• No results found

Bounding the Čebyšev function for a differentiable function whose derivative is h or λ-convex in absolute value and applications

N/A
N/A
Protected

Academic year: 2020

Share "Bounding the Čebyšev function for a differentiable function whose derivative is h or λ-convex in absolute value and applications"

Copied!
30
0
0

Loading.... (view fulltext now)

Full text

(1)

doi: 10.4418/2016.71.1.13

BOUNDING THE ˇCEBY ˇSEV FUNCTIONAL FOR A DIFFERENTIABLE FUNCTION WHOSE DERIVATIVE IShOR

λ-CONVEX IN ABSOLUTE VALUE AND APPLICATIONS SILVESTRU SEVER DRAGOMIR

Some bounds for the ˇCebyˇsev functional of a differentiable function whose derivative ishorλ-convex in absolute value and applications for

functions of selfadjoint operators in Hilbert spaces via the spectral repre-sentation theorem are given.

1. Introduction

For two Lebesgue integrable functions f,g:[a,b]→C, in order to compare the integral mean of the product with the product of the integral means, we consider theCebyˇsev functionalˇ defined by

C(f,g):= 1 b−a

Z b a

f(t)g(t)dt− 1

b−a

Z b a

f(t)dt· 1

b−a

Z b a

g(t)dt.

In 1934, G. Gr¨uss [55] showed that

|C(f,g)| ≤1

4(M−m) (N−n), (1)

Entrato in redazione: 12 maggio 2015

AMS 2010 Subject Classification:26D15, 47A63, 47A99.

(2)

providedm,M,n,Nare real numbers with the property that

−∞<m≤f ≤M<∞, −∞<n≤g≤N<∞ a.e. on [a,b]. (2)

The constant 14 is best possible in (1) in the sense that it cannot be replaced by a smaller one.

Another lesser known inequality forC(f,g)was derived in [14] under the assumption that f0,g0exist and are continuous on[a,b],and is given by

|C(f,g)| ≤ 1 12

f0

g0

∞(b−a)

2, (3)

wherekf0k:=supt∈[a,b]|f0(t)|<∞.

The constant 121 cannot be improved in general in (3). ˇ

Cebyˇsev’s inequality (3) also holds if f,g:[a,b]→R are assumed to be absolutely continuous and f0,g0∈L∞[a,b].

In 1970, A.M. Ostrowski [69] proved, amongst others, the following result that is in a sense a combination of the ˇCebyˇsev and Gr¨uss results:

|C(f,g)| ≤1

8(b−a) (M−m)

g0

∞, (4)

provided fis Lebesgue integrable on[a,b]and satisfying (2) whileg:[a,b]→R is absolutely continuous andg0∈L[a,b].Here the constant18 is also sharp.

In 1973, A. Lupas¸ [61] (see also [66, p. 210]) obtained the following result as well:

|C(f,g)| ≤ 1

π2

f0

2

g0

2(b−a), (5)

provided f,gare absolutely continuous and f0,g0∈L2[a,b]. Here the constant 1

π2 is the best possible as well.

In [11], P. Cerone and S.S. Dragomir proved the following inequalities:

|C(f,g)| (6)

      

      

inf

γ∈R

kg−γk·b−a1 Rab

f(t)−

1

b−a Rb

a f(s)ds

dt

inf

γ∈R

kg−γkq·b−a1

Rb a

f(t)−

1

b−a Rb

a f(s)ds

p

dt 1p

wherep>1,1/p+1/q=1.

Forγ=0,we get from the first inequality in (6)

|C(f,g)| ≤ kgk· 1

b−a

Z b a

f(t)− 1

b−a

Z b a

f(s)ds

(3)

for which the constant 1 cannot be replaced by a smaller constant. Ifm≤g≤Mfor a.e. x∈[a,b],theng−m+M2

∞≤

1

2(M−m)and by the

first inequality in (6) we can deduce the following result obtained by Cheng and Sun [15]

|C(f,g)| ≤ 1

2(M−m)· 1

b−a

Z b a

f(t)− 1

b−a

Z b a

f(s)ds

dt. (8)

The constant 12 is best in (8) as shown by Cerone and Dragomir in [12]. The following result holds [33].

Theorem 1.1. Let f :[a,b]→Cbe of bounded variation on[a,b]and g:[a,b]→

Ca Lebesgue integrable function on[a,b].Then

|C(f,g)| ≤1 2

b _

a

(f)· 1

b−a

Z b a

g(t)− 1

b−a

Z b a

g(s)ds

dt (9)

where

b _

a

(f)denotes the total variation of f on the interval[a,b].

The constant 12 is best possible in (9).

We denote the variance of the function f :[a,b]→CbyD(f)and defined as

D(f) =

C f,f¯1/2= "

1

b−a

Z b a

|f(t)|2dt−

1

b−a

Z b a

f(t)dt

2#1/2

, (10)

where ¯f denotes the complex conjugate function of f. We have [33]:

Corollary 1.2. If the function f :[a,b]→Cis of bounded variation on [a,b],

then

D(f)≤1 2

b _

a

(f). (11)

The constant 12 is best possible in (11).

Now we can state the following result when both functions are of bounded variation [33]:

Corollary 1.3. If f,g:[a,b]→Care of bounded variation on[a,b],then

|C(f,g)| ≤ 1 4

b _

a

(f)

b _

a

(g). (12)

(4)

Remark 1.4. We can consider the following quantity associated with a complex valued function f :[a,b]→C,

E(f):=|C(f,f)|1/2= 1

b−a

Z b a

f2(t)dt−

1

b−a

Z b a

f(t)dt 2

1/2

.

Utilising the above results we can state that

E2(f)≤1 2

b _

a

(f)· 1

b−a

Z b a

f(t)− 1

b−a

Z b a

f(s)ds dt (13) ≤1 2 b _ a

(f)D(f)≤1 4

"b

_ a

(f) #2

.

If we consider

G(f):=|C(f,|f|)|1/2

= 1

b−a

Z b a

f(t)|f(t)|dt− 1

b−a

Z b a

f(t)dt· 1

b−a

Z b a

|f(t)|dt

1/2

,

then we also have

G2(f)≤1 2

b _

a

(f)· 1

b−a

Z b a

|f(t)| − 1

b−a

Z b a

|f(s)|ds dt (14) ≤1 2 b _ a

(f)D(|f|)≤ 1 4

b _

a

(f)

b _

a

(|f|)≤ 1 4

"b

_ a

(f) #2

and

G2(f)≤1 2

b _

a

(|f|)· 1

b−a

Z b a

f(t)− 1

b−a

Z b a

f(s)ds dt (15) ≤1 2 b _ a

(|f|)D(f)≤1 4

b _

a

(f)

b _

a

(|f|)≤1 4

"b

_ a

(f) #2

.

Motivated by the results presented above, we establish in this paper some new bounds for the magnitude ofC(f,g) in the case when one of the com-plex valued function, say f, is differentiable and the derivative ish-convex or

λ-convex in absolute value while the other is Lebesgue integrable on[a,b].

Ap-plications for functions of selfadjoint operators in Hilbert spaces via the spectral representation theorem are also given.

(5)

2. h-Convex andλ-Convex Functions

2.1. h-Convex Functions

We recall here some concepts of convexity that are well known in the literature. LetI be an interval inR.

Definition 2.1([54]). We say that f :I →Ris a Godunova-Levin function or that f belongs to the class Q(I) if f is non-negative and for all x,y∈I and

t∈(0,1)we have

f(tx+ (1−t)y)≤1

t f(x) +

1

1−tf(y). (16)

Some further properties of this class of functions can be found in [43], [44], [46], [63], [73] and [74]. Among others, its has been noted that non-negative monotone and non-negative convex functions belong to this class of functions.

Definition 2.2([46]). We say that a function f:I→Rbelongs to the classP(I) if it is nonnegative and for allx,y∈Iandt∈[0,1]we have

f(tx+ (1−t)y)≤ f(x) +f(y). (17)

Obviously Q(I) containsP(I) and for applications it is important to note that alsoP(I)contain all nonnegative monotone, convex andquasi convex func-tions,i. e. nonnegative functions satisfying

f(tx+ (1−t)y)≤max{f(x),f(y)} (18) for allx,y∈Iandt∈[0,1].

For some results onP-functions see [46] and [71] while for quasi convex functions, the reader can consult [45].

Definition 2.3([7]). Letsbe a real number,s∈(0,1].A function f :[0,∞)→ [0,∞)is said to bes-convex (in the second sense) or Breckners-convex if

f(tx+ (1−t)y)≤tsf(x) + (1−t)sf(y)

for allx,y∈[0,∞)andt∈[0,1].

For some properties of this class of functions see [1], [2], [7], [8], [41], [42], [57], [59] and [76].

In order to unify the above concepts for functions of real variable, Varoˇsanec introduced the concept ofh-convex functions as follows.

(6)

Definition 2.4([80]). Leth:J→[0,∞)withhnot identical to 0. We say that

f:I→[0,∞)is anh-convex function if for allx,y∈I we have

f(tx+ (1−t)y)≤h(t)f(x) +h(1−t)f(y) (19)

for allt∈(0,1).

For some results concerning this class of functions see [80], [6], [61], [77], [75] and [79].

This concept can be extended for functions defined on convex subsets of lin-ear spaces in the same way as above replacing the intervalIbe the corresponding convex subsetCof the linear spaceX.

We can introduce now another class of functions.

Definition 2.5. We say that the function f :C⊆X →[0,∞)is ofs -Godunova-Levin type, withs∈[0,1],if

f(tx+ (1−t)y)≤ 1

tsf(x) + 1

(1−t)sf(y), (20)

for allt∈(0,1)andx,y∈C.

We observe that fors=0 we obtain the class ofP-functions while fors=1 we obtain the class of Godunova-Levin. If we denote by Qs(C) the class of

s-Godunova-Levin functions defined onC, then we obviously have

P(C) =Q0(C)⊆Qs1(C)⊆Qs2(C)⊆Q1(C) =Q(C)

for 0≤s1≤s2≤1.

For different inequalities related to these classes of functions, see [1]-[4], [6], [9]-[52], [58]-[60] and [71]-[79].

A functionh:J→Ris said to besupermultiplicativeif

h(ts)≥h(t)h(s) for anyt,s∈J. (21)

If the inequality (21) is reversed, thenhis said to be submultiplicative. If the equality holds in (21) thenhis said to be a multiplicative function onJ.

In [80] it has been noted that ifh:[0,∞)→[0,∞)withh(t) = (x+c)p−1,

then forc=0 the functionhis multiplicative. Ifc≥1,then for p∈(0,1)the functionhis supermultiplicative and forp>1 the function is submultiplicative. We observe that, ifh,gare nonnegative and supermultiplicative, the same is their product. In particular, ifhis supermultiplicative then its product with a power function`r(t) =tr is also supermultiplicative.

(7)

Theorem 2.6. Assume that the function f:C⊆X→[0,∞)is an h-convex func-tion with h∈L[0,1]. Let y,x ∈C with y6=x and assume that the mapping [0,1]3t7→ f[(1−t)x+ty]is Lebesgue integrable on[0,1].Then

1 2h 12f

x+y

2

≤ Z 1

0

f[(1−t)x+ty]dt≤[f(x) + f(y)]

Z 1

0

h(t)dt. (22)

Remark 2.7. If f :I →[0,∞) is anh-convex function on an intervalI of real numbers withh∈L[0,1]and f∈L[a,b]witha,b∈I,a<b,then from (22) we get the Hermite-Hadamard type inequality obtained by Sarikaya et al. in [75]

1 2h 12f

a+b

2

≤ Z b

a

f(u)du≤[f(a) +f(b)]

Z 1

0

h(t)dt. (23)

If we write (22) forh(t) =t,then we get the classical Hermite-Hadamard inequality for convex functions.

If we write (22) for the case ofP-type functions f :C→[0,∞), i.e.,h(t) =

1,t∈[0,1],then we get the inequality

1 2f

x+y

2

≤ Z 1

0

f[(1−t)x+ty]dt≤ f(x) + f(y), (24)

that has been obtained for functions of real variable in [46].

If f is Breckner s-convex onC,fors∈(0,1),then by takingh(t) =ts in (22) we get

2s−1f

x+y

2

≤ Z 1

0

f[(1−t)x+ty]dt≤ f(x) +f(y)

s+1 , (25)

that was obtained for functions of a real variable in [41].

If f:C→[0,∞)is ofs-Godunova-Levin type, withs∈[0,1), then

1 2s+1f

x+y

2

≤ Z 1

0

f[(1−t)x+ty]dt≤ f(x) +f(y)

1−s . (26)

We notice that fors=1 the first inequality in (26) still holds, i.e.

1 4f

x+y

2

≤ Z 1

0

f[(1−t)x+ty]dt. (27)

(8)

2.2. λ-Convex Functions

We start with the following definition (see [37]):

Definition 2.8. Let λ :[0,∞)→ [0,∞) be a function with the property that

λ(t)>0 for allt>0.A mapping f :C→R defined on convex subsetC of a linear spaceX is calledλ-convex onCif

f

αx+βy α+β

≤ λ(α)f(x) +λ(β)f(y)

λ(α+β) (28)

for allα,β≥0 withα+β >0 andx,y∈C.

We observe that if f :C→R isλ-convex onC,then f ish-convex on C

withh(t) =λλ((t1)),t∈[0,1].

If f :C→[0,∞)ish-convex function withhsupermultiplicative on[0,∞),

then f isλ-convex withλ=h.

Indeed, ifα,β ≥0 withα+β>0 andx,y∈Cthen

f

αx+βy α+β

≤h

α

α+β

f(x) +h

β

α+β

f(y)

≤h(α)f(x) +h(β)f(y)

h(α+β) .

The following proposition contain some properties of λ-convex functions

[37].

Proposition 2.9. Let f :C→Rbe aλ-convex function on C.

(i) Ifλ(0)>0,then we have f(x)≥0for all x∈C;

(ii) If there exists x0∈C so that f(x0)>0,then

λ(α+β)≤λ(α) +λ(β)

for allα,β>0,i.e. the mappingλ is subadditive on(0,∞). (iii) If there exists x0,y0∈C with f(x0)>0and f(y0)<0,then

λ(α+β) =λ(α) +λ(β)

for allα,β>0,i.e. the mappingλ is additive on(0,∞).

(9)

Theorem 2.10([37]). Let h(z) =∑∞n=0anzn a power series with nonnegative

coefficients an≥0for all n∈Nand convergent on the open disk D(0,R) with

R>0or R=∞.If r∈(0,R)then the functionλr:[0,∞)→[0,∞)given by

λr(t):=ln

h(r) h(rexp(−t))

(29)

is nonnegative, increasing and subadditive on[0,∞).

Remark 2.11. Now, if we takeh(z) =1−z1 ,z∈D(0,1),then

λr(t) =ln

1−rexp(−t)

1−r

(30)

is nonnegative, increasing and subadditive on[0,∞)for anyr∈(0,1). If we takeh(z) =exp(z),z∈Cthen

λr(t) =r[1−exp(−t)] (31)

is nonnegative, increasing and subadditive on[0,∞)for anyr>0.

Corollary 2.12([37]). Let h(z) =∑∞n=0anzn a power series with nonnegative

coefficients an≥0for all n∈Nand convergent on the open disk D(0,R) with

R>0or R=∞and r∈(0,R).For a mapping f :C→R defined on convex

subset C of a linear space X,the following statements are equivalent: (i) The function f isλr-convex withλr:[0,∞)→[0,∞),

λr(t):=ln

h(r) h(rexp(−t))

;

(ii) We have the inequality

h(r) h(rexp(−α−β))

f

αx+βy

α+β

h(r) h(rexp(−α))

f(x)

h(r) h(rexp(−β))

f(y)

(32)

for anyα,β ≥0withα+β >0and x,y∈C.

(iii) We have the inequality

[h(rexp(−α))]f(x)[h(rexp(−β))]f(y)

[h(rexp(−α−β))]f

αx+βy

α+β

≤[h(r)]

f(x)+f(y)−fαx+βy

α+β

(33)

(10)

Remark 2.13. We observe that, in the case when

λr(t) =r[1−exp(−t)],t≥0

then the function f isλr-convex on convex subsetCof a linear spaceX iff

f

αx+βy

α+β

≤[1−exp(−α)]f(x) + [1−exp(−β)]f(y)

1−exp(−α−β) (34)

for anyα,β≥0 withα+β >0 andx,y∈C.

We observe that this definition is independent ofr>0. The inequality (34) is equivalent with

f

αx+βy α+β

≤exp(β) [exp(α)−1]f(x) +exp(α) [exp(β)−1]f(y)

exp(α+β)−1 (35)

for anyα,β≥0 withα+β >0 andx,y∈C.

We also can introduce the following mappingkx,y:[0,1]→R

kx,y(t):= 1

2[f(tx+ (1−t)y) +f((1−t)x+ty)] forx,y∈C,x6=y.

The following result holds [37]:

Theorem 2.14. Let f :C→[0,∞)be a λ-convex function on C.Assume that x,y∈C with x6=y.

(i) We have the equality

kx,y(1−t) =kx,y(t) for all t∈[0,1];

(ii) The mapping kx,y isλ-convex on[0,1];

(iii) One has the inequalities

kx,y(t)≤

λ(t) +λ(1−t)

λ(1) ·

f(x) +f(y)

2 (36)

and

λ(2α)

2λ(α)f

x+y

2

≤kx,y(t) (37)

for all t∈[0,1]andα>0.

(iv) Let y,x∈C with y6=x and assume that the mappings [0,1]3t7→

f[(1−t)x+ty]andλ are Lebesgue integrable on[0,1],then we have the

Hermite-Hadamard type inequalities

λ(2α)

2λ(α)f

x+y

2

≤ Z 1

0

f((1−t)x+ty)dt≤ f(x) + f(y)

λ(1)

Z 1

0

λ(t)dt (38)

(11)

Corollary 2.15. If f :I→[0,∞)is anλ-convex function on an interval I of real numbers withλ∈L[0,1]and f∈L[a,b]with a,b∈I,a<b,then

λ(2α)

2λ(α)f

a+b

2

≤ 1

b−a

Z b a

f(u)du≤ f(a) +f(b)

λ(1)

Z 1

0

λ(t)dt (39)

for anyα>0.

3. New Results for ˇCebyˇsev Functional

We have the following result:

Theorem 3.1. Let f:I→Cbe a differentiable function onI˚,the interior of the interval I,[a,b]⊂I and g˚ :[a,b]→Cis an integrable function on[a,b].

(i) If|f0|isλ-convex integrable on[a,b]andλ is integrable on[0,1],then

|C(f,g)| ≤ |f

0(a)|+|f0(b)| 2λ(1)

Z 1

0

λ(t)dt

Z b a

g(t)− 1

b−a

Z b a

g(s)ds

dt. (40)

(ii) If|f0|is h-convex integrable on[a,b]and h is integrable on(0,1),then

|C(f,g)| ≤|f

0(a)|+|f0(b)| 2

Z 1

0

h(t)dt

Z b a

g(t)− 1

b−a

Z b a

g(s)ds

dt. (41)

Proof. We use Sonin’s identity

C(f,g) = 1 b−a

Z b a

(f(t)−µ)

g(t)− 1

b−a

Z b a

g(s)ds

dt, (42)

forµ= f(a)+f2 (b) to get

C(f,g) = 1 b−a

Z b a

f(t)− f(a) +f(b)

2 g(t)−

1

b−a

Z b a

g(s)ds

dt. (43)

Since f is differentiable, then we have

f(t)− f(a) +f(b)

2 =

1

2[f(t)−f(a) +f(t)−f(b)]

=1

2

Z t a

f0(s)ds− Z b

t

f0(s)ds

=1

2 Z b

a

sgn(t−s)f0(s)ds

(12)

Therefore we have the representation:

C(f,g) = 1

2(b−a)

Z b a

Z b a

sgn(t−s)f0(s)ds

(44)

×

g(t)− 1

b−a

Z b a

g(s)ds

dt.

Taking the modulus we have

|C(f,g)| ≤ 1 2(b−a)

Z b a Z b a

sgn(t−s)f0(s)ds (45) ×

g(t)− 1

b−a

Z b a

g(s)ds dt ≤ 1

2(b−a)

Z b a

Z b a

sgn(t−s)f0(s)ds ×

g(t)− 1

b−a

Z b a

g(s)ds dt = 1

2(b−a)

Z b a

Z b a

f0(s)

ds

g(t)− 1

b−a

Z b a

g(s)ds dt = 1

2(b−a)

Z b a

f0(s)

ds Z b a

g(t)− 1

b−a

Z b a

g(s)ds

dt.

(i) Since|f0|isλ-convex integrable on[a,b],then by Corollary 2.15 we have

1

b−a

Z b a

f0(s)

ds≤

|f0(a)|+|f0(b)| 2λ(1)

Z 1

0

λ(t)dt

and by (45) we get (40).

(ii) Follows by (23) and the details are omitted.

With the notations from the introduction we have:

Corollary 3.2. Let f :I→Cbe a differentiable function onI and˚ [a,b]⊂I˚.If

|f0|isλ-convex integrable on[a,b]andλ is integrable on[0,1],then

D2(f),E2(f)≤|f

0(a)|+|f0(b)| 2λ(1)

Z 1

0

λ(t)dt (46)

× Z b a

f(t)− 1

b−a

Z b a

f(s)ds dt, and

G2(f)≤|f

0(a)|+|f0(b)| 2λ(1)

Z 1

0

λ(t)dt

Z b a

|f(t)| − 1

b−a

Z b a

|f(s)|ds

dt. (47)

(13)

Remark 3.3. Let f :I →C be a differentiable function on ˚I, [a,b]⊂I˚and

g:[a,b]→Cis an integrable function on[a,b]. If|f0|is convex on[a,b],then

|C(f,g)| ≤|f

0(a)|+|f0(b)| 4

Z b a

g(t)− 1

b−a

Z b a

g(s)ds

dt. (48)

If|f0|is ofP-type on[a,b],then

|C(f,g)| ≤|f

0(a)|+|f0(b)| 2

Z b a

g(t)− 1

b−a

Z b a

g(s)ds

dt. (49)

If|f0|is Breckners-convex on[a,b],fors∈(0,1),then

|C(f,g)| ≤|f

0(a)|+|f0(b)| 2(s+1)

Z b a

g(t)− 1

b−a

Z b a

g(s)ds

dt. (50)

If|f0|is ofs-Godunova-Levin type, withs∈[0,1), then

|C(f,g)| ≤|f

0(a)|+|f0(b)| 2(1−s)

Z b a

g(t)− 1

b−a

Z b a

g(s)ds

dt. (51)

Remark 3.4. We notice, from the proof of Theorem 3.1, if|f0| satisfies the second Hemite-Hadamard inequality with a certain term R(|f0(a)|,|f0(b)|), i.e.

1

b−a

Z b a

f0(u)

du≤R

f0(a)

,

f0(b)

,

then we have the inequality

|C(f,g)| ≤R f0(a) ,

f0(b)

Z b a

g(t)− 1

b−a

Z b a

g(s)ds

dt. (52)

The case of p-normof the deviation

f− 1

b−a

Z b a

f(s)ds

is as follows:

Theorem 3.5. Let f :I →C be a differentiable function on I˚,[a,b]⊂I and˚

g:[a,b]→Cis an integrable function on[a,b].Assume that p,q>1with 1p+ 1

(14)

(i) If|f0|qisλ-convex integrable on[a,b]andλ is integrable on[0,1],then

|C(f,g)| ≤(b−a)

|f0(a)|q+|f0(b)|q 2λ(1)

Z 1

0

λ(t)dt

1/q

(53)

×

1

b−a

Z b a

g(t)− 1

b−a

Z b a

g(s)ds p dt 1/p

.

(ii) If|f0|qis h-convex integrable on[a,b]and h is integrable on(0,1),then

|C(f,g)| ≤(b−a) |

f0(a)|q+|f0(b)|q 2

Z 1

0

h(t)dt 1/q

(54)

×

1

b−a

Z b a

g(t)− 1

b−a

Z b a

g(s)ds p dt 1/p

.

Proof. Making use of H¨older’s integral inequality, we have

|C(f,g)| ≤ 1 2(b−a)

Z b a Z b a

sgn(t−s)f0(s)ds (55) ×

g(t)− 1

b−a

Z b a

g(s)ds dt ≤ 1

2(b−a) Z b a Z b a

sgn(t−s)f0(s)ds q dt 1/q

× Z b a

g(t)− 1

b−a

Z b a

g(s)ds p dt 1/p

.

Observe that, by Jensen’s integral inequality forq>1,we have

Rb

asgn(t−s)f0(s)ds

b−a

q ≤ Rb

a|sgn(t−s)f0(s)| q

ds b−a

=

Rb a|f

0(s)|q

ds

b−a ,

which shows that

Z b a

sgn(t−s)f0(s)ds

q

≤(b−a)q−1

Z b a

f0(s)

q

ds

for anyt∈[a,b]. Therefore, Z b a Z b a

sgn(t−s)f0(s)ds q dt 1/q

(b−a)q

Z b a

f0(s)

q

ds 1/q

= (b−a) Z b

a

f0(s)

q

(15)

and by (55) we get

|C(f,g)| ≤1 2

Z b a

f0(s)

q

ds

1/qZ b

a

g(t)− 1

b−a

Z b a

g(s)ds

p

dt 1/p

.

(i) Since |f0|q isλ-convex integrable on[a,b],then by Corollary 2.15 we

have

1

b−a

Z b a

f0(s)

q

ds≤ |f 0(a)|q

+|f0(b)|q 2λ(1)

Z 1

0

λ(t)dt

and by (55) we get (53).

(ii) Follows by (23) and the details are omitted.

The casep=q=2 is of interest.

Corollary 3.6. Let f :I→Cbe a differentiable function on I˚,[a,b]⊂I and˚

g:[a,b]→Cis an integrable function on[a,b].

If|f0|2isλ-convex integrable on[a,b]andλ is integrable on[0,1],then

|C(f,g)| ≤(b−a) "

|f0(a)|2+|f0(b)|2 2λ(1)

Z 1

0

λ(t)dt #1/2

D(g), (56)

where

D(g) = [C(g,g)]¯ 1/2= "

1

b−a

Z b a

|g(t)|2dt−

1

b−a

Z b a

g(t)dt

2#1/2

.

If |f0|2 is h-convex integrable on [a,b] and h is integrable on (0,1), then a similar inequality is valid.

The following particular cases are of interest as well:

Corollary 3.7. Let f :I→Cbe a differentiable function onI˚,[a,b]⊂I and g˚ :

[a,b]→Cis an integrable function on[a,b].Assume that p,q>1and1p+1q=1.

If|f0|qisλ-convex integrable on[a,b]andλ is integrable on[0,1],then

D2(f),E2(f)≤(b−a)

|f0(a)|q+|f0(b)|q 2λ(1)

Z 1

0

λ(t)dt

1/q

(57)

×

1

b−a

Z b a

f(t)− 1

b−a

Z b a

f(s)ds

p

dt 1/p

(16)

and

G2(f)≤(b−a)

|f0(a)|q+|f0(b)|q 2λ(1)

Z 1

0

λ(t)dt

1/q

(58)

×

1

b−a

Z b a

|f(t)| − 1

b−a

Z b a

|f(s)|ds

p

dt 1/p

.

In particular, if|f0|2isλ-convex integrable on[a,b]andλ is integrable on

[0,1],then

D2(f),E2(f)≤(b−a) "

|f0(a)|2+|f0(b)|2 2λ(1)

Z 1

0

λ(t)dt

#1/2

D(f), (59)

and

G2(f)≤(b−a) "

|f0(a)|2+|f0(b)|2 2λ(1)

Z 1

0

λ(t)dt

#1/2

D(|f|). (60)

The first inequality in (59) is equivalent to:

D(f)≤(b−a) "

|f0(a)|2+|f0(b)|2 2λ(1)

Z 1

0

λ(t)dt #1/2

. (61)

Remark 3.8. Let f :I→Cbe a differentiable function on ˚I,[a,b]⊂I˚andg:

[a,b]→Cis an integrable function on[a,b].Assume thatp,q>1 and1p+1q=1. If|f0|qis convex on[a,b],then

|C(f,g)| ≤(b−a) |

f0(a)|q+|f0(b)|q 4

1/q

(62)

×

1

b−a

Z b a

g(t)− 1

b−a

Z b a

g(s)ds

p

dt 1/p

.

If|f0|qis ofP-type on[a,b],then

|C(f,g)| ≤(b−a)

|f0(a)|q+|f0(b)|q 2

1/q

(63)

×

1

b−a

Z b a

g(t)− 1

b−a

Z b a

g(s)ds

p

dt 1/p

(17)

If|f0|qis Breckners-convex on[a,b],fors∈(0,1),then

|C(f,g)| ≤(b−a) |

f0(a)|q+|f0(b)|q 2(s+1)

1/q

(64)

×

1

b−a

Z b a

g(t)− 1

b−a

Z b a

g(s)ds

p

dt 1/p

.

If|f0|qis ofs-Godunova-Levin type, withs∈[0,1), then

|C(f,g)| ≤(b−a) |

f0(a)|q+|f0(b)|q 2(1−s)

1/q

(65)

×

1

b−a

Z b a

g(t)− 1

b−a

Z b a

g(s)ds

p

dt 1/p

.

4. Application for Riemann-Stieltjes Integral

The following representation is of interest in itself. The result was firstly ob-tained in [27] (see also [29]). For the sake a completeness we give here a short proof as well.

Lemma 4.1. If v:[a,b]→Cis continuous (of bounded variation) on[a,b]and h:[a,b]→Cis of bounded variation (continuous) on[a,b], then we have the

identity

v(b)Rb

a(t−a)dh(t) +v(a) Rb

a(b−t)dh(t)

b−a −

Z b a

v(t)dh(t) (66)

=

Z b a

h(t)dv(t)−v(b)−v(a)

b−a

Z b a

h(t)dt.

Proof. Integrating by parts in the Riemann-Stieltjes integral we have

v(b)Rb

a(t−a)dh(t) +v(a) Rb

a(b−t)dh(t)

b−a −

Z b a

v(t)dh(t) (67)

=

Z b a

v(b) (t−a) +v(a) (b−t)

b−a −v(t)

dh(t)

=

(t−a)v(b) + (b−t)v(a)

b−a −v(t)

h(t)

b

a

− Z b

a

h(t)d

(t−a)v(b) + (b−t)v(a)

b−a −v(t)

(18)

= [v(b)−v(b)]h(b)−[v(a)−v(a)]h(a)

− Z b

a

h(t)

v(b)−v(a)

b−a dt−dv(t)

=

Z b a

h(t)dv(t)−v(b)−v(a)

b−a

Z b a

h(t)dt

and the identity is proven.

We can provide now the following application for Riemann-Stieltjes inte-gral:

Proposition 4.2. If v:I →Cis twice differentiable on the interior of the in-terval I denotedI and˚ [a,b]⊂I˚.If|v00|isλ-convex integrable on[a,b]andλ

is integrable on[0,1],then for h:[a,b]→Cintegrable on[a,b], we have the

inequalities

v(b)Rb

a(t−a)dh(t) +v(a) Rb

a(b−t)dh(t)

b−a −

Z b a

v(t)dh(t)

(68)

≤ |v

00(a)|+|v00(b)| 2λ(1) (b−a)

Z 1

0

λ(t)dt

Z b a

h(t)− 1

b−a

Z b a

h(s)ds

dt

Proof. From (66) we have

v(b)Rb

a (t−a)dh(t) +v(a) Rb

a(b−t)dh(t)

b−a −

Z b a

v(t)dh(t) (69)

=

Z b a

h(t)v0(t)dt−v(b)−v(a)

b−a

Z b a

h(t)dt= (b−a)C v0,h.

Since |v00| is λ-convex integrable on [a,b], then by applying Theorem 3.1for

f=v0andg=hwe deduce the desired result (68).

Remark 4.3. Ifv:I→Cis twice differentiable on ˚I,[a,b]⊂I˚andg:[a,b]→C is an integrable function on[a,b].

If|v00|is convex on[a,b],then

v(b)Rb

a(t−a)dh(t) +v(a) Rb

a(b−t)dh(t)

b−a −

Z b a

v(t)dh(t)

(70)

≤|v

00(a)|+|v00(b)|

4 (b−a)

Z b a

h(t)− 1

b−a

Z b a

h(s)ds

(19)

If|v00|is ofP-type on[a,b],then

v(b)Rb

a(t−a)dh(t) +v(a) Rb

a(b−t)dh(t)

b−a −

Z b a

v(t)dh(t)

(71)

≤|v

00(a)|+|v00(b)|

2 (b−a)

Z b a

h(t)− 1

b−a

Z b a

h(s)ds

dt.

If|v00|is Breckners-convex on[a,b],fors∈(0,1),then

v(b)Rb

a(t−a)dh(t) +v(a) Rb

a(b−t)dh(t)

b−a −

Z b a

v(t)dh(t)

(72)

≤|v

00(a)|+|v00(b)| 2(s+1) (b−a)

Z b a

h(t)− 1

b−a

Z b a

h(s)ds

dt.

If|v00|is ofs-Godunova-Levin type, withs∈[0,1), then

v(b)Rb

a(t−a)dh(t) +v(a) Rb

a(b−t)dh(t)

b−a −

Z b a

v(t)dh(t)

(73)

≤|v

00(a)|+|v00(b)| 2(1−s) (b−a)

Z b a

h(t)− 1

b−a

Z b a

h(s)ds

dt.

Similar results may be stated if|f0|q isλ-convex integrable on[a,b]andλ

is integrable on[0,1].However the details are not provided here.

5. Applications for Selfadjoint Operators

We denote byB(H) the Banach algebra of all bounded linear operators on a complex Hilbert space(H;h·,·i). LetA∈ B(H) be selfadjoint and let ϕλ be

defined for allλ ∈Ras follows

ϕλ(s):= 

1,for −∞<s≤λ,

0,forλ <s<+∞.

Then for everyλ∈Rthe operator

Eλ :=ϕλ(A) (74)

is a projection which reducesA.

(20)

Theorem 5.1 (Spectral Representation Theorem). Let A be a bounded self-adjoint operator on the Hilbert space H and let m=min{λ|λ∈Sp(A)}=:

minSp(A) and M =max{λ|λ∈Sp(A)} =: maxSp(A).Then there exists a

family of projections{Eλ}λ

R, called the spectral family of A,with the follow-ing properties

a) Eλ≤Eλ0 forλλ0;

b) Em−0=0,EM=I and Eλ+0=Eλ for allλ∈R;

c) We have the representation

A=

Z M m−0

λdEλ. (75)

More generally, for every continuous complex-valued functionϕdefined on

Rand for everyε>0there exists aδ >0such that

ϕ(A)−

n

k=1

ϕ λk0 Eλk−Eλk−1

≤ε (76)

whenever

    

    

λ0<m=λ1< ... <λn−1<λn=M,

λk−λk−1≤δ for1≤k≤n,

λk0∈[λk−1,λk]for1≤k≤n

(77)

this means that

ϕ(A) =

Z M m−0

ϕ(λ)dEλ, (78)

where the integral is of Riemann-Stieltjes type.

Corollary 5.2. With the assumptions of Theorem 5.1 for A,Eλ andϕ we have

the representations

ϕ(A)x=

Z M m−0

ϕ(λ)dEλx for all x∈H (79)

and

hϕ(A)x,yi=

Z M m−0

ϕ(λ)dhEλx,yi for all x,y∈H. (80)

In particular,

hϕ(A)x,xi=

Z M m−0

(21)

Moreover, we have the equality

kϕ(A)xk2=

Z M m−0

|ϕ(λ)|2dkEλxk2 for all x∈H. (82)

The next result shows that it is legitimate to talk about ”the” spectral family of the bounded selfadjoint operator A since it is uniquely determined by the requirements a), b) and c) in Theorem 5.1, see for instance [56, p. 258]:

Theorem 5.3. Let A be a bounded selfadjoint operator on the Hilbert space H and let m=minSp(A)and M=maxSp(A).If{Fλ}λ

Ris a family of projec-tions satisfying the requirements a), b) and c) in Theorem 5.1, then Fλ =Eλ for allλ ∈Rwhere Eλ is defined by (74).

By the above two theorems, the spectral family {Eλ}λ

R uniquely

deter-mines and in turn is uniquely determined by the bounded selfadjoint operator

A.

We have:

Theorem 5.4. Let A be a bounded selfadjoint operator on the Hilbert space H and let m=min{λ|λ ∈Sp(A)}=: minSp(A)and M=max{λ|λ ∈Sp(A)}

=: maxSp(A).Consider also the spectral family{Eλ}λ

Rof A.

If f :I→C is twice differentiable on I and˚ [m,M]⊂I˚,|f00|is λ-convex integrable on[m,M]andλ is integrable on[0,1], then we have the inequalities

f(m) (M1H−A) +f(M) (A−m1H) M−m

x,y

− hf(A)x,yi

(83)

≤|f

00(m)|+|f00(M)|

2λ(1) (M−m)

Z 1

0

λ(t)dt

× Z M

m−0

hEtx,yi − 1

M−m

Z M m−0

hEsx,yids

dt

≤|f

00(m)|+|f00(M)| 4λ(1) (M

−m)2kxk kyk Z 1

0

λ(t)dt

for any x,y∈H.

Proof. Letx,y∈Hand considerh:R→C,h(t):=hEtx,yi.

(22)

ε>0,we have

f(M)RM

m−ε(t−m+ε)dhEtx,yi+f(m−ε)

RM

m−ε(M−t)dhEtx,yi

M−m+ε (84)

− Z M

m−ε

f(t)dhEtx,yi

≤|f 00(m

ε)|+|f00(M)|

2λ(1) (M−m+ε)

2Z 1 0

λ(t)dt

× 1

M−m+ε

Z M m−ε

hEtx,yi − 1

M−m+ε

Z M m−ε

hEsx,yids

dt.

Taking the limit overε→0+and using the Spectral representation theorem, we

have

f(m) (M1H−A) +f(M) (A−m1H) M−m

x,y

− hf(A)x,yi

(85)

≤|f

00(m)|+|f00(M)|

2λ(1) (M−m) 2Z 1

0

λ(t)dt

× 1

M−m

Z M m−0

hEtx,yi − 1

M−m

Z M m−0

hEsx,yids

dt

for anyx,y∈H.

By the Schwarz inequality inHwe have that

Z M m−0

hEtx,yi − 1

M−m

Z M m−0

hEsx,yids

dt (86) = Z M m−0 Etx−

1

M−m

Z M m−0

Esxds

,y dt

≤ kyk Z M m−0

Etx− 1

M−m

Z M m−0

Esxds

dt

for anyx,y∈H.

On utilizing the Cauchy-Buniakovski-Schwarz integral inequality we may state that Z M m−0

Etx− 1

M−m

Z M m−0

Esxds

dt (87)

≤(M−m)1/2

Z M m−0

Etx− 1

M−m

Z M m−0

Esxds

2 dt !1/2

(23)

Observe that the following equalities of interest hold and they can be easily proved by direct calculations

1

M−m

Z M m−0

Etx− 1

M−m

Z M m−0

Esxds

2 dt (88) = 1

M−m

Z M m−0

kEtxk2dt−

1

M−m

Z M m−0

Esxds

2 and 1

M−m

Z M m−0

kEtxk2dt−

1

M−m

Z M m−0

Esxds

2 (89) = 1

M−m

Z M m−0

Etx−

1

M−m

Z M m−0

Esxds,Etx− 1 2x

dt

for anyx∈H.

By (87), (88) and (89) we get

Z M m−0

Etx− 1

M−m

Z M m−0

Esxds

dt (90)

≤(M−m)1/2 Z M

m−0

Etx− 1

M−m

Z M m−0

Esxds,Etx− 1 2x

dt

1/2

for anyx∈H.

On making use of the Schwarz inequality inHwe also have Z M

m−0

Etx− 1

M−m

Z M m−0

Esxds,Etx− 1 2x dt (91) ≤ Z M m−0

Etx− 1

M−m

Z M m−0

Esxds

Etx− 1 2x dt =1

2kxk Z M m−0

Etx− 1

M−m

Z M m−0

Esxds

dt,

where we used the fact thatEt are projectors, and in this case we have

Etx− 1 2x 2

=kEtxk2− hEtx,xi+ 1 4kxk

2= 1

4kxk

2

for anyt∈[m,M]for anyx∈H. From (90) and (91) we get

Z M m−0

Etx− 1

M−m

Z M m−0

Esxds

dt (92)

≤(M−m)1/2

1 2kxk

Z M m−0

Etx− 1

M−m

Z M m−0

Esxds

dt 1/2

(24)

which is clearly equivalent with the following inequality of interest in itself

Z M m−0

Etx− 1

M−m

Z M m−0

Esxds

dt≤1

2kxk(M−m) (93)

for anyx∈H.

From (86) we then get

1

M−m

Z M m−0

hEtx,yi − 1

M−m

Z M m−0

hEsx,yids

dt≤ 1

2kxk kyk

for anyx,y∈H.

Remark 5.5. If|f00|is convex on[m,M], then we have the inequalities

f(m) (M1H−A) +f(M) (A−m1H) M−m

x,y

− hf(A)x,yi

(94)

≤|f

00(m)|+|f00(M)|

4 (M−m)

× Z M

m−0

hEtx,yi − 1

M−m

Z M m−0

hEsx,yids

dt

≤|f

00(m)|+|f00(M)|

8 (M−m)

2k xk kyk,

for anyx,y∈H.

Example 5.6. a) LetAbe a bounded selfadjoint operator on the Hilbert spaceH

and letm=min{λ|λ ∈Sp(A)}andM=max{λ|λ ∈Sp(A)}=: maxSp(A). Consider also the spectral family{Eλ}λ

RofA.Then by Theorem 5.4 we have

for f(t) =tp,p≥3 that

mp(M1H−A) +Mp(A−m1H) M−m

x,y

− hApx,yi

(95)

≤p(p−1)m

p−2+Mp−2

4 (M−m)

× Z M

m−0

hEtx,yi − 1

M−m

Z M m−0

hEsx,yids

dt

≤p(p−1)m

p−2+Mp−2

8 (M−m)

2k xk kyk

(25)

b) With the assumptions of a) and ifm>0,then by Theorem 5.4 we have for f(t) =lnt,that

lnm(M1H−A) +lnM(A−m1H) M−m

x,y

− hlnAx,yi

(96)

≤m

2+M2

4m2M2 (M−m)

Z M m−0

hEtx,yi − 1

M−m

Z M m−0

hEsx,yids

dt

≤m

2+M2

8m2M2 (M−m) 2k

xk kyk

for anyx,y∈H.

Similar results may be stated if|f0|q isλ-convex integrable on[a,b]andλ

is integrable on[0,1].However the details are not provided here.

REFERENCES

[1] M. Alomari - M. Darus,The Hadamard’s inequality for s-convex function, Int. J. Math. Anal. (Ruse) 2 (13-16) (2008), 639–646.

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

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

[4] N. S. Barnett - P. Cerone - S. S. Dragomir - M. R. Pinheiro - A. Sofo,Ostrowski type inequalities for functions whose modulus of the derivatives are convex and applications, Inequality Theory and Applications, Vol. 2 (Chinju/Masan, 2001), 19–32, Nova Sci. Publ., Hauppauge, NY, 2003. Preprint: RGMIA Res. Rep. Coll. 5 (2002), No. 2, Art. 1.

[5] E. F. Beckenbach,Convex functions, Bull. Amer. Math. Soc. 54 (1948), 439–460.

[6] M. Bombardelli - S. Varoˇsanec,Properties of h-convex functions related to the Hermite-Hadamard-Fej´er inequalities. Comp. Math. Appl. 58 (9) (2009), 1869– 1877.

[7] W. W. Breckner, Stetigkeitsaussagen f¨ur eine Klasse verallgemeinerter kon-vexer Funktionen in topologischen linearen R¨aumen, (German) Publ. Inst. Math. (Beograd) (N.S.) 23 (37) (1978), 13–20.

[8] W. W. Breckner - G. Orb´an,Continuity properties of rationally s-convex mappings with values in an ordered topological linear space.Universitatea ”Babes¸-Bolyai”, Facultatea de Matematica, Cluj-Napoca, 1978.

(26)

[10] P. Cerone - S. S. Dragomir, New bounds for the three-point rule involving the Riemann-Stieltjes integrals, in Advances in Statistics Combinatorics and Related Areas, C. Gulati, et al. (Eds.), World Science Publishing, 2002, 53–62.

[11] P. Cerone - S. S. Dragomir,New bounds for the ˇCebyˇsev functional, Appl. Math. Lett. 18 (2005), 603–611.

[12] P. Cerone - S. S. Dragomir,A refinement of the Gr¨uss inequality and applications, Tamkang J. Math. 38 (1) (2007), 37–49. Preprint available at RGMIA Res. Rep. Coll., 5 (2) (2002), Art. 14.

[13] P. Cerone - S. S. Dragomir - J. Roumeliotis,Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Mathematica, 32 (2) (1999), 697—712.

[14] P. L. Chebychev,Sur les expressions approximatives des int`egrals d`efinis par les outres prises entre les mˆeme limites, Proc. Math. Soc. Charkov 2 (1882), 93–98.

[15] X.-L. Cheng - J. Sun,Note on the perturbed trapezoid inequality, J. Inequal. Pure Appl. Math. 3 (2) (2002), Art. 29.

[16] G. Cristescu,Hadamard type inequalities for convolution of h-convex functions, Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 8 (2010), 3–11.

[17] S. S. Dragomir, Ostrowski’s inequality for monotonous mappings and applica-tions, J. KSIAM 3 (1) (1999), 127–135.

[18] S. S. Dragomir,The Ostrowski’s integral inequality for Lipschitzian mappings and applications, Comp. Math. Appl. 38 (1999), 33–37.

[19] S. S. Dragomir,On the Ostrowski’s inequality for Riemann-Stieltjes integral, Ko-rean J. Appl. Math. 7 (2000), 477–485.

[20] S. S. Dragomir,On the Ostrowski’s inequality for mappings of bounded variation and applications, Math. Ineq. & Appl. 4 (1) (2001), 33–40.

[21] S. S. Dragomir,On the Ostrowski inequality for Riemann-Stieltjes integralRb a f(t)

du(t)where f is of H¨older type and u is of bounded variation and applications, J. KSIAM 5 (1) (2001), 35–45.

[22] S. S. Dragomir,Ostrowski type inequalities for isotonic linear functionals, J. In-equal. Pure & Appl. Math. 3 (5) (2002), Art. 68.

[23] 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 Appl. Math. 3 (2) (2002), Art. 31.

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

[25] S. S. Dragomir, A survey on Cauchy-Buniakowski-Schwarz’s type discrete in-equality, J. Ineq. Pure & Appl. Math. 4 (3) (2003), Art. 63.

(27)

[27] S. S. Dragomir,Inequalities of Gr¨uss type for the Stieltjes integral and applica-tions, Kragujevac J. Math. 26 (2004), 89–112.

[28] S. S. Dragomir,Bounds for the normalised Jensen functional, Bull. Austral. Math. Soc. 74 (2006), 471–478.

[29] S. S. Dragomir,Inequalities for Stieltjes integrals with convex integrators and ap-plications, Appl. Math. Lett. 20 (2007), 123–130.

[30] S. S. Dragomir,Cebyˇsev’s type inequalities for functions of selfadjoint operatorsˇ in Hilbert spaces, Preprint RGMIA Res. Rep. Coll. 11(e) (2008), Art. 9.

[31] S. S. Dragomir,Gr¨uss’ type inequalities for functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll. 11(e) (2008), Art. 11.

[32] S. S. Dragomir,Inequalities for the ˇCebyˇsev functional of two functions of selfad-joint operators in Hilbert spaces, Prerpint RGMIA Res. Rep. Coll. 11(e) (2008).

[33] S. S. Dragomir,New Gr¨uss’ type inequalities for iunctions of bounded variation and applications, Appl. Math. Lett. 25 (10) (2012), 1475–1479.

[34] S. S. Dragomir, Operator Inequalities of the Jensen, ˇCebyˇsev and Gr¨uss Type, Springer Briefs in Mathematics, Springer, New York, 2012.

[35] S. S. Dragomir,Operator Inequalities of Ostrowski and Trapezoidal Type, Sprin-ger Briefs in Mathematics. SprinSprin-ger, New York, 2012.

[36] S. S. Dragomir,Inequalities of Hermite-Hadamard type for h-convex functions on linear spaces, Preprint RGMIA Res. Rep. Coll. 16 (2013), Art. 75.

[37] S. S. Dragomir,Inequalities of Hermite-Hadamard type forλ-convex functions on linear spaces,Preprint RGMIA Res. Rep. Coll. 17 (2014), Art. 13.

[38] S. S. Dragomir, Discrete inequalities of Jensen type for λ-convex functions on linear spaces, Preprint RGMIA Res. Rep. Coll. 17 (2014), Art. 15.

[39] S. S. Dragomir, Integral inequalities of Jensen type for λ-convex functions, Preprint RGMIA Res. Rep. Coll. 17 (2014), Art. 18.

[40] S. S. Dragomir - P. Cerone - J. Roumeliotis - S. Wang, A weighted version of Ostrowski inequality for mappings of H¨older type and applications in numerical analysis, Bull. Math. Soc. Sci. Math. Romanie 42 (90) (4) (1999), 301–314.

[41] S. S. Dragomir - S. Fitzpatrick,The Hadamard inequalities for s-convex functions in the second sense, Demonstratio Math. 32 (4) (1999), 687–696.

[42] S. S. Dragomir - S. Fitzpatrick,The Jensen inequality for s-Breckner convex func-tions in linear spaces, Demonstratio Math. 33 (1) (2000), 43–49.

[43] S. S. Dragomir - B. Mond,On Hadamard’s inequality for a class of functions of Godunova and Levin, Indian J. Math. 39 (1) (1997), 1–9.

[44] S. S. Dragomir - C. E. M. Pearce,On Jensen’s inequality for a class of functions of Godunova and Levin, Period. Math. Hungar. 33 (2) (1996), 93–100.

(28)

[46] S. S. Dragomir - J. Peˇcari´c - L. Persson, Some inequalities of Hadamard type, Soochow J. Math. 21 (3) (1995), 335–341.

[47] S. S. Dragomir - J. Peˇcari´c - L. Persson,Properties of some functionals related to Jensen’s inequality, Acta Math. Hungarica 70 (1996), 129–143.

[48] S. S. Dragomir - Th. M. Rassias (Eds),Ostrowski Type Inequalities and Applica-tions in Numerical Integration, Kluwer Academic Publisher, 2002.

[49] S. S. Dragomir - S. Wang,A new inequality of Ostrowski’s type in L1-norm and applications to some special means and to some numerical quadrature rules, Tamkang J. of Math. 28 (1997), 239–244.

[50] S. S. Dragomir - S. Wang,Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and some numerical quadrature rules, Appl. Math. Lett. 11 (1998), 105–109.

[51] S. S. Dragomir - S. Wang,A new inequality of Ostrowski’s type in Lp-norm and

applications to some special means and to some numerical quadrature rules, In-dian J. of Math., 40 (3) (1998), 245–304.

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

[53] T. Furuta - J. Mi´ci´c Hot - J. Peˇcari´c - Y. Seo,Mond-Peˇcari´c Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.

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

[55] G. Gr¨uss,Uber das maximum des absoluten Betrages von¨ b−a1 Rb

a f(x)g(x)dx−

1

(b−a)2

Rb

a f(x)dx·

Rb

ag(x)dx, Math. Z. 39 (1934), 215–226.

[56] G. Helmberg, Introduction to Spectral Theory in Hilbert Space, John Wiley & Sons, Inc. New York, 1969.

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

[58] 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.

[59] U. S. Kirmaci - M. Klariˇci´c Bakula - M. E ¨Ozdemir - J. Peˇcari´c,Hadamard-type inequalities for s-convex functions, Appl. Math. Comput. 193 (1) (2007), 26–35.

[60] M. A. Latif, On some inequalities for h-convex functions, Int. J. Math. Anal. (Ruse) 4 (29-32) (2010), 1473–1482.

(29)

[62] D. S. Mitrinovi´c - I. B. Lackovi´c,Hermite and convexity,Aequationes Math. 28 (1985), 229–232.

[63] D. S. Mitrinovi´c - J. E. Peˇcari´c, Note on a class of functions of Godunova and Levin, C. R. Math. Rep. Acad. Sci. Canada 12 (1) (1990), 33–36.

[64] A. Matkovi´c - J. E. Peˇcari´c - I. Peri´c,A variant of Jensen’s inequality of Mercer’s type for operators with applications, Linear Algebra Appl. 418 (2-3) (2006), 551– 564.

[65] D. S. Mitrinovi´c - J. E. Peˇcari´c - A. M. Fink,Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.

[66] D. S. Mitrinovi´c - J. E. Peˇcari´c - A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.

[67] B. Mond - J. E. Peˇcari´c,Convex inequalities in Hilbert spaces, Houston J. Math., 19 (1993), 405–420.

[68] B. Mond - J. E. Peˇcari´c,Classical inequalities for matrix functions, Utilitas Math. 46 (1994), 155–166.

[69] A. M. Ostrowski, On an integral inequality, Aequationes Math. 4 (1970), 358– 373.

[70] J. E. Peˇcari´c - J. Mi´ci´c - Y. Seo, Inequalities between operator means based on the Mond-Peˇcari´c method, Houston J. Math. 30 (1) (2004), 191–207.

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

[72] J. E. Peˇcari´c - S. S. Dragomir, On an inequality of Godunova-Levin and some refinements of Jensen integral inequality, Itinerant Seminar on Functional Equa-tions, Approximation and Convexity (Cluj-Napoca, 1989), 263–268, Preprint, 89-6, Univ. ”Babes¸-Bolyai”, Cluj-Napoca, 1989.

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

[74] M. Radulescu - S. Radulescu - P. Alexandrescu, On the Godunova-Levin-Schur class of functions, Math. Inequal. Appl. 12 (4) (2009), 853–862.

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

[76] E. Set - M. E. ¨Ozdemir - 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.

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

(30)

[79] M. Tunc¸,Ostrowski-type inequalities via h-convex functions with applications to special means, J. Inequal. Appl. 2013 (2013):326, 10 pp.

[80] S. Varoˇsanec,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.

References

Related documents

Metastable antiprotonic helium (pHe + ) is an exotic atom consisting of a helium nucleus, an electron in the ground 1s state and an antiproton in a Rydberg state of large principal (

Net present value of after-tax costs and benefits of spotted knrp weed eradication calculated for 3 alternative rates of knapweed spread and forage replacement, on

From each of the group of consecutive blocks choose any unused z blocks and merge them to form a merged-block with no inter-block common key.. To deal with remaining blocks that

Die Spaltung des Proteins resultiert nicht in einer Aktivierung des NF- B Signalweges, vielmehr wurde eine Funktion des C-terminalen BCL10 Fragments bei der

The contribution of the capillary endothelium to blood clearance and tissue deposition of anionic quantum dots in vivo:... To assess blood and tissue kinetics, fluorescence

In situ hybridization analyses demonstrated that the Om(1E) gene is expressed ubiquitously in embryonic cells, imaginal discs, and the cortex of the central nervous system

The objective of this investigation is to study the effect of high volume of fly ash in the fresh and hardened properties of concrete and to study the flexural response of HVFAC beams

The downward movement of NO 3 - in agricultural and non-agricultural soils is studied using glass column method, The Effect of Ca hardness of extractant, irrigation water