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(1)

doi:10.5937/MatMor1801065S

Vol. 22, No. 1 (2018), 65–79

Hermite-Hadamard type inequalities for

(

m, M

)

-

Ψ

-convex functions when

Ψ =

ln

Silvestru Sever Dragomir and Ian Gomm

Abstract. In this paper we establish some Hermite-Hadamard type inequalities for(m, M)-Ψ-convex functions whenΨ =−ln. Applicati-ons for power functiApplicati-ons and weighted arithmetic mean and geometric mean are also provided.

1. Introduction

The following integral inequality

(1) f

a+b

2

≤ 1

b−a

Z b

a

f(t)dt≤ f(a) +f(b)

2 ,

which holds for any convex function f : [a, b] → R, is well known in the

literature as the Hermite-Hadamard inequality.

There is an extensive amount of literature devoted to this simple and nice result which has many applications in the Theory of Special Means and in Information Theory for divergence measures, for which we would like to refer the reader to [1]-[4], [15]-[30], the monograph [13] and the references therein. Assume that the functionΨ :I ⊆RR(I is an interval) is convex onI

and m∈R. We shall say that the functionΦ :I →Ris m-Ψ-lower convex

if Φ−mΨ is a convex function on I. We may introduce (see [6]) the class of functions

(2) L(I, m,Ψ) :={Φ :I →R|Φ−mΨ is convex onI}.

Similarly, for M ∈R and Ψ as above, we can introduce the class ofM-Ψ -upper convex functions (see [6])

(3) U(I, M,Ψ) :={Φ :I →R|MΨ−Φ is convex onI}.

2010Mathematics Subject Classification. Primary: 26D15; Secondary: 26D10.

Key words and phrases. Convex functions, special convexity, weighted arithmetic and geometric means, logarithmic function.

Full paper. Received 12 January 2018, revised 17 April 2018, accepted 11 May 2018, available online 25 June 2018.

c

2018 Mathematica Moravica

(2)

The intersection of these two classes will be called the class of (m, M)-Ψ -convex functions and will be denoted by [6]

(4) B(I, m, M,Ψ) :=L(I, m,Ψ)∩ U(I, M,Ψ).

Remark 1.1. IfΨ∈ B(I, m, M,Ψ), thenΦ−mΨand MΨ−Φare convex and then(Φ−mΨ)+(MΨ−Φ)is also convex which shows that(M−m) Ψ

is convex, implying thatM ≥m(asΨis assumed not to be the trivial convex function Ψ (t) = 0,t∈I).

The above concepts may be introduced in the general case of a convex subset in a real linear space, but we do not consider this extension here.

In [12], S. S. Dragomir and N. M. Ionescu introduced the concept of g -convex dominated functions, for a function f : I → R. We recall this, by saying, for a given convex function g : I → R, the function f : I → R is

g-convex dominated iffg+f andg−f are convex functions onI. In [12], the authors pointed out a number of inequalities for convex dominated functions related to Jensen’s, Fuch’s, Pečarić’s, Barlow-Proschan and Vasić-Mijalković results, etc.

We observe that the concept of g-convex dominated functions can be obtained as a particular case from (m, M)-Ψ-convex functions by choosing

m=−1,M = 1 and Ψ =g. The following lemma holds [6].

Lemma 1.1. Let Ψ,Φ : I ⊆ RR be differentiable functions on ˚I, the interior of I and Ψis a convex function on ˚I.

(i) For m∈R, the functionΦ∈ L˚I, m,Ψ

if and only if

(5) m

Ψ (t)−Ψ (s)−Ψ0(s) (t−s)

≤Φ (t)−Φ (s)−Φ0(s) (t−s),

for all t, s∈˚I.

(ii) For M ∈R, the function Φ∈ U˚I, M,Ψ if and only if

(6) Φ (t)−Φ (s)−Φ0(s) (t−s)≤MΨ (t)−Ψ (s)−Ψ0(s) (t−s),

for all t, s∈˚I.

(iii) For M, m∈Rwith M ≥m, the function Φ∈ B˚I, m, M,Ψ

if and

only if both (5)and (6)hold.

Another elementary fact for twice differentiable functions also holds [6].

Lemma 1.2. Let Ψ,Φ : I ⊆ RR be twice differentiable on ˚I and Ψ is convex on˚I.

(i) For m∈R, the functionΦ∈ L

˚

I, m,Ψ

if and only if

(3)

(ii) For M ∈R, the function Φ∈ U˚I, M,Ψ if and only if

(8) Φ00(t)≤MΨ00(t) for all t∈˚I.

(iii) For M, m∈Rwith M ≥m, the function Φ∈ B˚I, m, M,Ψif and only if both (7)and (8)hold.

For various inequalities concerning these classes of function, see the survey paper [11].

In what follows, we are considering the class of functionsB(I, m, M,−ln)

for M, m ∈ R with M ≥ m that is obtained for Ψ : I ⊆ (0,∞)→R,

Ψ (t) =−lnt.

IfΦ :I ⊆(0,∞)→Ris a differentiable function on˚I then by Lemma 1.1 we haveΦ∈ B(I, m, M,−ln) if and only if

m

lns−lnt−1

s(s−t)

≤Φ (t)−Φ (s)−Φ0(s) (t−s)

(9)

≤M

lns−lnt−1

s(s−t)

,

for any s, t∈˚I.

If Φ : I ⊆ (0,∞)→R is a twice differentiable function on ˚I then by

Lemma 1.2 we have Φ∈ B(I, m, M,−ln)if and only if

(10) m≤t2Φ00(t)≤M,

which is a convenient condition to verify in applications.

In this paper we establish some Hermite-Hadamard type inequalities for

(m, M)-Ψ-convex functions when Ψ =−ln.Applications for power functi-ons and weighted arithmetic mean and geometric mean are also provided.

2. Hermite-Hadamard Type Inequalities

In 2002, Barnett, Cerone and Dragomir [5] obtained the following refi-nement of the Hermite-Hadamard inequality for the convex function f : [a, b]→R:

f

a+b

2

≤νf

a+νb−a

2

+ (1−ν)f

a+b

2 +ν

b−a

2

(11)

≤ 1

b−a

Z b

a

f(u)du

≤ 1

2[f((1−ν)a+νb) +νf(a) + (1−ν)f(b)]≤

f(a) +f(b)

2 ,

for allν ∈[0,1].

(4)

Applying the Hermite-Hadamard inequality for the convex functionf on each subinterval [a,(1−ν)a+νb],[(1−ν)a+νb, b],ν ∈(0,1),we have,

f

a+ (1−ν)a+νb

2

[(1−ν)a+νb−a]

Z (1−ν)a+νb

a

f(u)du

≤ f((1−ν)a+νb) +f(a)

2 [(1−ν)a+νb−a]

and

f

(1−ν)a+νb+b

2

[b−(1−ν)a−νb]

Z b

(1−ν)a+νb

f(u)du

≤ f(b) +f((1−ν)a+νb)

2 [b−(1−ν)a−νb],

which are clearly equivalent to

νf

a+νb−a

2

≤ 1

b−a

Z (1−ν)a+νb

a

f(u)du

(12)

≤ νf((1−ν)a+νb) +νf(a) 2

and

(1−ν)f

a+b

2 +ν

b−a

2

≤ 1

b−a

Z b

(1−ν)a+νb

f(u)du

(13)

≤ (1−ν)f(b) + (1−ν)f((1−ν)a+νb) 2

respectively.

Summing (12) and (13), we obtain the second and third inequality in (11). By the convexity property of f, we obtain

νf

a+νb−a

2

+ (1−ν)f

a+b

2 +ν

b−a

2

≥f

ν

a+νb−a

2

+ (1−ν)

a+b

2 +ν

b−a

2

=f

a+b

2

and the first inequality in (11) is proved.

(5)

(i) The arithmetic mean

A=A(a, b) := a+b

2 , a, b≥0;

(ii) The geometric mean:

G=G(a, b) :=√ab, a, b≥0;

(iii) The harmonic mean:

H =H(a, b) := 2 1

a+

1

b

, a, b≥0;

(iv) The logarithmic mean:

L=L(a, b) :=

  

 

a, if a=b;

b−a

lnb−lna, if a6=b;

a, b >0;

(v) The identric mean:

I :=I(a, b) =

   

  

a, if a=b;

1

e

bb aa

b−a1

, if a6=b;

a, b >0;

(vi) The p-logarithmic mean:

Lp =Lp(a, b) :=    

  

bp+1−ap+1

(p+ 1) (b−a)

1p

, if a6=b;

a, if a=b;

wherep∈R\ {−1,0}and a, b >0.

It is well known that Lp is monotonic nondecreasing over p ∈ R with

L−1 :=L and L0:=I.

In particular, we have the inequalities H ≤ G ≤ L ≤ I ≤ A. We also notice that

1

b−a

Z b

a

tpdt=Lpp(a, b), p∈R\ {−1,0}, 1

b−a

Z b

a

dt t =L

−1(a, b)

and

1

b−a

Z b

a

lntdt= lnI(a, b).

We define the weighted arithmetic and geometric means

Aν(a, b) := (1−ν)a+νband Gν(a, b) :=a1−νbν

where ν ∈ [0,1] and a, b > 0. If ν = 12, then we recapture A(a, b) and

(6)

Theorem 2.1. LetM, m∈Rwith M > mandΦ∈ B((0,∞), m, M,−ln).

Then for anya, b >0 and ν∈[0,1]we have

ln

"

a+νb−2aν a+2b +νb−2a1−ν I(a, b)

#m (14)

≤ 1

b−a

Z b

a

Φ (u)du−

νΦ

a+νb−a

2

+ (1−ν) Φ

a+b

2 +ν

b−a

2

≤ln

"

a+νb−2a+b

2 +ν

b−a

2

1−ν

I(a, b)

#M

and

ln

"

I(a, b)

p

Aν(a, b)G1−ν(a, b) #m (15)

≤ 1

2[Φ ((1−ν)a+νb) +νΦ (a) + (1−ν) Φ (b)]− 1

b−a

Z b

a

Φ (u)du

≤ln

"

I(a, b)

p

Aν(a, b)G1−ν(a, b) #M

.

Proof. SinceΦ∈ B((0,∞), m, M,−ln),thenΦm := Φ +mlnis convex and by the second inequality in (11) we have

νΦ

a+νb−a

2

+ (1−ν) Φ

a+b

2 +ν

b−a

2

(16)

+mln

"

a+νb−a

2

ν

a+b

2 +ν

b−a

2

1−ν#

≤ 1

b−a

Z b

a

f(u)du+m 1 b−a

Z b

a

lnudu

= 1

b−a

Z b

a

f(u)du+mlnI(a, b),

while from the third inequality in (11) we have

1

b−a

Z b

a

f(u)du+m 1 b−a

Z b

a

lnudu

(17)

≤ 1

2[Φ ((1−ν)a+νb) +νΦ (a) + (1−ν) Φ (b)]

+1

2mln [Aν(a, b)G1−ν(a, b)],

(7)

Since Φ∈ B((0,∞), m, M,−ln), then also fM :=−Φ−Mln is convex and by the second inequality in (11) we have

−νΦ

a+νb−a

2

−(1−ν) Φ

a+b

2 +ν

b−a

2

(18)

−Mln

"

a+νb−a

2

ν

a+b

2 +ν

b−a

2

1−ν#

≤ − 1

b−a

Z b

a

Φ (u)du−MlnI(a, b),

while from the third inequality in (11) we have

− 1

b−a

Z b

a

Φ (u)du−MlnI(a, b)

(19)

≤ −1

2[Φ ((1−ν)a+νb) +νΦ (a) + (1−ν) Φ (b)]

−1

2Mln [Aν(a, b)G1−ν(a, b)],

for any a, b >0 andν ∈[0,1].

Making use of (16)-(19) we deduce the desired results (14) and (15).

Remark 2.1. If we write the second inequality in (11) for the convex

function −lnwe have

lnI(a, b)≤ln

"

a+νb−a

2

ν

a+b

2 +ν

b−a

2

1−ν#

,

which implies that

I(a, b)≤

a+νb−a

2

ν

a+b

2 +ν

b−a

2

1−ν

showing that

ln

"

a+νb−2aν a+2b +νb−2a1−ν I(a, b)

#

≥0.

If we use the third inequality in (11) for the convex function −lnwe have

lnpAν(a, b)G1−ν(a, b)≤lnI(a, b),

which implies that

p

Aν(a, b)G1−ν(a, b)≤I(a, b),

showing that

ln

"

I(a, b)

p

Aν(a, b)G1−ν(a, b) #

(8)

Corollary 2.1. With the assumptions of Theorem 2.1 we have

(20) ln

A(a, b)

I(a, b)

m

≤ 1

b−a

Z b

a

Φ (u)du−Φ

a+b

2

≤ln

A(a, b)

I(a, b)

M

and

(21) ln

I(a, b)

G(a, b)

m

≤ Φ (a) + Φ (b)

2 −

1

b−a

Z b

a

Φ (u)du≤ln

I(a, b)

G(a, b)

M

.

The inequality (20) was obtained in 2002 by Dragomir in [7], see also [11, p. 197].

Corollary 2.2. With the assumptions of Theorem 2.1 we have

ln

 q

3a+b

4

a+3b

4

I(a, b)

 m

(22)

≤ 1

b−a

Z b

a

Φ (u)du−1 2

Φ

3a+b

4

+ Φ

a+ 3b

4 ≤ln   q

3a+b

4

a+3b

4

I(a, b)

  M and ln "

I(a, b)

p

A(a, b)G(a, b)

#m (23) ≤ 1 2 Φ

a+b

2

+ Φ (a) + Φ (b) 2

− 1

b−a

Z b

a

Φ (u)du

≤ln

"

I(a, b)

p

A(a, b)G(a, b)

#M

.

For related results see [8] and [11, p. 197].

Theorem 2.2. LetM, m∈Rwith M > mandΦ∈ B((0,∞), m, M,−ln).

Then for anya, b >0 and ν∈[0,1]we have

m

"

(b−a)2 8ab −ln

I(a, b)

G(a, b)

# (24)

≤ 1

8[Φ−(b)−Φ+(a)] (b−a)−

Φ (a) + Φ (b)

2 −

1

b−a

Z b

a

Φ (u)du

≤M

"

(b−a)2 8ab −ln

I(a, b)

G(a, b)

(9)

and

m

"

(b−a)2 8ab −ln

A(a, b)

I(a, b)

# (25)

≤ 1

8[Φ−(b)−Φ+(a)] (b−a)−

1

b−a

Z b

a

Φ (x)dx−Φ

a+b

2

≤M

"

(b−a)2 8ab −ln

A(a, b)

I(a, b)

#

.

Proof. The following reverses of the Hermite-Hadamard inequality hold [9] and [10]: Let h: [a, b]→R be a convex function on[a, b].Then

(0≤)h(a) +h(b)

2 −

1

b−a

Z b

a

h(x)dx

(26)

≤ 1

8[h−(b)−h+(a)] (b−a)

and

(0≤) 1

b−a

Z b

a

h(x)dx−h

a+b

2

(27)

≤ 1

8[h−(b)−h+(a)] (b−a).

The constant 18 is best possible in (26) and (27).

SinceΦ∈ B((0,∞), m, M,−ln),thenΦm := Φ +mlnis convex and by (26) we have

Φ (a) + Φ (b)

2 −

1

b−a

Z b

a

Φ (x)dx+mln

G(a, b)

I(a, b)

≤ 1

8[Φ−(b)−Φ+(a)] (b−a)−

m

8ab(b−a) 2,

which proves the first inequality in (24).

Since Φ∈ B((0,∞), m, M,−ln), then also fM :=−Φ−Mln is convex and by (26) we have

(0≤)−Φ (a)−Φ (b)

2 +

1

b−a

Z b

a

Φ (x)dx−Mln

G(a, b)

I(a, b)

≤ 1

8[−Φ−(b) + Φ+(a)] (b−a) +

M

8ab(b−a) 2,

which proves the second inequality in (24).

Further on, since Φ ∈ B((0,∞), m, M,−ln), then Φm := Φ +mln is convex and by (27) we have

(0≤) 1

b−a

Z b

a

Φ (x)dx−Φ

a+b

2

−mln

A(a, b)

I(a, b)

(10)

≤ 1

8[Φ−(b)−Φ+(a)] (b−a)−

m

8ab(b−a) 2,

which is equivalent to the first inequality in (25).

Since Φ∈ B((0,∞), m, M,−ln), then also fM :=−Φ−Mln is convex and by (27) we have

(0≤)− 1

b−a

Z b

a

Φ (x)dx+ Φ

a+b

2

+Mln

A(a, b)

I(a, b)

≤ 1

8[−Φ−(b) + Φ+(a)] (b−a) +

M

8ab(b−a) 2

,

which is equivalent to the second inequality in (25).

Remark 2.2. If we write the inequality (26) for the convex function −ln

we have

(0≤) lnI(a, b)−lnG(a, b)≤ 1 8

1

a −

1

b

(b−a),

which shows that

(b−a)2 8ab −ln

I(a, b)

G(a, b)

≥0.

Also, if we write the inequality (27) for the convex function−lnwe have

(0≤) lnA(a, b)−lnI(a, b)≤ 1 8

1

a−

1

b

(b−a),

which shows that

(b−a)2 8ab −ln

A(a, b)

I(a, b)

≥0.

3. Applications for Special Means

Form, M withM > m >0 we define

(28) Mp :=  

Mp if p >1

mp if p <0

and mp :=  

mp if p >1

Mp if p <0

.

Consider the function Φ (t) = tp, p ∈ (−∞,0)∪(1,∞). This is a convex function and Φ00(t) = p(p−1)tp−2, t > 0. Consider κ(t) := t2Φ00(t) =

p(p−1)tp. We observe that

max

t∈[m,M]κ(t) =p(p

−1)Mp and min

t∈[m,M]κ(t) =p(p

−1)mp.

By making use of the inequalities (14) and (15) for the functionΦ (t) =tp,

p∈(−∞,0)∪(1,∞),then for any a, b∈[m, M]and ν∈[0,1]we have

ln

"

a+νb−2a+b

2 +ν

b−a

2

1−ν

I(a, b)

#p(p−1)mp

(11)

≤Lpp(a, b)−

ν

a+νb−a

2

p

+ (1−ν)

a+b

2 +ν

b−a

2

p

≤ln

"

a+νb−2aν a+2b +νb−2a1−ν I(a, b)

#p(p−1)Mp

and

ln

"

I(a, b)

p

Aν(a, b)G1−ν(a, b)

#p(p−1)mp

(30)

≤ 1

2[((1−ν)a+νb)

p+νap+ (1ν)bp]Lp p(a, b)

≤ln

"

I(a, b)

p

Aν(a, b)G1−ν(a, b)

#p(p−1)Mp

,

wheremp andMp are defined by (28). If we takep= 2 in (30), then we get

ln

"

I(a, b)

p

Aν(a, b)G1−ν(a, b) #2m2 (31)

≤ 1 2

h

((1−ν)a+νb)2+νa2+ (1−ν)b2i−L22(a, b)

≤ln

"

I(a, b)

p

Aν(a, b)G1−ν(a, b) #2M2

.

Since

1 2

h

((1−ν)a+νb)2+νa2+ (1−ν)b2

i

−L22(a, b)

= 1 2

h

(1−ν)2a2+ 2ν(1−ν)ab+ν2b2+νa2+ (1−ν)b2

i

− 1

b−a

Z b

a

t2dt

= 1 2

h

(1−ν)2a2+ 2ν(1−ν)ab+ν2b2+νa2+ (1−ν)b2i

− 1 3 a

2+ab+b2

= 1

6(b−a)

2 3ν23ν+ 1

,

then by (31) we get

ln

"

I(a, b)

p

Aν(a, b)G1−ν(a, b) #2m2

≤ 1

6(b−a)

(12)

≤ln

"

I(a, b)

p

Aν(a, b)G1−ν(a, b) #2M2

,

which is equivalent to

exp " 1 12 − 1

4ν(1−ν)

(b−a)2

M2

#

p I(a, b)

Aν(a, b)G1−ν(a, b) (32) ≤exp " 1 12 − 1

4ν(1−ν)

(b−a)2

m2

#

,

for any a, b∈[m, M]and ν∈[0,1].

If we take in (32) ν= 0,then we get

(33) exp

"

1 12

(b−a)2

M2

#

≤ I(a, b)

G(a, b) ≤exp

"

1 12

(b−a)2

m2

#

,

for any a, b∈[m, M].

If we take in (32) ν= 12,then we get

(34) exp

"

1 48

(b−a)2

M2

#

p I(a, b)

A(a, b)G(a, b) ≤exp

"

1 48

(b−a)2

m2

#

,

for any a, b∈[m, M].

If a, b > 0 then my taking M = max{a, b} and m = min{a, b} in (33) and (34) and since

(b−a)2 max2{a, b} =

min{a, b} max{a, b}−1

2

and (b−a)

2

min2{a, b} =

max{a, b} min{a, b} −1

2

,

for any a, b >0,then we have

exp

"

1 12

min{a, b} max{a, b} −1

2#

≤ I(a, b)

G(a, b)

(35) ≤exp " 1 12

max{a, b} min{a, b} −1

2# and exp " 1 48

min{a, b} max{a, b}−1

2#

p I(a, b)

A(a, b)G(a, b)

(36) ≤exp " 1 48

max{a, b} min{a, b} −1

2#

,

(13)

By making use of the inequalities (24) and (25) for the functionΦ (t) =tp,

p∈(−∞,0)∪(1,∞),then for any a, b∈[m, M]and ν∈[0,1]we have

p(p−1)mp "

(b−a)2 8ab −ln

I(a, b)

G(a, b)

# (37)

≤ 1 8p b

p−1ap−1

(b−a)−

ap+bp

2 −L

p p(a, b)

≤p(p−1)Mp "

(b−a)2 8ab −ln

I(a, b)

G(a, b)

#

and

p(p−1)mp "

(b−a)2 8ab −ln

A(a, b)

I(a, b)

# (38)

≤ 1 8p b

p−1ap−1

(b−a)−

Lpp(a, b)−Ap(a, b)

≤p(p−1)Mp "

(b−a)2 8ab −ln

A(a, b)

I(a, b)

#

,

wheremp andMp are defined by (28).

The case p = 2 provides some simpler inequalities, however the details are left to the interested reader.

References

[1] A. Akkurt, H. Yldirim, On Hermite-Hadamard-Fejér inequality type for convex functions via fractional integrals, Mathematica Moravica, 21 (1) (2017), 105–123. [2] M. Alomari, M. Darus, The Hadamard’s inequality for s-convex function, Int. J.

Math. Anal. (Ruse), 2 (13-16) (2008), 639–646.

[3] M. Alomari, M. Darus,Hadamard-type inequalities for s-convex functions, Int. Math. Forum, 3 (37-40) (2008), 1965–1975.

[4] G. A. Anastassiou,Univariate Ostrowski inequalities, revisited, Monatsh. Math., 135 (3) (2002), 175–189.

[5] N. S. Barnett, P. Cerone and S. S. Dragomir, Some new inequalities for Hermite-Hadamard divergence in information theory, Preprint RGMIA Res. Rep. Coll. 5

(2002), No. 4, Art. 8, Published in Y. J. Cho, J. K. Kim and Y. K. Choi (Eds.),

Stochastic Analysis and Applications, Voulme 3, Nova Science Publishers, 2003, pp. 7-19.

[6] S. S. Dragomir,On a reverse of Jessen’s inequality for isotonic linear functionals, J. Ineq. Pure & Appl. Math., 2 (3) (2001), Article ID: 36.

[7] S. S. Dragomir,On the Jessen’s inequality for isotonic linear functionals, Nonlinear Anal. Forum, 7 (2) (2002), 139–151.

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[9] S. S. Dragomir,An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure and Appl. Math., 3 (2) (2002), Article ID: 31.

[10] S. S. Dragomir,An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure and Appl. Math., 3 (3) (2002), Article ID: 35.

[11] S. S. Dragomir,A survey on Jessen’s type inequalities for positive functionals, in P. M. Pardalos et al. (eds.),Nonlinear Analysis, Springer Optimization and Its Applications 68, In Honor of Themistocles M. Rassias on the Occasion of his 60th Birthday, DOI 10.1007/978-1-4614-3498-6_12, cSpringer Science+Business Media, LLC 2012. [12] S. S. Dragomir, N. M. Ionescu,On some inequalities for convex-dominated functions,

L’Anal. Num. Théor. L’Approx., 19 (1) (1990), 21–27.

[13] S. S. Dragomir, C. E. M. Pearce,Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, 2000.

[14] A. El Farissi,Simple proof and refinement of Hermite-Hadamard inequality, J. Math. Inequal., 4 (3) (2010), 365–369.

[15] E. K. Godunova, V. I. Levin, Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions, Numerical mathematics and mathematical physics, Moskov. Gos. Ped. Inst., Moscow, 166 (1985), 138–142 (in Russian).

[16] H. Hudzik, L. Maligranda,Some remarks on s-convex functions, Aequationes Math., 48 (1) (1994), 100–111.

[17] E. Kikianty, S. S. Dragomir,Hermite-Hadamard’s inequality and the p-HH-norm on the Cartesian product of two copies of a normed space, Math. Inequal. Appl., 13 (1) (2010), 1–32.

[18] U. S. Kirmaci, M. Klaričić Bakula, M. E. Özdemir, J. Pečarić,Hadamard-type ine-qualities for s-convex functions, Appl. Math. Comput., 193 (1) (2007), 26–35. [19] M. A. Latif,On some inequalities for h-convex functions, Int. J. Math. Anal. (Ruse),

4 (29-32) (2010), 1473–1482.

[20] D. S. Mitrinović, I. B. Lacković, Hermite and convexity, Aequationes Math., 28 (1985), 229–232.

[21] D. S. Mitrinović, J. E. Pečarić,Note on a class of functions of Godunova and Levin, C. R. Math. Rep. Acad. Sci. Canada, 12 (1) (1990), 33–36.

[22] C. E. M. Pearce, A. M. Rubinov,P-functions, quasi-convex functions, and Hadamard-type inequalities, J. Math. Anal. Appl., 240 (1) (1999), 92–104.

[23] J. E. Pečarić, S. S. Dragomir, On an inequality of Godunova-Levin and some refi-nements of Jensen integral inequality, Itinerant Seminar on Functional Equations, Approximation and Convexity, Univ. "Babeş-Bolyai”, Cluj-Napoca, 89 (6) (1989), 263–268.

[24] J. Pečarić, S. S. Dragomir, A generalization of Hadamard’s inequality for isotonic linear functionals, Radovi Mat. (Sarajevo), 7 (1991), 103–107.

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[26] M. Z. Sarikaya, A. Saglam, H. Yildirim, On some Hadamard-type inequalities for h-convex functions, J. Math. Inequal., 2 (3) (2008), 335–341.

[27] E. Set, M. E. Özdemir, M. Z. Sarıkaya, New inequalities of Ostrowski’s type for s-convex functions in the second sense with applications, Facta Univ. Ser. Math. Inform., 27 (1) (2012), 67–82.

[28] M. Z. Sarikaya, E. Set, M. E. Özdemir,On some new inequalities of Hadamard type involving h-convex functions, Acta Math. Univ. Comenian. (N.S.), 79 (2) (2010), 265–272.

[29] M. Tunç,Ostrowski-type inequalities via h-convex functions with applications to spe-cial means, J. Inequal. Appl., 2013, 2013:326.

[30] S. Varošanec,On h-convexity, J. Math. Anal. Appl., 326 (1) (2007), 303–311.

Silvestru Sever Dragomir

Mathematics

College of Engineering & Science Victoria University

PO Box 14428 Melbourne City, MC 8001 Australia

Secondary address:

School of Computer Science & Applied Mathematics University of the Witwatersrand

Private Bag 3, Johannesburg 2050 South Africa

E-mail address: sever.dragomir@vu.edu.au

Ian Gomm

Mathematics

College of Engineering & Science Victoria University

PO Box 14428 Melbourne City, MC 8001 Australia

References

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