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Abstract  

The Abel’s and Dirichlet’s criterions for convergence of series in analysis are very basic classical results and both require the monotonicity condition. In this note we show that the monotonicity condition in these criterions can be generalized to RBV condition, while cannot be generalized to quasimonotonicity.

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Acta Mathematica Hungarica
Authors: C. L. Belna, M. J. Evans, and P. D. Humke
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Summary  

To verify the universal validity of the ``two-sided'' monotonicity condition introduced in [4], we will apply it to include more classical examples. The present paper selects the L p convergence case for this purpose. Furthermore, Theorem 3 shows that our improvements are not trivial.

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Abstract  

The psi function ψ(x) is defined by ψ(x) = Γ′(x)/Γ(x) and ψ (i)(x), for i ∈ ℕ, denote the polygamma functions, where Γ(x) is the gamma function. In this paper, we prove that the functions
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$[\psi '(x)]^2 + \psi ''(x) - \frac{{x^2 + 12}} {{12x^4 (x + 1)^2 }}$$ \end{document}
and
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{{x + 12}} {{12x^4 (x + 1)}} - \{ [\psi '(x)]^2 + \psi ''(x)\}$$ \end{document}
are completely monotonic on (0,∞).
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Abstract  

Chaundry and Jolliffe [1] proved that if a k is a nonnegative sequence tending monotonically to zero, then a necessary and sufficient condition for the uniform convergence of the series Σk=1 a k sin kx is limk→∞ ka k = 0. Lately, S. P. Zhou, P. Zhou and D. S. Yu [4] generalized this classical result. In this paper we propose new classes of sequences for which we get the extended version of their results. Moreover, we generalize the results of S. Tikhonov [2] on the L 1-convergence of Fourier series.

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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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