On Conditions of the Average Convergence (Upper Boundedness) of Trigonometric Series

Let $c_n=\widehat f(n)$ be Fourier coefficients of a function $f\in L_{2\pi}$. We prove that the condition
$$ \sum_{k=\left[\frac n2\right]}^{2n}\frac{|c_k|+|c_{-k}|}{|n-k|+1}=o(1) \quad \big(=O(1)\big) $$
is necessary for the convergence of the Fourier series of $f$ in the $L$-metric; moreover, this condition is sufficient under some additional hypothesis for Fourier coefficients of $f$.