Quasiconvex uniform-convergence factors for Fourier series of functions with a given modulus of continuity

It is proved that a quasiconvex sequencelambda $\{\lambda_\nu\}$ of convergence factors transforms Fourier series of functions whose moduli of continuity do not exceed a given modulus of continuity $\omega(\delta)$ into uniformly convergent series if and only iflambda $\lambda_n\omega(1/n)\log n\to0$. The sufficiency of this condition is already known.