Summary: Suppose that a sequence of multiplicative arithmetical functions \(h_ n(m)\) belongs to the class \(D^+=D^+(\phi,\psi,\rho,r,\alpha,c_ 2)\), \(\alpha >0\), there exist nonnegative strictly decreasing functions \(\phi\) (n), \(\psi\) (n), \(\rho\) (n), a real number r \((r>2)\) and a constant \(c_ 2>0\) such, that \[ \sum_{p\leq n,\quad | \ln | h_ n(p)| | >\phi (n)}| \ln | h_ n(p)| |^ rp^{-1}\leq \phi^ r(n),\sum_{p>n,\quad h_ n(p)\leq 0}p^{-1}\leq \rho (n), \] \[ \sum_{p\leq n,| \ln | h_ n(p^ k)| | >\phi (n)}\sum_{k\geq 2}| \ln | h_ n(p^ k)| |^ rp^{- 1}\leq \phi^ r(n), \] \[ \sum_{p\leq n,h_ n(p)\leq 0}\ln p/(p \ln n)\leq \psi (n). \] Theorems of deviations for functions of this class are obtained in the zone \[ 1\leq x\leq a(\ln (1/r(n)))^{1/2},\quad 0