Abstract
Let be a function defined by power series with complex coefficients and convergent on the open disk , and , a Banach algebra, with . In this paper we establish some upper bounds for the norm of the Čebyšev type difference , provided that the complex number λ and the vectors 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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1 Introduction
For two Lebesgue integrable functions , consider the Čebyšev functional:
In 1935, Grüss [1] showed that
provided that there exist real numbers m, M, n, N such that
The constant is best possible in (1.1) in the sense that it cannot be replaced by a smaller quantity.
Another, however, less known result, even though it was obtained by Čebyšev in 1882 [2], states that
provided that , exist and are continuous on and . The constant cannot be improved in the general case.
The Čebyšev inequality (1.4) also holds if are assumed to be absolutely continuous and , while .
A mixture between Grüss’ result (1.2) and Čebyšev’s one (1.4) is the following inequality obtained by Ostrowski in 1970 [3]:
provided that f is Lebesgue integrable and satisfies (1.3), while g is absolutely continuous and . The constant is best possible in (1.5).
The case of Euclidean norms of the derivative was considered by Lupaş in [4] in which he proved that
provided that f, g are absolutely continuous and . The constant is the best possible.
Recently, Cerone and Dragomir [5] have proved the following results:
where and or and , and
provided that and (, ; , or , ).
Notice that for , in (1.7) we obtain
and if g satisfies (1.3), then
The inequality between the first and the last term in (1.10) has been obtained by Cheng and Sun in [6]. However, the sharpness of the constant , a generalization for the abstract Lebesgue integral, and the discrete version of it have been obtained in [7].
For other recent results on the Grüss inequality, see [8–22], and the references therein.
In order to consider a Čebyšev type functional for functions of vectors in Banach algebras, we need some preliminary definitions and results as follows.
2 Some facts on Banach algebras
Let ℬ be an algebra. An algebra norm on ℬ is a map such that is a normed space, and, further
for any . The normed algebra is a Banach algebra if is a complete norm.
We assume that the Banach algebra is unital, this means that ℬ has an identity 1 and that .
Let ℬ be a unital algebra. An element is invertible if there exists an element with . The element b is unique; it is called the inverse of a and written or . The set of invertible elements of ℬ is denoted by . If then and .
For a unital Banach algebra we also have:
-
(i)
if and , then ;
-
(ii)
;
-
(iii)
is an open subset of ℬ;
-
(iv)
the map is continuous.
For simplicity, we denote λ 1, where and 1 is the identity of ℬ, by λ. The resolvent set of is defined by
the spectrum of a is , the complement of in ℂ, and the resolvent function of a is , . For each we have the identity
We also have . The spectral radius of a is defined as .
If a, b are commuting elements in ℬ, i.e. , then
Let ℬ a unital Banach algebra and . Then
-
(i)
the resolvent set is open in ℂ;
-
(ii)
for any bounded linear functionals , the function is analytic on ;
-
(iii)
the spectrum is compact and nonempty in ℂ;
-
(iv)
for each and , we have
-
(v)
we have .
Let f be an analytic functions on the open disk given by the power series . If , then the series converges in the Banach algebra ℬ because , and we can define to be its sum. Clearly is well defined 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 defined as
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:
In a general Banach algebra ℬ it is difficult to determine the elements in the range of the exponential map , i.e. the element which have a ‘logarithm’. However, it is easy to see that if a is an element in B such that , then a is in . That follows from the fact that if we set
then the series converges absolutely and, as in the scalar case, substituting this series into the series expansion for yields .
It is well known that if x and y are commuting, i.e. , then the exponential function satisfies the property
Also, if x is invertible and with then
Moreover, if x and y are commuting and is invertible, then
Let be a function defined by power series with complex coefficients and convergent on the open disk , and with . In this paper we establish some upper bounds for the norm of the Čebyšev type difference
provided that the complex number λ and the vectors are such that the series in (2.1) 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 [23–26], the recent monographs [27–29], and the references therein.
3 The results
We denote by ℂ the set of all complex numbers. Let be nonzero complex numbers and let
Clearly , but we consider only the case .
Denote by
consider the functions
and
Let ℬ be a unital Banach algebra and 1 its unity. Denote by
We associate to f the map
Obviously, is correctly defined because the series is absolutely convergent, since .
In addition, we assume that . Let and .
With the above assumptions we have the following.
Theorem 1 Let such that and let with , and . Then:
-
(i)
We have
(3.1)
where
-
(ii)
We also have
(3.3)
Proof For and since we have
for any .
Taking the norm in (3.4) we have
for any and .
Observe that
for any and .
We have
and then
From the first inequality in (3.7) and since we have
-
(i)
Using the Cauchy-Bunyakovsky-Schwarz inequality for double sums,
where for , we have
for any and .
From (3.8) and (3.9) we get the inequality
Since the series
are convergent in ℬ, is convergent and the limit
exists, then by letting in (3.10) we deduce the desired result in (3.1) 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,
where for , we also have
for any and .
From (3.8) and (3.11) we have
for any and .
If we denote , then for we have
and
However
and then
Therefore
and
for .
Since all the series whose partial sums are involved in (3.12) are convergent, then by letting in (3.12) we deduce the desired inequality (3.3) 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 1 If , Theorem 1 holds true. Moreover, in this case the restrictions need no longer be imposed.
Remark 2 We observe that if the power series has the radius of convergence , then
and
In this case ψ is finite and
Therefore, if with , then from (3.1) we have
Corollary 1 Under the assumptions of Theorem 1 we have the inequalities
provided with , and
provided with .
Theorem 2 Let be a function defined by power series with complex coefficients and convergent on the open disk , , and with and .
If with , then
where
has the radius of convergence .
Proof As pointed out in (3.6), we have
for any and .
Denote
We obviously have
From (3.8) and (3.18) we get the inequality
for any and .
Since all the series whose partial sums are involved in (3.19) are convergent, then by letting in (3.19) we deduce the desired inequality (3.16) 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 3 Since the power series is not easy to compute, we can provide some bounds for the quantity
where , as follows.
If and , then
and by taking in this inequality we get
for .
If and
then
and by taking in this inequality we get
for .
If the series and are convergent, then
for .
If , with , and
then by Hölder’s inequality we have
and by taking in this inequality we get
for .
If the series and are convergent, then
for .
The following result also holds.
Theorem 3 Let be a function defined by power series with complex coefficients and convergent on the open disk , , and with and .
If with and with , then
where
is assumed to exist and be finite.
Proof Using Hölder’s inequality for with and (3.6), we have
for any and .
Applying Hölder’s inequality once more we have
for any and .
From (3.8) and (3.28) we get the inequality
for any and .
Since all the series whose partial sums are involved in (3.29) are convergent, then by letting in (3.29) we deduce the desired inequality (3.25) 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 4 Observe that
and then further bounds for the inequality (3.25) may be provided by the use of Remark 3. However the details are not mentioned here.
We can obtain a simpler upper bound for φ as follows.
Using the Cauchy-Bunyakovsky-Schwarz inequality for double sums
where for , we have
for .
If the series is finite and ψ, defined by (3.2), is finite, then from (3.30) we have
We observe that, if the power series has the radius of convergence , then ψ is finite and
We have from (3.31) the inequality
4 Some examples
As some natural examples that are useful for applications, we can point out that, if
then the corresponding functions constructed by the use of the absolute values of the coefficients are
Other important examples of functions as power series representations with nonnegative coefficients are
where Γ is the Gamma function.
If we apply the inequality (3.13) to the exponential function, then we have
for any with , , and .
If we take in (4.4), then we get
for any with and .
If we apply the inequality (3.3) for the exponential functions we also have
for any with , , and .
If we take in (4.6), then we get
Now, consider the function , . If we apply the inequality (3.3) for this function, then we get the result
for any with , , and with .
We have in particular the inequalities
and
for any with and with .
Now, if we take with then we get from (4.8) the inequality
which is equivalent with
for any with , , and with .
If we use the resolvent function notation, then we have the following inequality:
for any with , , and with .
In particular, we have
for any with and with .
Remark 5 Similar inequalities may be stated for the other power series mentioned at the beginning of this paragraph. However, the details are not presented here.
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Dragomir, S.S., Boldea, M.V., Buşe, C. et al. Norm inequalities of Čebyšev type for power series in Banach algebras. J Inequal Appl 2014, 294 (2014). https://doi.org/10.1186/1029-242X-2014-294
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DOI: https://doi.org/10.1186/1029-242X-2014-294