Norm inequalities of Cebysev type for power series in Banach algebras

Abstract

Let [InlineEquation not available: see fulltext.] be a function defined by power series with complex coefficients and convergent on the open disk [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.], a Banach algebra, with [InlineEquation not available: see fulltext.]. In this paper we establish some upper bounds for the norm of the Cebysev type difference[InlineEquation not available: see fulltext.], provided that the complex number [lambda] and the vectors [InlineEquation not available: see fulltext.] are such that the series in the above expression are convergent. Applications for some fundamental functions such as the exponential function and the resolvent function are provided as well.

MSC: 47A63, 47A99.

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R E S E A R C H Open Access

Norm inequalities ofebyev type for power series in Banach algebras

Silvestru S Dragomir1,2*, Marius V Boldea3, Constantin Buse4 and Mihail Megan4

*Correspondence: mailto:[email protected]

Web End [email protected] ; http://rgmia.org/dragomir

Web End =http://rgmia.org/dragomir

1Mathematics, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia

2School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg, 2050, South Africa Full list of author information is available at the end of the article

Abstract

Let f() =

n=0 nn be a function dened by power series with complex coecients and convergent on the open disk D(0, R)

C, R > 0 and x, y B, a Banach algebra,

with xy = yx. In this paper we establish some upper bounds for the norm of the ebyev type dierence f()f(xy) f(x)f(y), provided that the complex number and the vectors x, y B are such that the series in the above expression are

convergent. Applications for some fundamental functions such as the exponential function and the resolvent function are provided as well.

MSC: 47A63; 47A99

Keywords: Banach algebras; power series; exponential function; resolvent function; norm inequalities

1 Introduction

For two Lebesgue integrable functions f , g : [a, b]

R, consider the ebyev functional:

C(f , g) :=

b a

ba f (t)g(t) dt

(b a)

ba f (t) dt

ba g(t) dt. (.)

In , Grss [] showed that

C(f

, g)

(M m)(N n), (.)

provided that there exist real numbers m, M, n, N such that

m f (t) M and n g(t) N for a.e. t [a, b]. (.)

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

Another, however, less known result, even though it was obtained by ebyev in [], states that

C(f

, g)

f

(b a), (.) provided that f , g exist and are continuous on [a, b] and f = supt[a,b] |f (t)|. The constant

cannot be improved in the general case.

2014 Dragomir et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons

Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction

in any medium, provided the original work is properly cited.

g

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R are assumed to be absolutely continuous and f , g L[a, b], while f = ess supt[a,b] |f (t)|.

A mixture between Grss result (.) and ebyevs one (.) is the following inequality obtained by Ostrowski in []:

C(f

, g)

The ebyev inequality (.) also holds if f , g : [a, b]

(b a)(M m) g

, (.)

provided that f is Lebesgue integrable and satises (.), while g is absolutely continuous and g L[a, b]. The constant is best possible in (.).

The case of Euclidean norms of the derivative was considered by Lupa in [] in which he proved that

C(f

, g)

(b a), (.) provided that f , g are absolutely continuous and f , g L[a, b]. The constant

is the

f

g

best possible.Recently, Cerone and Dragomir [] have proved the following results:

C(f

, g)

inf

R

g q

b a

b a

f

(t)

b a

ba f (s) ds

p

dt

p, (.)

where p > and p + q = or p = and q = , and

C(f

, g)

inf

R

g

b a

ess sup

f

(t)

b a

t[a,b]

ba f (s) ds ,

(.)

provided that f Lp[a, b] and g Lq[a, b] (p > , p + q = ; p = , q = or p = , q = ).Notice that for q = , p = in (.) we obtain

C(f

, g)

inf

R

g

b a

b a

f

(t)

b a

ba f (s) ds

dt

g

b a

b a

f

(t)

b a

ba f (s) ds

dt (.)

and if g satises (.), then

C(f

, g)

inf

R

g

b a

b a

f

(t)

b a

ba f (s) ds

dt

g

n + N

b a

b a

f

(t)

b a

ba f (s) ds

dt

(N n)

b a

b a

f

(t)

b a

dt. (.)

The inequality between the rst and the last term in (.) has been obtained by Cheng and Sun in []. However, the sharpness of the constant , a generalization for the abstract

Lebesgue integral, and the discrete version of it have been obtained in [].

ba f (s) ds

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For other recent results on the Grss inequality, see [], and the references therein. In order to consider a ebyev type functional for functions of vectors in Banach algebras, we need some preliminary denitions and results as follows.

2 Some facts on Banach algebras

Let B be an algebra. An algebra norm on B is a map : B[, ) such that (B, ) is a normed space, and, further

ab a b

for any a, b B. The normed algebra (B, ) is a Banach algebra if is a complete norm.

We assume that the Banach algebra is unital, this means that B has an identity and that = .

Let B be a unital algebra. An element a B is invertible if there exists an element b B with ab = ba = . The element b is unique; it is called the inverse of a and written a or a.

The set of invertible elements of B is denoted by Inv B. If a, b Inv B then ab Inv B and (ab) = ba.

For a unital Banach algebra we also have:(i) if a B and limn an /n < , then a Inv B;(ii) {a B : b < } Inv B;(iii) Inv B is an open subset of B;(iv) the map Inv B a a Inv B is continuous. For simplicity, we denote , where

C and is the identity of B, by . The resolvent

set of a B is dened by

C : a Inv B};

the spectrum of a is (a), the complement of (a) in C, and the resolvent function of a is Ra : (a) Inv B, Ra() := ( a). For each , (a) we have the identity

Ra( ) Ra() = ( )Ra()Ra( ).

We also have (a) {

C : || a }. The spectral radius of a is dened as (a) =

sup{|| : (a)}.

If a, b are commuting elements in B, i.e. ab = ba, then

(ab) (a)(b) and (a + b) (a) + (b).

Let B a unital Banach algebra and a B. Then(i) the resolvent set (a) is open in C;(ii) for any bounded linear functionals : B

C, the function Ra is analytic on (a);(iii) the spectrum (a) is compact and nonempty in C;(iv) for each n

N and r > (a), we have

||=rn( a) d;

(v) we have (a) = limn an /n.

(a) := {

an =

i

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Let f be an analytic functions on the open disk D(, R) given by the power series f () :=

j= jj (|| < R). If (a) < R, then the series

j= jaj converges in the Banach algebra

B because

j= |j| aj < , and we can dene f (a) to be its sum. Clearly f (a) is well dened and there are many examples of important functions on a Banach algebra B that can be constructed in this way. For instance, the exponential map on B denoted exp and dened as

exp a :=

j=j!aj for each a B.

If B is not commutative, then many of the familiar properties of the exponential function from the scalar case do not hold. The following key formula is valid, however, with the additional hypothesis of commutativity for a and b from B:

exp(a + b) = exp(a) exp(b).

In a general Banach algebra B it is dicult to determine the elements in the range of the exponential map exp(B), i.e. the element which have a logarithm. However, it is easy to see that if a is an element in B such that a < , then a is in exp(B). That follows from the fact that if we set

b =

n=n( a)n,

then the series converges absolutely and, as in the scalar case, substituting this series into the series expansion for exp(b) yields exp(b) = a.

It is well known that if x and y are commuting, i.e. xy = yx, then the exponential function satises the property

exp(x) exp(y) = exp(y) exp(x) = exp(x + y).

Also, if x is invertible and a, b

R with a < b then

b a

exp(tx) dt = x

exp(bx) exp(ax)

.

Moreover, if x and y are commuting and y x is invertible, then

exp

( s)x + sy ds =

exp

s(y x) exp(x) ds

=

exp

s(y x) ds

exp(x)

= (y x)

exp(y x) I exp(x)

= (y x)

exp(y) exp(x)

.

n= nn be a function dened by power series with complex coecients and convergent on the open disk D(, R)

C, R > and x, y B with xy = yx. In this paper

Let f () =

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we establish some upper bounds for the norm of the ebyev type dierence

f ()f (xy) f (x)f (y) (.)

provided that the complex number and the vectors x, y B are such that the series in (.) are convergent. Applications for some fundamental functions such as the exponential function and the resolvent function are provided as well.

Inequalities for functions of operators in Hilbert spaces may be found in [], the recent monographs [], and the references therein.

3 The results

We denote by C the set of all complex numbers. Let n be nonzero complex numbers and let

R :=

lim sup |n|

n .

Clearly R , but we consider only the case < R .Denote by

D(, R) =

{z

C : |z| < R}, if R < , C, if R = ,

consider the functions

f () : D(, R)

C, f () :=

n= nn

and

fA() : D(, R)

C, fA() :=

n= | n|n.

Let B be a unital Banach algebra and its unity. Denote by

B(, R) =

{x B : x < R}, if R < ,

B, if R = .

We associate to f the map

x f(x) : B(, R) B, f(x) :=

n= nxn.

Obviously, f is correctly dened because the series

n= nxn is absolutely convergent,

since

n= |n| x n.In addition, we assume that s :=

n= n|n| < . Let s :=

n= nxn

n= |n| < and s :=

n= n|n| < .

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With the above assumptions we have the following.

Theorem Let

C such that max{||, ||} < R < and let x, y B with x , y , and xy = yx. Then:(i) We have

f(

)f(xy) f(x)f(y)

min

x , y

fA

||

(.)

where

:= ss s. (.)

(ii) We also have

f(

)f(xy) f(x)f(y)

min

x , y

fA

||

fA

|| ||f A

|| + ||f A

||

||f A

||

/.

(.)

Proof For m and since xy = yx we have

m

n=

m

j= njnj

xn xj

yn

m

m

=

m

n=

j=njnjxnyn

m

n=

j= njnjxjyn

=

m

j=jj

m

n=nnxnyn

m

j= jjxj

m

n= nnyn

m

=

j=jj

m

n=nn(xy)n

m

j= jjxj

m

n=nnyn (.)

C.

Taking the norm in (.) we have

m

for any

j=jj

m

n=nn(xy)n

m

j= jjxj

m

n= nnyn

m

n=

m

j= | n||j|||n||j

xn

xj

yn

m

n=

m

j= | n||j|||n||j

xn

xj

yn

m

n=

m

j= | n||j|||n||j

xn

xj

y n

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m

n=

m

j= | n||j|||n||j

xn

xj

=

j<nm

|n||j|||n||j

xn

xj

,

(.)

for any

C and m .

Observe that

L := j<nm

|n||j|||n||j

xn

xj

=

|n||j|||n||j

n

=j

x + x

j<nm

=

|n||j|||n||j

n

=jx (x )

j<nm

n

=j x (.)

for any

C and m .

We have

n

=j x (n j) max {j,...,n} x (n j)max {,...,m} x

and then

L x max

{,...,m} x

j<nm

x

|n||j|||n||j

j<nm

|n||j|||n||j(n j). (.)

From the rst inequality in (.) and since x < we have

m

j=jj

m

n=nn(xy)n

m

j= jjxj

m

n= nnyn

x

j<nm

|n||j|||n||j(n j)

= x

m

n=

m

j= |n||j|||n||j|n j|. (.)

(i) Using the Cauchy-Bunyakovsky-Schwarz inequality for double sums,

m

n=

m

j=pn,jan,jbn,j

m

n=

m

j= pn,jan,j

/ m

n=

m

j= pn,jbn,j

/,

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where pn,j, an,j, bn,j for n, j {, . . . , m}, we have

m

n=

m

j= |n||j|||n||j|n j|

m

n=

j= | n||j|||n||j /

m

n=

j= |n||j||n j| /

m

m

=

n= | n|||n

m

n= | n|

m

n=n|n|

m

m

/

n= n|n|

(.)

for any

C and m .

From (.) and (.) we get the inequality

m

j=jj

m

n=nn(xy)n

m

j= jjxj

m

n= nnyn

x

n= | n|||n

m

n=n|n|

m

m

n= | n|

m

/. (.)

n= n|n|

Since the series

j= jj,

n= nn(xy)n,

j= jjxj,

n= nnyn

are convergent in B,

n= |n|||n is convergent and the limit

lim

m

n=n|n|

m

m

n= | n|

m

/

n= n|n|

exists, then by letting m in (.) we deduce the desired result in (.) for x. Due to the commutativity of x with y, a similar result can be stated for y, and taking the minimum, we deduce the desired result.(ii) Using the Cauchy-Bunyakovsky-Schwarz inequality for double sums,

m

n=

j=pn,jan,j

m

n=

m

j=pn,j

/ m

n=

m

m

j= pn,jan,j

/

where pn,j, an,j for n, j {, . . . , m}, we also have

m

n=

m

j= |n||j|||n||j|n j|

m

n=

j= | n||j|||n||j /

m

n=

j= |n||j|||n||j|n j| /

m

m

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=

n= | n|||n

m

m

n= | n|||n

m

n=n|n|||n

m

n=n|n|||n / (.)

for any

C and m .

From (.) and (.) we have

m

j=jj

m

n=nn(xy)n

m

j= jjxj

m

n= nnyn

x

n= | n|||n

m

m

n= | n|||n

m

n=n|n|||n

m

n=n|n|||n / (.)

for any

C and m .

If we denote f (u) :=

n= nun, then for |u| < R we have

n=nnun = uf (u)

and

n=nnun = u

ug (u)

.

However

u

ug (u)

= ug (u) + ug (u) and then

n=nnun = ug (u) + ug (u).

Therefore

n=n|n|||n = ||f A

|| + ||f A

||

and

m

n=n|n|||n = ||f

||

for || < R.

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Since all the series whose partial sums are involved in (.) are convergent, then by letting m in (.) we deduce the desired inequality (.) for x. Due to the commutativity of x with y, a similar result can be stated for y, and taking the minimum, we deduce the desired result.

Remark If R = , Theorem holds true. Moreover, in this case the restrictions x , y need no longer be imposed.

Remark We observe that if the power series f () =

n= nn has the radius of conver-

gence R > , then

n= |n| = fA(),

n=n|n| = f A() + f A()

and

n=n|n| = f A().

In this case is nite and

= lim

m

n=n|n|

m

m

n= | n|

m

/

n= n|n|

=

fA()

f A() + f A()

f A()

/.

Therefore, if

C with ||, ||, || x , || y < R, then from (.) we have

f(

)f(xy) f(x)f(y)

fA()

f A() + f A()

f A()

/

min

x , y

fA

||

. (.)

Corollary Under the assumptions of Theorem we have the inequalities

f(

)f

x f(x)

x fA

||

(.)

provided

C with ||, ||, || x < R, and

f(

)f

x f(x)

x fA

||

fA

|| ||f A

|| + ||f A

||

||f A

||

/

(.)

provided

C with ||, || x < R.

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Theorem Let f () =

n= nn be a function dened by power series with complex coecients and convergent on the open disk D(, R)

C, R > , and x, y B with xy = yx and

x , y < .

If

C with ||, || x , || y < R, then

f(

)f(xy) f(x)f(y)

min

x x , y y

f A

|| fA

|| ,(.)

where

fA () :=

n= |n|n (.)

has the radius of convergence R.

Proof As pointed out in (.), we have

L x j<nm

n

=j x

|n||j|||n||j

m

= x

x

j<nm

|n||j|||n||j (.)

for any

C and m .

Denote

Km :=

j<nm

|n||j|||n||j.

We obviously have

Km =

m

n,j= | n||j|||n||j

m

n= | n|||n

=

n= | n|||n

m

n= | n|||n .

From (.) and (.) we get the inequality

m

m

j=jj

m

n=nn(xy)n

m

j= jjxj

m

n= nnyn

m

= x

x

n= | n|||n

m

n= |n|||n , (.)

m

for any

C and m .

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Since all the series whose partial sums are involved in (.) are convergent, then by letting m in (.) we deduce the desired inequality (.) for x. Due to the commutativity of x with y, a similar result can be stated for y, and taking the minimum, we deduce the desired result.

Remark Since the power series fA () :=

n= |n|n is not easy to compute, we can provide some bounds for the quantity

Df

|| := f A || fA

||

,

where || < R, as follows.If || < and a := supnN{|an|} < , then

Km a

j<nm

||n||j

a

n= | |n

m

n= | |n

and by taking m in this inequality we get

Df

|| a ||

||

=

m

(.)

for || < .If || < and

a := lim

m

m

n= |n| <

m

n= | n|

then

Km max

n{,...,m} |

|n

|n||j|

j<nm

n= | n|

and by taking m in this inequality we get

Df

|| a (.)

for || < .If the series

n= |n| and

m

m

n= | n|

n= |n| are convergent, then

Df

||

n= | n|

n= |n| (.)

for || < .

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If || < , p, q > with p + q = , and

a q := lim

m

n= | n|q

m

n= |n|q <

m

then by Hlders inequality we have

Km

|n|q|j|q /q

||pn||pj /p

j<nm

j<nm

n= | n|q

m

n= | n|q /q

m

n= | |pn /p

and by taking m in this inequality we get

Df

|| a/q q ||p

||p

n= | |pn

m

m

/p

(.)

for || < .If the series

n= |n|q and

n= |n|q are convergent, then

Df

||

n= | n|q

n= |n|q /p ||p

||p

/p(.)

for || < .

The following result also holds.

Theorem Let f () =

n= nn be a function dened by power series with complex coecients and convergent on the open disk D(, R)

C, R > , and x, y B with xy = yx and

x , y < .

If p, q > with p + q = and

C with ||, ||p, || x , || y < R, then

f(

)f(xy) f(x)f(y)

min

x ( x p)/p

, y

( y p)/p

/q

f A

||p fA

||p

/p,

(.)

where

:= lim

m

m

n,j= |n||j||n j| (.)

is assumed to exist and be nite.

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Proof Using Hlders inequality for p, q > with p + q = and (.), we have

L x j<nm

|n||j|||n||j(n j)/q

=j x p /p

n

m

x

= x p /p j<nm

|n||j|||n||j(n j)/q (.)

for any

C and m .

Applying Hlders inequality once more we have

j<nm

|n||j|||n||j(n j)/q

|n||j|||n(n j) /q

|n||j|||pn||pj /p

j<nm

j<nm

/q

=

m

n,j= |n||j||n j|

n= | n|||np

m

n= | n|||np /p

=

m

m

n,j= |n||j||n j|

/q

n= | n|||np

m

n= |n|||np /p (.)

for any

C and m .

From (.) and (.) we get the inequality

m

m

j=jj

m

n=nn(xy)n

m

j= jjxj

m

n= nnyn

x m

=

x p /p

m

n,j= |n||j||n j|

/q

n= | n|||np

m

n= |n|||np /p, (.)

for any

C and m .

Since all the series whose partial sums are involved in (.) are convergent, then by

letting m in (.) we deduce the desired inequality (.) for x. Due to the commutativity of x with y, a similar result can be stated for y, and taking the minimum, we deduce

the desired result.

m

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Remark Observe that

f A

||p fA ||p /p= D/pf

||p

and then further bounds for the inequality (.) may be provided by the use of Remark . However the details are not mentioned here.

We can obtain a simpler upper bound for as follows.

Using the Cauchy-Bunyakovsky-Schwarz inequality for double sums

m

n=

j=pi,jai,j

m

n=

m

j=pi,j

/ m

n=

m

m

j= pi,jai,j

/,

where pi,j, ai,j for i, j {, . . . , m}, we have

m

n,j= |n||j||n j|

/ m

n,j= |n||j||n j| /

m

n,j= | n||j|

m

n= | n|

=

n=n|n|

m

m

n= | n|

m

/(.)

n= n|n|

for m .If the series

n= |n| is nite and , dened by (.), is nite, then from (.) we have

n= |n|. (.)

We observe that, if the power series f () =

n= nn has the radius of convergence R > ,

then is nite and

=

fA()

f A() + f A()

f A()

/.

We have from (.) the inequality

fA()

fA()

f A() + f A()

f A()

/. (.)

4 Some examples

As some natural examples that are useful for applications, we can point out that, if

f () =

n=()nnn = ln + , D(, );

g() =

n=()n(n)!n = cos ,

C;

h() =

(.)

n=()n(n + )!n+ = sin ,

C;

l() =

n=()nn = + , D(, ),

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then the corresponding functions constructed by the use of the absolute values of the coecients are

fA() =

n=nn = ln , D(, );

gA() =

n=(n)!n = cosh ,

C;

hA() =

(.)

n=(n + )!n+ = sinh ,

C;

lA() =

n=n = , D(, ).

Other important examples of functions as power series representations with nonnegative coecients are

exp() =

n=n!n,

C,

ln

+ =

n=n n, D(, );

sin() =

n= (n + )

(n + )n!

n+, D(, );

tanh() =

n=n n, D(, );

F(, , , ) =

n= (n + ) (n + ) ( )n! () () (n + )n, , , > , D(, );

(.)

where is the Gamma function.If we apply the inequality (.) to the exponential function, then we have

exp (

+ xy)

exp

(x + y)

e min

x , y exp

||

(.)

for any x, y B with xy = yx, x , y < , and

C.

If we take y = x in (.), then we get

exp

x

e min

x , x + exp

||

(.)

for any x B with x < and

C.

If we apply the inequality (.) for the exponential functions we also have

exp (

+ xy)

exp

(x + y)

min

, (.)

for any x, y B with xy = yx, x , y < , and

C.

x , y

||/ exp

||

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If we take y = x in (.), then we get

exp

x

min

x , x +

||/ exp

||

. (.)

n= n = , D(, ). If we apply the inequality (.) for this function, then we get the result

(

)( xy) ( x)( y)

Now, consider the function f () :=

min

x , y

||/

||

(.) for any x, y B with xy = yx, x , y < , and

C with || < .

We have in particular the inequalities

(

)

x

( x)

x ||/

||

(.)

and

(

)

+ x

x

min

x , x +

||/

||

(.)

C with || < .

Now, if we take = with | | > then we get from (.) the inequality

( )( xy) ( x)( y)

for any x B with x < and

min

x , y

| |/

| |

| |,

which is equivalent with

(

)( xy) ( x)( y)

min

x , y

| |/

| |

C with | | > .

If we use the resolvent function notation, then we have the following inequality:

(

)Rxy( ) Rx( )Ry( )

for any x, y B with xy = yx, x , y < , and

min

x , y

| |/

| |

(.) for any x, y B with xy = yx, x , y < , and

C with | | > .

In particular, we have

(

)Rx ( ) Rx( )

x | |/

| |

(.)

for any x B with x < and

C with | | > .

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Remark Similar inequalities may be stated for the other power series mentioned at the beginning of this paragraph. However, the details are not presented here.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally and signicantly in writing this article. All authors read and approved the nal manuscript.

Author details

1Mathematics, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia.

2School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg, 2050, South Africa. 3Mathematics and Statistics, Banat University of Agricultural Sciences and Veterinary Medicine Timisoara, 119 Calea Aradului, Timisoara, 300645, Romnia. 4Department of Mathematics, West University of Timisoara, B-dul V. Prvan 4, Timisoara, 1900, Romnia.

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments that have been implemented in the nal version of the paper.

Received: 20 February 2014 Accepted: 27 June 2014 Published: 18 August 2014

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doi:10.1186/1029-242X-2014-294Cite this article as: Dragomir et al.: Norm inequalities ofebyev type for power series in Banach algebras. Journal of Inequalities and Applications 2014 2014:294.