Summary: For a measure \(P\) defined on the \(\sigma\)-algebra \(B\) of Borel sets of the real line with Lebesgue measure \(L\), the concentration functions \[ Q(P,z) = \sup_{x \in R} {\mathbf P} ([x,x + z)),\qquad \widehat{Q}(P,z) = \sup\{ {\mathbf P}(A): L(A) \leq z,\;A \in {\mathcal B}\} \] and the concentration function of the decomposition \(\widehat{P}\): \[ \widehat{P} ([-z,0)) = \widehat{P}((0,z]) = (\widehat{Q} (P,2z) - \widehat{Q}(P,0))/2,\;z > 0, \qquad \widehat{P}(\{0\}) = \widehat{Q}(P,0) \] are introduced. It is proved that if the finite measures \(P_ k\) and \(T_ k\) satisfy \(\widehat{Q}(P_ k,z) \leq \widehat{Q}(T_ k, z)\), \(k = 1,\dots,n\), then \[ \widehat{Q}(P_ 1 * \cdots * P_ n, z) \leq Q(\widehat{P}_ 1 * \cdots * \widehat{P}_ n,z) \leq Q(\widehat{T}_ 1 * \cdots * \widehat{T}_ n, z). \]