Norm inequalities of Čebyšev type for power series in Banach algebras

Let f(λ)=∑n=0∞αnλn be a function defined by power series with complex coefficients 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 Čebyšev type differencef(λ)f(λxy)−f(λx)f(λy), provided that the complex number λ 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 exponentialfunction and the resolvent function are provided as well. MSC: 47A63, 47A99.


Introduction
For two Lebesgue integrable functions f , g : (.) In , Grüss [] showed that provided that there exist real numbers m, M, n, N such that 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 Čebyšev in  [], states that provided that f , g exist and are continuous on [a, b] and f ∞ = sup t∈ [a,b] |f (t)|.The constant   cannot be improved in the general case.http://www.journalofinequalitiesandapplications.com/content/2014/1/294 The Čebyšev inequality (.) also holds if f , g : [a, b] → R are assumed to be absolutely continuous and f , g ∈ L ∞ [a, b], while f ∞ = ess sup t∈ [a,b] |f (t)|.
A mixture between Grüss' result (.) and Čebyšev's one (.) is the following inequality obtained by Ostrowski in  []: provided that f is Lebesgue integrable and satisfies (.), 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 Lupaş in [] in which he proved that provided that f , g are absolutely continuous and f , g ∈ L  [a, b].The constant  π  is the best possible.
Recently, Cerone and Dragomir [] have proved the following results: where p >  and  p +  q =  or p =  and q = ∞, and and if g satisfies (.), then The inequality between the first 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 [].http://www.journalofinequalitiesandapplications.com/content/2014/1/294 For other recent results on the Grüss inequality, see [-], 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 • : B→[, ∞) such that (B, • ) is a normed space, and, further 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 For a unital Banach algebra we also have: For simplicity, we denote λ, where λ ∈ C and  is the identity of B, by λ.The resolvent set of a ∈ B is defined by the spectrum of a is σ (a), the complement of ρ(a) in C, and the resolvent function of a is R a : ρ(a) → Inv B, R a (λ) := (λa) - .For each λ, γ ∈ ρ(a) we have the identity Let f be an analytic functions on the open disk D(, R) given by the power series f (λ) := ∞ j= α j λ j (|λ| < R).If ν(a) < R, then the series ∞ j= α j a j converges in the Banach algebra B because ∞ j= |α j | a j < ∞, and we can define f (a) to be its sum.Clearly f (a) 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 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 = - 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 satisfies the property Moreover, if x and y are commuting and yx is invertible, then we establish some upper bounds for the norm of the Čebyšev type difference provided that the complex number λ 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.

The results
We denote by C the set of all complex numbers.Let α n be nonzero complex numbers and let Let B be a unital Banach algebra and  its unity.Denote by We associate to f the map Obviously, f ˜is correctly defined because the series With the above assumptions we have the following.
We have However and then 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 (λ) = ∞ n= α n λ n has the radius of convergence R > , then In this case ψ is finite and Therefore, if λ ∈ C with |λ|, |λ|  , |λ| x , |λ| y < R, then from (.) we have Corollary  Under the assumptions of Theorem  we have the inequalities Theorem  Let f (λ) = ∞ n= α n λ n be a function defined by power series with complex coefficients and convergent on the open disk D(, R) ⊂ C, R > , and x, y ∈ B with xy = yx and x , y < .
has the radius of convergence R  .
Proof As pointed out in (.), we have for any λ ∈ C and m ≥ .Denote We obviously have From (.) and (.) we get the inequality for any λ ∈ 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.
Remark  Since the power series f A  (λ) := ∞ n= |α n |  λ n is not easy to compute, we can provide some bounds for the quantity where |λ| < R, as follows.
and by taking m → ∞ in this inequality we get and by taking m → ∞ in this inequality we get for |λ| < . http://www.journalofinequalitiesandapplications.com/content/2014/1/294 If |λ| < , p, q >  with  p +  q = , and then by Hölder's inequality we have and by taking m → ∞ in this inequality we get The following result also holds.
n= α n λ n be a function defined by power series with complex coefficients and convergent on the open disk D(, R) ⊂ C, R > , and x, y ∈ B with xy = yx and x , y < .Proof Using Hölder's inequality for p, q >  with  p +  q =  and (.), we have Remark  Observe that 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 ϕ as follows.
Using the Cauchy-Bunyakovsky-Schwarz inequality for double sums We observe that, if the power series f (λ) = ∞ n= α n λ n has the radius of convergence R > , then ψ is finite and We have from (.) the inequality

Some examples
As some natural examples that are useful for applications, we can point out that, if Other important examples of functions as power series representations with nonnegative coefficients are where is the Gamma function.
http://www.journalofinequalitiesandapplications.com/content/2014/1/294 then the corresponding functions constructed by the use of the absolute values of the coefficients are ), (.)