Abstract
Let f ( λ ) = ∑ n = 0 ∞ α n λ n Open image in new window be a function defined by power series with complex coefficients and convergent on the open disk D ( 0 , R ) ⊂ C Open image in new window, R > 0 Open image in new window and x , y ∈ B Open image in new window, a Banach algebra, with x y = y x Open image in new window. In this paper we establish some upper bounds for the norm of the Cebysev type difference f ( λ ) f ( λ x y ) − f ( λ x ) f ( λ y ) Open image in new window, provided that the complex number λ and the vectors x , y ∈ B Open image in new window 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.
Highlights
For two Lebesgue integrable functions f, g : [a, b] → R, consider the Čebyšev functional: b b bC(f, g) := b–a a f (t)g(t) dt – (b – a)f (t) dt a g(t) dt. a ( . )In, Grüss [ ] 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, less known result, even though it was obtained by Čebyšev in [ ], states that C(f, g) ≤ f
Is best possible in in the sense that it cannot be replaced by a smaller quantity
Less known result, even though it was obtained by Čebyšev in
Summary
For two Lebesgue integrable functions f , g : [a, b] → R, consider the Čebyšev functional: b b b. ∞(b – a) , provided that f , g exist and are continuous on [a, b] and f ∞ = supt∈[a,b] |f (t)|. In order to consider a Čebyšev type functional for functions of vectors in Banach algebras, we need some preliminary definitions and results as follows. We assume that the Banach algebra is unital, this means that B has an identity and that =. The resolvent set of a ∈ B is defined by ρ(a) := {λ ∈ C : λ – a ∈ Inv B}; the spectrum of a is σ (a), the complement of ρ(a) in C, and the resolvent function of a is Ra : ρ(a) → Inv B, Ra(λ) := (λ – a)–.
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