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# Arithmetic functions monotonic at consecutive arguments

Studia Scientiarum Mathematicarum Hungarica
Authors: Jean-Marie Koninck and Florian Luca

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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# Iterative Einschliessung von Lösungen Nichtlinearer Operatorgleichungen und Bestimmung von Startnäherungen

Periodica Mathematica Hungarica
Author: R. Voller

## 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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# Monotonicity of the Complementary Vector Field of a Nonexpansive Map

Acta Mathematica Hungarica
Author: S. Németh
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# Some monotonicity properties of the zeros of ultraspherical polynomials

Acta Mathematica Hungarica
Authors: Á. Elbert and A. Laforgia
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# Continuity of the Entropy for Monotonic Mod One Transformations

Acta Mathematica Hungarica
Author: P. Raith
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# On the intersection of monotonicity preserving linear extensions

Acta Mathematica Hungarica
Author: J. Szigeti
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# Some monotonicity properties of polynomials with equally spaced zeros

Acta Mathematica Hungarica
Author: L. Lorch
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# On a monotonic property of certain Sturm-Liouville functions

Acta Mathematica Hungarica
Author: E. Makai
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# On Walsh series with coefficients tending monotonically to zero

Acta Mathematica Hungarica
Author: F. Móricz
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# Error Bounds Related to Midpoint and Trapezoid Rules for the Monotonic Integral Transform of Positive Operators in Hilbert Spaces

Mathematica Pannonica
Author: Silvestru Sever Dragomir

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

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

$0≤124δM″(w,μ)(β)≤M(w,μ)A+B2−∫01M(w,μ)((1−t)A+tB)dt≤−124ΔM″(w,μ)(α)$

and

$0≤−112δM″(w,μ)(β)≤∫01M(w,μ)((1−t)A+tB)dt−M(w,μ)(A)+M(w,μ)(B)2≤112ΔM″(w,μ)(α),$

where$M″(w,μ)$ is the second derivative of$M(w,μ)$ as a real function.

Applications for power function and logarithm are also provided.

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