Theorem 1. Letf WI^ıR!Rbe a differentiable mapping onI^ı,a; b2I^ıwith a < b. Ifjf⁰jis convex onŒa; b, then the following inequality holds:

ˇ ˇ ˇ

f .a/Cf .b/

2 .b a/

Z b a

f .x/dx ˇ ˇ

ˇ .b a/² 8

jf⁰.a/j C jf⁰.b/j

: (1.2)

c 2019 Miskolc University Press

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The striped area shown in Figure 1, is equivalent to the difference between the area of trapezoid abcd and the area under the graph off which is estimated by

.b a/²

8 jf⁰.a/j C jf⁰.b/j

. Also U. S. Kirmaci in [5], obtained the mid-point type

FIGURE1.

inequality related to (1.1) as the following:

Theorem 2. ConsiderIas the interior of intervalI R. Letf WI!Rbe a differentiable mapping onI,a; b2Iwitha < b. Ifjf⁰jis convex onŒa; b, then we have

ˇ ˇ ˇ

Z b a

f .x/dx .b a/f

aCb 2

ˇ ˇ

ˇ .b a/² 8

jf⁰.a/j C jf⁰.b/j

: (1.3)

In Figure2, it is shown that the difference between the area under the graph of f and the area of rectangleabcd can be estimated by ^{.b a/}₈ ² jf⁰.a/j C jf⁰.b/j

. Recently In [8], the authors obtained Hermite-Hadamard’s inequality related to frac-

FIGURE2.

tional integrals as the following:

Theorem 3. Letf WŒa; b!Rbe convex function with0a < bandf 2LŒa; b.

Iff⁰2LŒa; b;then the following equality for fractional integrals holds.

faCb 2

.˛C1/

2.b a/^˛

J_a^˛_Cf .b/CJ_b^˛ f .a/

f .a/Cf .b/

2 ; (1.4)

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whereJ_a^˛_Cf andJ_b^˛ f are the Riemann-Liouville integrals of order˛ > 0defined by

J_a^˛_Cf .x/D 1 .˛/

Z x a

.x t /^{˛ 1}f .t /dt; x > a;

and

J_b^˛ f .x/D 1 .˛/

Z b x

.t x/^{˛ 1}f .t /dt; x < b;

such that

.˛/D Z ₁

0

e ^tt^{˛ 1}dt;

is Gamma function andJ_a⁰_Cf .x/DJ_b⁰ f .x/Df .x/.

The trapezoid and mid-point type inequalities related to (1.4) have been obtained in [8] and [4] respectively.

Theorem 4. Letf WŒa; b!Rbe a differentiable mapping on.a; b/witha < b. If jf⁰jis convex function onŒa; b, then the following inequality for fractional integrals holds:

ˇ ˇ ˇ ˇ

f .a/Cf .b/

2

.˛C1/

2.b a/^˛

ˇ ˇ ˇ ˇ

(1.5) b a

2.˛C1/

1 1

2^˛ h

jf⁰.a/j C jf⁰.b/ji :

Theorem 5. Letf WŒa; b!Rbe a differentiable mapping on.a; b/witha < b. If jf⁰jis convex function onŒa; b, then the following inequality for Riemann-Liouville fractional integrals holds for0 < ˛1:

ˇ ˇ ˇ ˇ

faCb 2

.˛C1/

2.b a/^˛

ˇ ˇ ˇ

ˇ b a 2^˛^C¹.˛C1/

h

jf⁰.a/j C jf⁰.b/ji : (1.6) On the other hand in [7], we can find two results related to the convexity of a function as the following:

(a) Any convex function defined on a closed intervalŒa; bis bounded.

(b) If a real valued function defined on the intervalI is convex, then it satisfies a Lipschitz condition on any closed intervalŒa; b there is a constantKso that for any two pointsx; y2Œa; b,jf .x/ f .y/j Kjx yj

contained in the interiorI^ıofI. Motivated by above works and results, we obtain some trapezoid and mid-point type inequalities related to (1.4) where the convexity condition for the absolute value of the derivative of considered function is replaced by boundedness and a Lipschit- zian condition for the derivative. In fact we obtain new bounds for the left side of inequalities (1.5) and (1.6) which give some refinements for these inequalities. Also by

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the use of fractional integrals we present a refinement of Hermite-Hadamard inequality for nonnegative monotone convex functions. Finally we give some applications of our results in connection with special means.

2. TRAPEZOID AND MID-POINT TYPE INEQUALITIES

In this section, we obtain some Hermite-Hadamard’s trapezoid and mid-point type inequalities via fractional integrals where the derivative of considered function is bounded and satisfies a Lipschitz condition. The following result has been obtained in [8] and we use it to obtain trapezoid type inequalities.

Lemma 1. Letf WŒa; b!Rbe a differentiable mapping on.a; b/witha < b. If f⁰2LŒa; b, then the following equality for fractional integral holds:

f .a/Cf .b/

2

.˛C1/

2.b a/^˛

Db a 2

Z 1 0

t^˛ .1 t /^˛

f⁰ t aC.1 t /b dt:

In the following theorem we consider that the derivative of considered function is bounded.

Theorem 6. Suppose thatf WI!Ris a differentiable function onI^ı. Consider a; b2I^ıwitha < bsuch thatf⁰2LŒa; b. If there exist constantsl < Lsuch that

1< lf⁰.x/L <1 f or al l x2Œa; b;then ˇ

ˇ ˇ ˇ ˇ

f .a/Cf .b/

2

.˛C1/

2.b a/^˛ h

J_a^˛_Cf .b/CJ_b^˛ f .a/i ˇ ˇ ˇ ˇ ˇ

(2.1) .b a/.L l/

2.˛C1/

1 1

2^˛

: Proof. From Lemma1we have

J D ˇ ˇ ˇ ˇ

f .a/Cf .b/

2

.˛C1/

2.b a/^˛

ˇ ˇ ˇ ˇ

Db a 2

Z 1 0

h.t / h

f⁰ t aC.1 t /b lCL

2 ClCL 2

i dt;

whereh.t /Dt^˛ .1 t /^˛;for allt2Œ0; 1. Since Z 1

0

h.t /dtD0;

Z 1

0 jh.t /jdtD 2

˛C1.1 1 2^˛/;

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and

ˇ ˇ

ˇf⁰ t aC.1 t /b lCL 2

ˇ ˇ

ˇL l 2 ; then

jJj b a 2

Z 1 0 jh.t /j

ˇ ˇ

ˇ ˇ ˇdt .b a/.L l/

4

Z 1

0 jh.t /jdtD.b a/.L l/

2.˛C1/ .1 1 2^˛/:

Remark1. Ifjf⁰jis convex onŒa; b, then there existl; Lwith

0lD2 ˇ ˇ

ˇf⁰ aCb 2

ˇ ˇ

ˇ L jf⁰.x/j LDmaxfjf⁰.a/j;jf⁰.b/jg<1 for allx2Œa; bwhich implies thatL l jf⁰.a/j C jf⁰.b/j, since

jf⁰.b/j>jf⁰.a/j !LD jf⁰.b/jand ll jf⁰.a/j !L l jf⁰.a/j C jf⁰.b/j; jf⁰.a/j>jf⁰.b/j !LD jf⁰.a/jand ll jf⁰.b/j !L l jf⁰.a/j C jf⁰.b/j: So (2.1) gives a refinement for (1.5).

In the following theorem we consider that the derivative of considered function satisfies a Lipschitz condition.

Theorem 7. Suppose thatf WI!Ris a differentiable function onI^ı. Consider a; b2I^ı with a < b such thatf⁰2LŒa; band satisfies a Lipschitz condition for someK > 0. Then

ˇ ˇ ˇ ˇ ˇ

f .a/Cf .b/

2

.˛C1/

2.b a/^˛ h

K˛.b a/² 2.˛C1/.˛C2/: Proof. From Lemma1we have

J D ˇ ˇ ˇ ˇ

f .a/Cf .b/

2

.˛C1/

2.b a/^˛

ˇ ˇ ˇ ˇ

D b a 2

Z 1 0

h.t /h

f⁰ t aC.1 t /b

f⁰aCb 2

Cf⁰aCb 2

i dt D b a

2 Z 1

0

h.t / h

f⁰ t aC.1 t /b f⁰

aCb 2

i dt;

whereh.t /Dt^˛ .1 t /^˛;for allt2Œ0; 1and Z 1

0

h.t /f⁰aCb 2

dtD0:

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Sincef⁰satisfies a Lipschitz condition for someK > 0, then jJj K.b a/²

2

Z 1

0 jh.t /jjt 1 2jdt:

The definition ofh.t /implies that jJj K.b a/²

2

Z 1 0

t^˛ .1 t /^˛ .t 1

2/dt D K˛.b a/² 2.˛C1/.˛C2/:

The following lemma has been proved in [4] and we use it to obtain mid-point type inequalities.

Lemma 2. Letf WŒa; b!Rbe a differentiable function on.a; b/. Iff⁰2LŒa; b, then the following identity for Riemann-Liouville fractional integrals holds:

faCb 2

.˛C1/

2.b a/^˛ h

J_a^˛_Cf b/CJ_b^˛ f .a/i

Db a 2

4

X

kD1

I_k; where

I1D Z ¹₂

0

t^˛f⁰ t bC.1 t /a

dt; I2D Z ¹₂

0

. t^˛/f⁰ t aC.1 t /b dt;

I3D Z 1

1 2

.t^˛ 1/f⁰ t bC.1 t /a

dt; I4D Z 1

1 2

.1 t^˛/f⁰ t aC.1 t /b dt:

If we consider the boundedness of the derivative of considered function we get the following mid-point type inequality.

Theorem 8. Suppose thatf WI!Ris a differentiable function onI^ı. Consider a; b2I^ıwitha < bsuch thatf⁰2LŒa; b. If there exist constantsl < Lsuch that

1< lf⁰.x/L <1 f or al l x2Œa; b;then ˇ

ˇ ˇ ˇ

faCb 2

.˛C1/

2.b a/^˛ h

J_a^˛_Cf b/CJ_b^˛ f .a/iˇ ˇ ˇ

ˇ M.b a/

2.˛C1/

h

˛ 1C 1 2

˛ 1i

; whereM Dmaxf l; Lg.

Proof. If we consider I1D

Z ¹₂

0

t^˛h

f⁰ t aC.1 t /b lCL

2 ClCL 2

i dt;

then

ˇ ˇ ˇ ˇ

I1

lCL 2

Z ¹₂

0

t^˛ ˇ ˇ ˇ ˇ

Z ¹₂

0

t^˛ ˇ ˇ

ˇ ˇ ˇdt

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L l 2

1

˛C1.1 2/^˛^C¹

; which implies that

ˇˇI1

ˇ ˇ

L l

2 C

ˇ ˇ ˇ

lCL 2

ˇ ˇ ˇ

1

˛C1.1 2/^˛^C¹

: Moreover it is not hard to see that

M Dmaxf l; Lg D L l

2 Cˇ ˇ ˇ

lCL 2

ˇ ˇ ˇ

: So

ˇ ˇI1

ˇ ˇM

1

˛C1.1 2/^˛^C¹

: Similarly we can obtain that

ˇˇI2

ˇ

ˇM 1

˛C1.1 2/^˛^C¹

: It follows that

ˇˇI3

ˇ

ˇMh1 2

1

˛C1

1 1

2

˛C1i

; and

ˇˇI4

ˇ

ˇMh1 2

1

˛C1

1 1

2

˛C1i

; Now by adding all of above inequalities we get

4

X

iD1

jIij M

˛C1

˛ 1C 1 2

˛ 1

;

which implies the desired result.

Remark2. In proof of Theorem8if we consider0 < ˛1, then by the use of the fact that

jt₁^˛ t₂^˛j jt1 t2j^˛; for anyt1; t22Œ0; 1, we obtain that

ˇ ˇ ˇ ˇ

faCb 2

.˛C1/

2.b a/^˛ h

J_a^˛_Cf b/CJ_b^˛ f .a/iˇ ˇ ˇ

ˇ M.b a/

2^˛.˛C1/: (2.2) whereM Dmaxf l; Lg. Furthermore in the case that

2M jf⁰.a/j C jf⁰.b/j; inequality (2.2) gives a refinement for (1.6).

If the derivative of considered function satisfies a Lipschitz condition, then the following inequality holds.

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Theorem 9. Suppose thatf WI!Ris a differentiable function onI^ı. Consider a; b2I^ı with a < b such thatf⁰2LŒa; band satisfies a Lipschitz condition for someK > 0. Then

ˇ ˇ ˇ ˇ

faCb 2

.˛C1/

.b a/^˛ h

J_a^˛_Cf b/CJ_b^˛ f .a/iˇ ˇ ˇ ˇ

b a 2^˛.˛C1/

K.b a/

˛C2 Cˇ ˇ

ˇf⁰aCb 2

ˇ ˇ ˇ

; for0 < ˛1:

Proof. With some calculations similar to the proof of Theorem7we deduce that jI1j D jI2j D jI3j D jI4j 1

2^˛^C¹.˛C1/

h_{K.b a/}

˛C2 Cˇ

ˇf⁰.^a^C₂^b/ˇ ˇ i

;which along with Lemma2imply the result. Note that injI3jandjI4jwe used the fact that

jt₁^˛ t₂^˛j jt1 t2j^˛;

for anyt1; t22Œ0; 1and0 < ˛1.

Corollary 1. If we consider˛D1in (i) Theorem6, then we get

ˇ ˇ ˇ ˇ ˇ

f .a/Cf .b/

2

1 .b a/

Z b a

f .x/dx ˇ ˇ ˇ ˇ ˇ

.b a/.L l/

8 ;

If there exist constantsl < Lsuch that 1< lf⁰.x/L <1 f or al l x2Œa; b.

(ii) Theorem7, then we obtain ˇ

ˇ ˇ ˇ ˇ

f .a/Cf .b/

2

1 .b a/

Z b a

f .x/dx ˇ ˇ ˇ ˇ ˇ

K.b a/²

12 ;

if f⁰ satisfies a Lipschitz condition for some K > 0on Œa; b. Comparing this inequality with inequality (2.6) in [2], shows that the existence of a Lipschitz condition forf⁰gives a better estimation rather than the existence of a Lipschitz condition for f.

(iii) Theorem8, then we have ˇ

ˇ ˇ ˇ

faCb 2

1

.b a/

Z b a

f .x/dx ˇ ˇ ˇ

ˇ M.b a/

4 ;

if there exist constantsl < Lsuch that 1< lf⁰.x/L <1 f or al l x2Œa; b

andM Dmaxf l; Lg.

(iv) Theorem9, then we deduce that ˇ

ˇ ˇ ˇ

faCb 2

1

.b a/

Z b a

f .x/dx ˇ ˇ ˇ

ˇ K.b a/²

12 C1

4 ˇ ˇ

ˇf⁰aCb 2

ˇ ˇ ˇ:

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iff⁰satisfies a Lipschitz condition for someK > 0onŒa; b.

3. REFINEMENT OFHERMITE-HADAMARD INEQUALITY

In this section we give a refinement of Hermite-Hadamard inequality and some new inequalities in connection with fractional integrals related to the nonnegative monotone convex functions.

Theorem 10. Letf WŒa; b!Rbe a convex function.

(i) For˛1, the following refinement of Hermite-Hadamard inequality holds if f is nonnegative and increasing.

f

aCb 2

1

b a

Z b a

f .x/dx .˛C1/

2.b a/^˛ h

J_a^˛_Cf .b/CJ_b^˛ f .a/i

f .a/Cf .b/

2 :

(3.1) (ii) For any˛ > 0we have

f

aCb 2

.˛C1/

2.b a/^˛ h

J_a^˛_Cf .b/CJ_b^˛ f .a/i ˛

b a Z b

a

f .x/dx: (3.2) (iii) If˛1andf is nonnegative and increasing, then

1 b a

Z b a

f .x/dx .˛C1/

2.b a/^˛ h

dx ˛

b a Z b

a

f .x/dx:

(3.3) Proof. From convexity off we have

f t bC.1 t /a

tf .b/C.1 t /f .a/;

and

f t aC.1 t /b

tf .a/C.1 t /f .b/:

By adding these inequalities we get f t bC.1 t /a

Cf t aC.1 t /b

f .a/Cf .b/: (3.4) Multiplying both sides of (3.4) byt^{˛ 1}and integrating the resulting inequality with respect tot overŒ0; 1and using Theorem 1 in [1], we obtain

Z 1 0

t^{˛ 1}dt Z 1

0

f t bC.1 t /a dtC

Z 1 0

t^{˛ 1}dt Z 1

0

f t aC.1 t /b dt

Z 1 0

t^{˛ 1}f t bC.1 t /a dtC

Z 1 0

t^{˛ 1}f t aC.1 t /b dt

f .a/Cf .b/

Z 1 0

t^{˛ 1}dt:

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Using the changes of variablexDt bC.1 t /aandxDt aC.1 t /b, respectively, in above integrals we have

1

b a

Z 1 0

t^{˛ 1}dt:

Z b a

f .x/dxC Z b

a

f .aCb x/dx

.˛/

.b a/^˛ h

J_a^˛_Cf .b/CJ_b/^˛ f .a/i

f .a/Cf .b/

Z 1 0

t^{˛ 1}dt:

Since

Z 1 0

t^{˛ 1}dt D 1

˛; and

Z b a

f .x/dxD Z b

a

f aCb x dx;

then we obtain 1 b a

Z b a

f .x/dx .˛C1/

2.b a/^˛ h

f .a/Cf .b/

2 : (3.5)

On the other hand sincef is convex, then f

aCb 2

1

b a Z b

a

f .x/dx;

which along with (3.5), implies the inequality (3.1).

To obtain inequality (3.2), we consider that f

aCb 2

1

2 h

f t aC.1 t /b

Cf t bC.1 t /ai :

Multiplying both sides byt^{˛ 1}and integrating the resulting inequality with respect totoverŒ0; 1we get

2

˛f

aCb 2

.˛/

.b a/^˛ h

Z 1 0

f t aC.1 t /b dtC

Z 1 0

f t bC.1 t /a dt

which implies the inequality (3.2). Finally inequality (3.3) comes from (3.1) and

(3.2).

Remark3. Iff andgare nonnegative decreasing functions defined onŒ0; 1and B is an upper bound for them, then B f andB g are nonnegative increasing functions and so

Z 1 0

.B f .x//dx Z 1

0

.B g.x//dx Z 1

0

.B f .x//.B g.x//dx;

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which gives again Z 1

0

f .x/dx Z 1

0

g.x/dx Z 1

0

f .x/g.x/dx:

This implies that inequalities (3.1) and (3.3) of Theorem10can be obtained if f W Œa; b!Rbe a decreasing nonnegative convex function and0 < ˛1.

4. APPLICATION TO SPECIAL MEANS

The following means for real numbersa; b2Rare known:

A.a; b/DaCb

2 ari t hmet i c mean;

Ln.a; b/Dhbⁿ^C¹ aⁿ^C¹ .nC1/.b a/

i¹_n

ge nerali´ed log mean; n2N; a < b:

Considerf .x/Dxⁿforx0,n2N. Ifx2Œa; b,

lDna^{n 1}f⁰.x/Dnx^{n 1}nb^{n 1}DL:

So from Theorem6, we obtain ˇ

ˇ ˇ ˇ ˇ

aⁿCbⁿ 2

.˛C1/

2.b a/^˛ h

n.b a/.b^{n 1} a^{n 1}/ 2.˛C1/

1 1

2^˛

; where

J_a^˛_Cf .b/D Z b

a

.b t /^{˛ 1}tⁿdtD

n

X

kD0

a^{n k}.b a/^˛^C^kP .n; k/

Qk

iD0.˛Ci /

;

J_b^˛ f .a/D Z b

a

.t a/^{˛ 1}tⁿdtD

n

X

kD0

. 1/^kb^{n k}.b a/^˛^C^kP .n; k/

Qk

iD0.˛Ci / ; and

P .n; k/D nŠ .n k/Š;

which is the number of possible permutations ofkobjects from a set ofn.

In special case if we consider˛D1, then we have

J_a^˛_Cf .b/CJ_b^˛ f .a/D2.bⁿ^C¹ aⁿ^C¹/

nC1 :

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So

ˇ ˇ ˇ ˇ ˇ

aⁿCbⁿ 2

bⁿ^C¹ aⁿ^C¹ .nC1/.b a/

ˇ ˇ ˇ ˇ ˇ

n.b a/.b^{n 1} a^{n 1}/

8 ;

or equivalently ˇ ˇ ˇ ˇ

A.aⁿ; bⁿ/ Lⁿ_n.a; b/

ˇ ˇ ˇ

ˇ n.b a/.b^{n 1} a^{n 1}/

8 (4.1)

n.b a/.b^{n 1}Ca^{n 1}/

8 Dn.b a/A.a^{n 1}; b^{n 1}/

4 :

Inequality (4.1) gives a refinement for inequality (1.5) in the case that˛D1and f .x/Dxⁿ

that turns to the inequality obtained in Proposition 3.1 in [3], where 0a < b,n2Nandn2.

It follows thatf⁰.x/Dnx^{n 1}, satisfies a Lipschitz condition for KD sup

x2Œa;b

n.n 1/x^{n 2}Dn.n 1/b^{n 2}: So from Theorem7we have

ˇ ˇ ˇ ˇ

A.aⁿ; bⁿ/ Lⁿ_n.a; b/

ˇ ˇ ˇ

ˇ n.n 1/b^{n 2}.b a/²

12 :

At last, Theorem8and9imply the following inequalities.

ˇ ˇ ˇ ˇ

f A.a; b/

Lⁿ_n.a; b/

ˇ ˇ ˇ

ˇ nb^{n 1}.b a/

4 ;

and ˇ ˇ ˇ ˇ

f A.a; b/

Lⁿ_n.a; b/

ˇ ˇ ˇ

ˇ n.n 1/b^{n 1}.b a/²

12 Cn

4A^{n 1}.a; b/:

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Author’s address

M. Rostamian Delavar

Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P. O. Box 1339, Bojnord 94531, Iran

E-mail address:m.rostamian@ub.ac.ir