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For a large class of arithmetic functions f, it is possible to show that, given an arbitrary integer κ ≤ 2, the string of inequalities f(n + 1) < f(n + 2) < … < f(n + κ) holds for in-finitely many positive integers n. For other arithmetic functions f, such a property fails to hold even for κ = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of n satisfying σ2(n − 1) < σ2 < σ2(n + 1), where σ2(n) = ∑d|n d 2. On the other hand, we prove that for the function f(n) := ∑p|n p 2, we do have f(n − 1) < f(n) < f(n + 1) in finitely often.

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Abstract  

In the present paper some Newton-like iteration methods are developed to enclose solutions of nonlinear operator equations of the kindF(x)=0. HereF maps a certain subset of a partially ordered vector space into another partially ordered vector space. The obtained results are proved without any special properties of the orderings by taking use of a new kind of a generalized divided difference operator, so that they even hold for nonconvex operators. Furthermore a method for constructing including starting points is presented and two examples are given.

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For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the followingmonotonic integral transform

M(w,μ)(T) :=0w(λ)T(λ+T)1dμ(λ),

where the integral is assumed to exist forT a positive operator on a complex Hilbert spaceH. We show among others that, if β ≥ A, B ≥ α > 0, and 0 < δ ≤ (B − A)2 ≤ Δ for some constants α, β, δ, Δ, then

0124δM(w,μ)(β)M(w,μ)A+B201M(w,μ)((1t)A+tB)dt124ΔM(w,μ)(α)

and

0112δM(w,μ)(β)01M(w,μ)((1t)A+tB)dtM(w,μ)(A)+M(w,μ)(B)2112ΔM(w,μ)(α),

whereM(w,μ) is the second derivative ofM(w,μ) as a real function.

Applications for power function and logarithm are also provided.

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