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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 σ_{2}(*n* − 1) < σ_{2} < σ_{2}(*n* + 1), where σ_{2}(*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.

## Abstract

In the present paper some Newton-like iteration methods are developed to enclose solutions of nonlinear operator equations
of the kind*F(x)*=0. Here*F* 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.

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

where the integral is assumed to exist for*T* a positive operator on a complex Hilbert space*H*. 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

and

where

Applications for power function and logarithm are also provided.