Power-monotone sequences and Fourier series with positive coefficients (Q2740869)

The author generalizes two of his theorems [\textit{J. Németh}, Acta Sci. Math. 54, No. 3/4, 291-304 (1990; Zbl 0762.42003)] and a result of \textit{M. Izumi} and \textit{S.-I. Izumi} [J. Math. Mech. 18, 857-870 (1969; Zbl 0177.09002)] by using the concept of quasi-power-monotonicity. One of his theorems is as follows.NEWLINENEWLINENEWLINETheorem. If \(\{n^\varepsilon \gamma_n\}\) is quasi-monotonically decreasing and \(\{n^{2-\varepsilon}\gamma_n\}\) is quasi-monotonically increasing for some \(\varepsilon> 0\), then NEWLINE\[NEWLINE\omega^{(2)}\Biggl(\varphi,{1\over n}\Biggr)= O(\gamma_n)NEWLINE\]NEWLINE if and only if \(\sum^\infty_{k=n} \lambda_k= O(\gamma_n)\), \(\lambda_n\geq 0\),where \(\varphi(x)\) is a continuous \(2\pi\) periodic function which is either odd or even, \(\lambda_n\) is its Fourier coefficient and \(\omega^{(2)}(\varphi,\delta)\) is its modulus of continuity of second order.