INTEGRAL INEQUALITIES OF HERMITE-HADAMARD TYPE FOR ((α;M); LOG)-CONVEX FUNCTIONS ON CO-ORDINATES Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 4(22), No. 2, 2015, pp. 73-92

DOI: 10.15393/j3.art.2015.2829

73

UDC 517.51

Bo-Yan Xi and Feng Qi

INTEGRAL INEQUALITIES OF HERMITE - HADAMARD TYPE FOR ((a, m), log)-CONVEX FUNCTIONS ON CO-ORDINATES

Abstract. The convexity of functions is a basic concept in mathematics and it has been generalized in various directions. Establishing integral inequalities of Hermite - Hadamard type for various convex functions is one of the main topics in the theory of convex functions and attracts a number of mathematicians for several centuries. Currently an amount of literature on integral inequalities of Hermite - Hadamard type for various convex functions has been accumulated. In the paper the authors introduce a new concept "((a,m), log)-convex functions on the co-ordinates on the rectangle of the plane" and establish new integral inequalities of the Hermite-Hadamard type for ((a, m), log)-convex functions on the co-ordinates on the rectangle of the plane.

Keywords: convex function, ((a,m), log)-convex function, coordinates, integral inequality of the Hermite - Hadamard type

2010 Mathematical Subject Classification: 26A51, 26D15, 26D20, 26E60, 41A55

1. Introduction. The convexity of functions is a very important and fundamental concept in mathematics and mathematical sciences. It has been being generalized to various forms and there is an amount of literature on integral inequalities of the Hermite - Hadamard type for various convex functions.

Let us recall some definitions and related conclusions. Definition 1. Let I C R = (-to, to) be an interval. A function f : I ^ R is said to be convex if

f (Ax + (1 - A)y) < Af (x) + (1 - A)f (y) holds for all x,y E I and A E [0,1].

©Petrozavodsk State University, 2015

[MglHl

Let f : I C R ^ R be a convex function and a, b G I with a < b. Then

b

f (a+b) <_!_ i f (x)dx < f (a) + f (b)

2 J - b - a J J K ' ~ 2

a

is valid. This double inequality is well known in the literature as the Hermite - Hadamard integral inequality for convex functions.

Definition 2. If a positive function f : I C R ^ R+ = (0, to) satisfies f (Ax + (1 - A)y) < fx(x)f 1-A(y)

for all A G [0,1], then we call f a logarithmically convex function on I.

Remark 1. It is well known that logarithmic convexity of a function f is equivalent to convexity of the function ln f or loga f for a > 1.

Definition 3. [1] Let f : [0, b] ^ R and m G (0,1]; if

f (Ax + m(1 - A)y) < Af (x) + m(1 - A)f (y)

is valid for all x,y G [0, b] and A G [0,1], then we say that f is an m-convex function on [0, b].

Remark 2. The 1-convexity is equivalent to the ordinary convexity defined by Definition 1.

Theorem 1. [2] Let f : Ro = [0, to) ^ R be m-convex and m G (0,1]. If f G L1([a, b]) for 0 < a < b < to, then

b

1

f (x)dx < mini" f (a)+ mf (b/m), mf (a/m) + f (b)).

b - a 2 2

a

Definition 4. [3] Let f : [0, b] ^ R and (a,m) G (0,1] x (0,1]; if

f (Ax + m(1 - A)y) < Aaf (x) + m(1 - Aa)f (y)

is valid for all x, y G [0, b] and A G [0,1], then we say that f is an (a, m)-convex function on [0, b].

Remark 3. The (1, m)-convexity is equivalent to m-convexity. Any convex function for which f (0) < 0 is m-convex for any m G (0,1].

Theorem 2. [4, Theorem 3.1] Let I ^ R0 be an open real interval and let f : I ^ R be a differentiable function on I such that f' G L([a, b]) for 0 < a < b < to. If (f')q is (a, m)-convex on [a, b] for some given numbers a, m G (0,1] and q > 1, then

f (a) + f (b)

1

2 b - a

+ v2 m

f '( ^

m

f (x)dx 1/q

<

b — a /1

1-1/q

mm

v2m

f M ^^

m

vi[f '(a)]9+

1/q -

+ vi [f ' (b)]9

Where V1 = (a+1)(a+2) (« + 2a ) and v2 = (a+1)V+2) (

a2 +a+2 2

2a

Definition 5. [5, 6] A function f : A ^ R is said to be convex on the co-ordinates A = [a, b] x [c, d] C R2 with a < b and c < d, if the partial mappings

fy : [a, b] ^ R, fy (u) = f (u, y) and fx : [c, d] ^ R, fx(v) = f (x, v) are convex for all fixed x G (a, b) and y G (c, d).

Remark 4. A formal definition for convex functions on the co-ordinates may be restated [5, 6] as follows. A function f : A ^ R is said to be convex on the co-ordinates A = [a, b] x [c, d] with a < b and c < d, if

b

2

9

f (tx + (1 - t)z, Ay + (1 - A)w) < tAf (x, y) + t(1 - A)f (x, w)+

+ (1 - t)Af (z,y) + (1 - t)(1 - A)f (z,w)

holds for all t, A G [0,1] and (x, y), (z, w) G A.

Definition 6. [7] For some (ai,mi), (a2,m2) G (0,1]2, a function f : [0, b] x [0, d] ^ R is said to be (a1, m1)-(a2, m2)-convex on the co-ordinates [0, b] x [0, d], if

f (ta + mi(1 - t)b, Ac + m2(1 - A)d) < tai Aa2f (a,c)+

+ m2tai (1 - Aa2)f (a, d) + mi(1 - tai)Aa2f (b, c) +

+ mim2(1 - tai)(1 - Aa2)f (b, d)

holds for all t, A G [0,1] and (x,y), (z,w) G [0, b] x [0, d].

Definition 7. [8] For mi,m2, a E (0,1] and s E [-1,1], a function f : [0,6] x [0, d] ^ R0 is called (a,m1)-(s, m2)-convex on co-ordinates if

f (tx + m1(1 - t)z, Ay + m2 (1 - A)w) < taAsf (x,y)+

+mi(1 - ta)Asf (z,y) + m2ta(1 - A)sf (x,w) +

+mim2(1 - ta)(1 - A)sf (z,w)

holds for all (t, A) E [0,1] x (0,1) and (x, y), (z, w) E [0, 6] x [0, d].

Theorem 3. [5,6, Theorem 2.2] Let f : A = [a, 6] x [c, d] ^ R be convex on the co-ordinates A = [a, 6] x [c, d] with a < 6 and c < d. Then

'a + b c + d\ 1 f 2 ' 2 / " 2

1

r-b

b - aJ a V 2 /

<

1 /a + b

+ ^ ^ T /

- (b - a)(d - c) i i f (x,y)dydx -

1

< -- 4

'a «/ c rb r b

' ' f (x,c)dx + I f (x,d)dx ) +

b - a

rd d

<

+ d-c^/ f(a,y)dy + J f(b,y)dy

- 1 [f (a, c) + f (b, c) + f (a, d) + f (b,d)].

For more information on integral inequalities of the Hermite - Hadamard type for convex functions on the co-ordinates, please refer to [9]—[12] and closely related references therein.

In this paper, we will introduce a new concept "((a,m), log)-convex functions on the co-ordinates on the rectangle of the plane" and establish some new integral inequalities of the Hermite — Hadamard type for ((a,m), log)-convex functions on the co-ordinates on the rectangle of the plane.

2. A definition and a lemma. Now we introduce the definition as follows.

Definition 8. A mapping f : [0, b] x [c, d] ^ R+ is said to be ((a, m), log)-convex on co-ordinates [0, b] x [c, d] with c, d G R, c < d, and b > 0 if

f (tx + m(1 - t)z, Ay + (1 - A)w) <

< ta[f (x,y)]A[f (x,w)]i-A + m(1 - ta)[f (z,y)]A[f (z,w)]i-A

holds for all t, A G [0,1] and (x, y), (z, w) G [0, b] x [c, d] and some m, a G G (0, 1].

Remark 5. The Definition 8 implies

f (tx + m(1 - t)z, Ay + (1 - A)w) < ta[f (x, y)]A[f (x, w)]i-A+ + m(1 - ta)[f (z,y)]A[f (z,w)]i-A < < taAf (x,y)+ ta(1 - A)f (x,w) +

+ m(1 - ta)Af (z, y) + m(1 - ta)(1 - A)f (z, w).

If the function f is ((a, m), log)-convex on co-ordinates [0, b] x [c, d], then it is (a, m)-(1,1)-convex on co-ordinates [0, b] x [c, d] (with (ai,mi) = = (a,m) and (a2,m2) = (1,1) in Definition\^or with (a,mi) = (a,m) and (s,m2) = (1,1) in Definition

In order to prove our main results, we need the following lemma.

Lemma 1. Let f : A = [a, b] x [c, d] C R2 ^ R have partial derivatives of the second order. If fX'y G Li(A), then

J(f, A) = (b - a)(d - c^4f (a, c) - 2f (a, d) - 2f (b, c) + f (b, d)-

1 fb 1 Z,d

' [2f (x,c) - f (x,d)]dx - [2f (a, y) - f (b,y)]dy+

b — a Ja d —

+ - a1(d - c)l L f (x'y)dxd4 =

= (1+ t)(l + A)/Xy(ta +(1 — t)b, Ac +(1 — A)d)dtdA. (1) Jo io

Proof. Integration by parts gives

I Î (1+ t)(1 + A)/"y (ta +(1 — t)b, Ac +(1 — A)d)dtdA = 'o io

^ T(1 + A)

a — b o

(1 + t)/y (ta + (1 — t)b, Ac + (1 — A)d) 11=0—

- f f (ta +(1 - t)b, Ac +(1 - A)d)dt

dA =

1

4f (a, c) - 2f (b,c) - 2f (a, d) + f (b,d)-

(a - b)(c - d)

- 2 f f (a, Ac +(1 - A)d)dA + f f (6, Ac + (1 - A)d)dA-

Jo Jo

- 2 f f (ta + (1 - t)b, c)dt + f f (ta + (1 - t)b, d)dt+

oo

+ 11 f (ta +(1 - t)b, Ac +(1 - A)d)dtdA

oo

Using substitutions x = ta + (1 -t)b and y = Ac + (1 - A)d for t, A G [0,1], we obtain (1). Lemma 1 is thus proved. □

3. Some integral inequalities of the Hermite — Hadamard type.

Now we are in a position to establish some new integral inequalities of the Hermite - Hadamard type for differentiable ((a, m), log)-convex functions on the co-ordinates on rectangle of the plane R0 x R. The first main result is Theorem 4.

Theorem 4. Let f : R0 x R ^ R be a partially differentiable function on R0 x R and f"y G Li([a, m] x [c, d]) with 0 < a < b, c < d, and m G (0,1]. If |f"y |q is ((a, m), log)-convex on co-ordinates [0, m] x [c, d] for q > 1 and some a G (0,1], then

3 \ 2( i-i/q)

ij (f, A)|< (

i /q

x

2(a + 1)(a + 2) _ x <| 2(2a + 3)Ff (a, c)|q, fy(a,d)|q) +

+ ma(3a + 5)F

fxy ( _„„, c

b

m

/iu m,d

q ^ i /q

where

F(u,v) = \ 3

L(u,v) - u L(u, v) —----, u = v,

ln v — ln u

2 U,

u = v.

l

0

1

q

Proof. By Lemma 1 and Holder's integral inequality, we have

|J(/, A)|</ I (1+ t)(1 + A) I /Xy (ta +(1 — t)b, Ac +(1 — A)d)|dtdA <

oo

1 rl

<

(1 + t)(1 + A)dtdA

oo

1-1/q

1 rl

(1 + t)x

oo

x(1 + A)|/"y/(ta + (1 — t)b, Ac + (1 — A)d) |qdtdA Using the ((a,m), log)-convexity of |/"y |q, we have

i/q

(2)

I Î (1+ t)(1 + A//(ta +(1 — t)b, Ac +(1 — A)d)|qdtdA < Jo Jo

oo

< /71(1+1)(1 + A)

oo

t/(a,c)|q/(a, d) |q(1-A) +

q(l-A)

+m(1 — ta)

/xy( m,c

qA

q(l-A)

dt dA =

(1 + t)tadt

+t)(1 — ta )dt

Note that

(1 + A) | /% (a, c) |qA | /% (a, d) |q(1-A) dA + m

q(i-A)

(1 + A)

m,c

qA

ci m,d

q(l-A)

dA.

(1+

(3)

ll

oo l

(1 + t)(1 + A)dtdA = ( - ) ,

î (1+t)tadt = 2a +3 ,

Jo (a + 1)(a + 2)'

^ (1+ t)(1 — t«)dt = a(3a + 5)

2(a + 1)(a + 2)'

1) If /y (a, c) |q = /y (a, d) |q and /y ( m ,c)|q = |/:

have

p// ( b jm q

' X y V m '

we

(1 + (a, c) |qA | /"y (a, d) |q(1-A) dA = 3 (a,c)|q,

q(l-A)

(1 + A)

m,c

qA

/x«l m,d

q(i-A)

dA = -2

ixy ( _ „„ , c

m

1

1

1

o

o

o

1

o

1

o

1

q

o

2) If £ 4 (a'c) /Xy («'d) we can write

= 1 and n =

( mb 'c)

(mm,d)

= 1, integrating by parts,

(1 + / (a, c)|qA|/(a,d)|q( 1 -A)dA =

|/xy (a,d)|

= |/x„(a,d)|q / (1 + A)£adA =

_|/x„(a, d) p

ln £

2£ - 1 - £AdA

ln £

2£ -1 -

£ -1 ln £

= F(j/iy(a, c) |q, /y(a,d)|q)

and

(1 + A)

fiy ( ___, c

m

qA

ixy( ___,d

m

q( 1 -A)

dA =

ci m,d

q r 1

(1 + A)nA dA =

/ (b,

K x y V m '

|f " (b,

1« y Vm-

ln £

2n - 1 -

£ -1 ln n

= F

ln n

fiy ( ZT,c

2£ - 1 - nAdA

m

fiy ( .__ , d

m

(4)

By utilizing (2), (3), and (4), we obtain

1 1

|J(/, A)|<y |(1 + t)(1 + A) | ///y (ta +(1 - t)b,Ac +(1 - A)d)|dtdA+

0 0

\ 1/q

+ (1 - t)b,Ac +(1 - A)d)|qdtdA] <

< 2

2( 1 - 1 /q)

1

2(a + 1)(a + 2)

\ 1 /u

J |2(2a + 3)F/ (a, c) |q,

|/,//y (a, d)|q) + ma(3a + 5)F

ixy( ___,c

m

/iy ( ___, d

m

q\ 11/q

This completes the proof of Theorem 4. □

If taking q = 1 in Theorem 4, we can derive the following corollary.

q

q

q

q

0

1

0

1

0

q

1

0

q

q

Corollary 1. Under the conditions of Theorem 4 if q = m = a = 1, then

|J (f, A)| < -{5F (|f''y (a, c) |9, f'y (a, d) |9 ) +

+ 4F (|f''y (b, c) |9 f (b,d)|9 )}.

Theorem 5. Let f : R0 x R ^ R be a partially differentiable function on R0 x R and f|'y G Li([a, mm] x [c, d]) with 0 < a < b, c < d, and m G (0,1]. If |f''y |q is ((a, m), log)-convex on co-ordinates [0, m] x [c, d] for q > 1 and a G (0,1], then

J(f, A)l< (23(2^ [2(29-1)/(9-1) -1])

1-1/9

1

2(a + 1)(a + 2)

1/9

x

x 2(2a + 3)L|f''y(a,c)|9, |f''y(a,d)|9) +

+ ma(3a + 5)L where L(u, v) is defined by

fxy ( ___, c

m

f"»( zr,d

m

9 \ Ï 1/9

v — u

L(u, v) = / Uv1 *dt = < ln v - ln u

u,

, U = v,

u = v.

(5)

Proof. By Lemma 1, Holder's integral inequality, and the ((a, m), log)-convexity of |f"y |q, it follows that

11

1-1/9

|J(f, A)| < I J (1+ t)(1 + A)9/(9-1)dtdA J x

x

1-1/9

î Î (1 + t)f'y f (ta +(1 - t)b, Ac +(1 - A)d)|<? dtdA Jo Jo

< Oo Jo (1 + t)(1 + A)9/(9-1)dtdA

tf (a, c)|9A|f^'y(a,d)|9(1-A)

1/9

<

x{ i! (1+

9

1

o

+m(1 - ta)

/xj/ ( ___, c

m

qA

/г// ( ___, d

m

q(1-A)n

dt dA

1/q

1 1/q

x

/ 3(q - 1) r2(2q 1)/(q 1) _ 1lV

\ r-1

(1+ t)tadtjj0 /г;(а,е)|/(a,d)|q(1-A)dA+

+m / (1+ t)(1 - ta)dt Jo Jo

/xy ( ___, c

m

qA

/a;j/ ( ___, d

m

q(1 a) ï 1/q dA

f 3(q - 1) r2(2q 1)/(q 1) _1Л 1 1/q (

V 2(2q - 1)^ ^ I

2(a + 1)(a + 2)

1/q

x

x 2(2a + 3)L(j/T'y(a, c)|q, /y(a,d)|q) +

+ma(3a + 5)L

b

/ry( m,c

b

f-I m,d

1/q

Theorem 5 is proved. □

Corollary 2. Under the conditions of Theorem 5 if m = a = 1, then

1-1/q

J (/,д)|< ( 1Г ( [2(2q-1)/(q-1) -ч)

X {5L(|/^Ty (a, c) |q / (a, d) |q ) +4L(|/^; (b,c)|q,/^ (b,d)|q )}

1/q

Theorem 6. Let / : R0 x R ^ R be a partially differentiable function on R0 X R and /Г/ G L1([a, m] x [c, d]) with 0 < a < b, c < d, and m G (0,1]. If |/Г/ |q is ((a, m), log)-convex on co-ordinates [0, m] x [c, d] for q > 1 and a G (0,1], then

| J (/,д) i< ( [2(2q-1)/(q-1) - 1l)1'1/\a+r )Vq x

{f(|/г;(a, c) |q/(a, d) |q) +

+ maF

/xy( m,c

x

q\ ï 1/q

/ry( m,d

1

1

1

q

where F(u, v) is given as in Theorem 4

Proof. By Lemma [I], Holder's inequality, and the ((a, m), log)-convexity

of fy|q, we get

|J(f, A)| < f Î (1+ t)(1 + A)|f''y (ta +(1 - t)b, Ac +(1 - A)d)|dtdA < Jo Jo

r1 (• 1 )1-1/9

< I / / (1 +1)9/(9-1)(1 + A)dtdA x

oo

x

11

(1 + A) | f''y f (ta + (1 - t)b, Ac + (1 - A)d) |9dtdA

oo

1/9

<

<

/ 3(q - 1) T2(29-1)/(9-1^ V2(2q - 1)^ ^

1-1/9

x

x

ta dt) / (1 + A)|f" (a, c) |9A | f''y (a,d)|9(1 A) dA+

oo +m I (1 - ta)dt f (1 + A)

f'y ( ___, c

m

9A

f'y ( ___, d

m

9(1-A) 1/9

dA

(2^ [2(29-1)/(9-1) - 1]) 1-1/9 (-+-1 )1/9 [F f (a, c) |9,

- 1)

|f''y (a, d) |9) + maF

b

f'y( m,c

b

m,d

1/9

The proof of Theorem 6 is complete. □

Corollary 3. Under the conditions of Theorem 6 if m = a = 1, then

|Jf A)l< (2)1/9 (g^ [2(29-1)/(9-1) - 1])1-1/9x

x | F f (a, c) |9 ,|f''y (a, d) |9 ) + F f (b, c) |9, fy (b, d) |9 )}

1/9

Theorem 7. Let f : R0 x R ^ R be a partially differentiable function on R0 x R and f|'y G Li([a, m] x [c, d]) with 0 < a < b, c < d, and m G (0,1]. If |f''y |q is ((a, m), log)-convex on co-ordinates [0, mm] x [c, d]

o

for q > 1 and a G (0,1], then

|J (/, A)|<

q - 1

2q - 1

2(1-1/q) / i \ 1/q [2(2q-1)/(q-1) - 1]) (—) x

a + 1

x l(|/;;(«,c)|q, |/;;(a,d)|q)+

+ maL

fxy ( ___, c

m

f/y ( ___, d

m

q\ 1/q

where L(u, v) is given by (5).

Proof. By Lemma [I], Holder's inequality, and the ((a,m), log)-convexity of | //y |q, we acquire

|J(/, A)| < f f (1 + t)(1 + A)|/x//y(ta + (1 - t)b,Ac +(1 - A)d)|dtdA <

00

{Jo Jo [(1 + t)(1 + A)]q/(q-1)dtdA^)

1 1/q

x

<

r f1

00

|/i/y/(ta + (1 - t)b, Ac + (1 - A)d)|qdtdA

1/q

<

< ( J_L [2(2q-1)/(q-1)

"1 \ r 1

x I ta dt

00 1

2(1-1/q)

x

d*) I |/iy(a,c)|qA|/iy(a,d)|q(1-A)dA+

(f (1 - ta ^iV- (m,c)r k"y (mi,d)r"A) dA}

q 1 \ 2(1-1/q)( i )1/q r

q - 1 [2<2q-1)/<q-1» - 1]) ({L/(a,c)|q,

2q - 1

|///y (a,d)D + maL

m

/xy ( .__ , d

m

q\ 1/q

The proof of Theorem 7 is complete. □

Corollary 4. Under the conditions of Theorem 7 if m = a = 1, then

|J(/, a)| < ( 1/q([2(2q-1)/(q-1)

V v2q -1

2(1 1/q)

x

x{ (a, c) |q f (a,d)|q) + L(|f^ (b, c) |q, fy (b,d)|q )} ^. S

Theorem 8. Let f : R0 x R ^ R+ be integrable on [a, m ] x [c, d] with 0 < a < b, c < d, and m G (0,1]. If f is ((a, m), log)-convex on co-ordinates [0, m ] x [c, d] for a G (0,1], then

Va + b c + dNj <

2 ' 2

<

1

1

2a+M b - a

^Tr/' c + d\ fx c + d

/ x,^T" + m(2a - 1)/ —,

+

1

d - c + m(2a - 1)

< 2a+1 j b - a +

2

/(^ ,c + d - y

m 2

dx+

f

2

a+b 2m

, y f

2

a+b 2m

1/2

+

, c + d - y

1/2

dy <

* f/fx,^ + m(2" - 1)/fV + d

2

dc

/(^,y) + m(2" - 1)^0+b,y

2

2m

m2

dy <

dx+

d /•&

<

22a(b - a)(d - c)

/(x,y) + 2m(2a - 1)/ m,y ) +

x

+ m2(*" - D2/{m,y)

dxdy,

where L(u, v) is the logarithmic mean.

Proof. Using the ((a, m), log)-convexity of f and by the GA inequality, we have

^a + b c + d^ - J1 f ^a + (1 - *)b + (1 - t)a + tb c + dN

dt <

1

<

¿ f (ta + (1 - t)b,c±^i +

. n /(1 - t)a + tb c + d

+m(1 - Ia )f

dt =

1

Ia (b - a) 1

m I

br-/ c + d\ . f x c + d

f x, —+ m(2a - 1)f —,

а

1 /• b

2a(b - a) У a

f x,

I ,

c+d 1

m I

+ m(2a - D/l^

dx =

dxdЛ <

<

32a (b - a) J a +Im(Ia - 1)

x

, Лc + (1 - Л)й)/(x, (1 - Л)c + Лй)] 1/2+

f( -^c +(1 - ЛШ/( -, (1 - Лк + Лй )

mm

1/2

+

+m2(Ia - l)2

f (m ^c+(1 - Л)й)/(m, (1 - ^c+11/2}

>dxdЛ =

1

I2a (b - a)(d - c) +Im(Ia - 1)

x,y)f (x,c + d - y)]1/2+

c «/a ^

fi^ff-c + d - y) |1/2 +

mm

1/2 }

dxdy <

<

+m2(Ia - l)2 1

I2a (b - a)(d - c)

Я ,Ил ,c + d - y

m2 m2

r.d Л b

f (x, y) + Im(Ia - 1)f -,y +

m

x

x

+m2 (2a - 1)2 fi— ,y

dxdy.

Similarly, we acquire a + b c + d

/a + b c + d\ f1 a + b Лc + (1 - Л)й + (1 - Л)c + 1Л

Ч—Ja Ч—,-i-Г <

< Ia

x V1 3a

, Лc +(1 - Л)/^, (1 - Л)c + лЛ

f

I

a+b Im

1/2

+ mx

a+b

, Лc +(1 - Л)^/^^-, (1 - Л^ + Лй

)Г"(

ЙЛ =

1

a

1

1

<

2a (d - c) +m(2a - 1)

2a(d - c) / 1

<

f

a+b

, y f

a+b

, c + d - y

f

a+b

, y f

a+b

,c+d-y

1/2

2m 2m

"r/ ( + m(2- - 1)/( o+i ,y

2

d /. 1

2m

1/2

+

dy < dy <

X/

22a (d - J o (1 - t)a + tb

m

f(ta + (1 - t)b, y) + 2m(2a - 1) X

y ) dtdy =

,y) + m2(2a - 1)2 / (1 - t)a + tb

m2

1

d /. b

22a (b - a)(d - c)

f(x,y) + 2m(2a - 1)f - y +

m

x

+m2(2a - 1)2f( ,y

x

m2

dxdy.

Combining the above inequalities leads to Theorem 8. □

Theorem 9. Let f : R0 x R ^ R+ be integrable on [a, m] x [c, d] with 0 < a < b, c < d, and m G (0,1]. If f is ((a, m), log)-convex on co-ordinates [0, m] x [c, d] for a G (0,1], then

1

db

(b - a)(d - c) b

f (x,y)dxdy <

1

2a+1 (b - a)

x

X / <L(f(x,c),f(x,d))+ m(2a - 1)L(f( JxAff^d

dx+

+

1

2(a + 1)(d - c)

f (a,y) + —,y

<

1

2a+2(b - a) Ja 1

+ 2(a + 1)(d - c) 1

G(x, c, d)dx+

f (a,y) + maf —,y

m

b

dy <

m

dy <

< 2a+2(a+G(a,c,d)+maG^,c,d) +2L(f (a,c),f (a,d))+

b

2

2

b

d

+ 2m(2a - 'K'fe^'fe,")) +2maM/(m4

/f-,^ + 2m(2a - A/fA,d <

m

<

1

m2

G(a, c, d) + maG( —, c, d

m2

2a+1(a + 1) where L(u, v) is the logarithmic mean and

G(x, c, d) = /(x, c) + /(x, d) + m(2a - 1) (/ (X, c) + / (XX,d) )

for x G [а, m].

Proof. Since L(x,y) < for x,y > 0, from the ((a,m), log)-convexity of / and by the GA inequality, we obtain

1

d ,-b

(6 - a)(d - c)

1 b

/ (x,y)dxdy =

6 — a

/(x, Ac + (1 - A)d)dxdA <

0 ./a

<

1

2a (6 - a) Jo

+ m(2a - 1) 1

2a (6 - a)

x,c)/(x,d)]A+

1A

/(-,c / -,d

mm

L(/(x,c),/ (x,d)) +

dxdA =

+ m(2a - 1)L/m,^,4m,d

dx

<

1

2a+1 (6 - a)

G(x, c, d)dx <

<

1

2®+1

taG(a, c, d) + m(1 - ta)G —, c, d

6

m

dt =

2a+1 (a + 1)

G(a, c, d) + maG —, c, d

6

0

and

1

d f-b

(b - a)(d - c) d r i

f (x,y)dxdy =

1

c J0 d i- i

<

d - c 1

d - c c 0

1

(a + 1)(d - c)

f (ta + (1 - t)b, y)dtdy <

ta f (a, y) + m(1 - ta )f(-,y

b

m

f (a,y) + maf ( — ,y

m

dtdy = dy <

<

1

2a(a + 1^0 + m(2a - 1)

[f (a, c)]A [f (a,.

li-A

+

m

fi-,««

m

i-A

+

+ ma

b

f ZT,c

m

m

i-A

+

+ m(2a - 1)

m2

f(i, d

m2

i-A-

dA =

l(/(a,c),f(a,d)) + m(2"" ^fô'Vfe") ' +

+ ma{L(f ( m,c),f ( m,d)) + + m(2a - WffA AffA, d <

m2

m2

<

1

2a+i(a + 1)

G (a, c, d) + maG —, c, d

b

m

Combining the above inequalities results in Theorem 9. □

Corollary 5. Under the conditions of Theorems 8 and 9 if a = m = 1, we have

f l'a + b c + d^ < 1 I 1

22

2] b - a

f (x —+- )dx+

A

A

A

+

1

< -< 2

<

dc

ba

d fi^ ,c+d - y

2

2

i/2

d^ <

fix, c + ) dx +

2

rd r b

dc

f (^t6 'dy

<

(b — a)(d — c)

f (x,y)dxdy <

1

< -< 4

1

ba

2L(f (x, c), f (x, d))dx +

dc

[f (a,y) + f (b,y)]dy

<

1

1

1

b

1

1

< -- 4

1 b d

[f (x,c) + f (x,d) dx — [f (a,y)T f (b,y^ dy

b — a J a d — c ,, c

<

< 1 [f (a, c) + f (a, d) + f (b, c) + f (b, d) +

+ 2L(f (a, c), f (a, d)) + 2L(f (b, c), f (b, d))] <

< 1 [f (a, c) + f (a, d) + f (b, c) + f (b,d)].

Acknowledgment. This work was partially supported by the National Natural Science Foundation of China under Grant no. 11361038, by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant no. NJZY14192, and by the Inner Mongolia Autonomous Region Natural Science Foundation Project under Grant no. 2015MS0123, China. The authors appreciate anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

References

[1] Toader G. Some generalizations of the convexity. Proceedings of the Colloquium on Approximation and Optimization, Univ. Cluj-Napoca, Cluj-Napoca, 1985, pp. 329-338.

[2] Dragomir S. S. and Toader G. Some inequalities for m-convex functions. Studia Univ. Babe§-Bolyai Math., 1993, vol. 38, no. 1, pp. 21-28.

[3] Mihe§an V. G. A generalization of the convexity. Seminar on Functional Equations, Approx. Convex, Cluj-Napoca, 1993. (Romania)

[4] Bakula M. K., Ozdemir M. E., and PeCariC J. Hadamard type inequalities for m-convex and (a, m)-convex functions. J. Inequal. Pure Appl. Math., 2008, vol. 9, no. 4, Art. 96, 12 pages. Available online at http://www.emis.de/journals/JIPAM/article1032.html.

[5] Dragomir S. S., On the Hadamard's inequality for convex functions on the co-ordinates in a rectangle from the plane. Taiwanese J. Math., 2001, vol. 5, no. 4, pp. 775-788.

[6] Dragomir S. S. and Pearce C. E. M. Selected Topics on Hermite-Hadamard Inequalities and Applications. RGMIA Monographs. Victoria University, 2000. Online at http:// www.staff.vu.edu.au/RGMIA/monographs/hermite_ hadamard.html.

[7] Chun L. Some new inequalities of Hermite-Hadamard type for (ai ,mi)-(a2, m2)-convex functions on coordinates. J. Funct. Spaces, 2014, vol. 2014, Article ID 975950, 7 pages. DOI: 10.1155/2014/975950.

[8] Xi B.-Y., Bai S.-P., and Qi F. Some new inequalities of Hermite - Hadamard type for (a, mi)-(s,m2)-convex functions on co-ordinates. Research Gate Dataset. DOI: 10.13140/2.1.2919.7126.

[9] Bai S.-P. and Qi F. Some inequalities for (si ,mi)-(s2,m2)-convex functions on the co-ordinates. Glob. J. Math. Anal., 2013, vol. 1, no. 1, pp. 22-28. DOI: 10.14419/gjma.v1i1.776.

[10] Bai S.-P., Wang S.-H., and Qi F. Some new integral inequalities of Hermite-Hadamard type for (a,m; P)-convex functions on co-ordinates. J. Appl. Anal. Comput., 2016, vol. 6, no. 1, pp. 171-178. DOI: 10.11948/2016014.

[11] Guo X.-Y., Qi F., and Xi B.-Y. Some new Hermite-Hadamard type inequalities for geometrically quasi-convex functions on co-ordinates. J. Nonlinear Sci. Appl., 2015, vol. 8, no. 5, pp. 740-749.

[12] Xi B.-Y., Hua J., and Qi F. Hermite-Hadamard type inequalities for extended s-convex functions on the co-ordinates in a rectangle. J. Appl. Anal., 2014, vol. 20, no. 1, pp. 29-39. DOI: 10.1515/jaa-2014-0004.

Received June 23, 2015. In revised form, October 26, 2015.

Inner Mongolia University for Nationalities

Tongliao City, Inner Mongolia Autonomous Region, 028043, China E-mail: [email protected], [email protected]

Tianjin Polytechnic University

Tianjin City, 300387, China;

Henan Polytechnic University

Jiaozuo City, Henan Province, 454010, China

E-mail: [email protected], [email protected]