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.
To verify the universal validity of the ``two-sided'' monotonicity condition introduced in , we will apply it to include
more classical examples. The present paper selects the Lp convergence case for this purpose. Furthermore, Theorem 3 shows that our improvements are not trivial.
Chaundry and Jolliffe  proved that if ak is a nonnegative sequence tending monotonically to zero, then a necessary and sufficient condition for the uniform convergence
of the series Σk=1∞ak sin kx is limk→∞kak = 0. Lately, S. P. Zhou, P. Zhou and D. S. Yu  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 
on the L1-convergence of Fourier series.
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|nd2. On the other hand, we prove that for the function f(n) := ∑p|np2, we do have f(n − 1) < f(n) < f(n + 1) in finitely often.