Further bounds for Cebysev functional for power series in banach algebras via Grüss-Lupas type inequalities for p-norms

Some Grüss-Lupas type inequalities for p-norms of sequences in Banach algebras are obtained. Moreover, if f(λ)=Σ^^∞__<n=0>α_nλ^n is 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, then we also establish some new upper bounds for the norm of the Cebysev type difference
f(λ)f(λxy) - f(λx)f(λy), λ ∈ D(0,R).
These results build upon the earlier results obtained by the authors. Applications for some fundamental functions such as the exponential function and the resolvent function are provided as well.