Inequalities for concentration of a decomposition

For a measure $P$ defined on the $\sigma $-algebra $B$ of Borel sets of the real line with Lebesgue measure $L$, the concentration functions
$$ Q({P,z})=\sup_{x \in R}\mathbf{P}({[{x,x + z})}), \qquad \widehat Q({P,z})=\sup\{{\mathbf{P}(A):L(A)\le z,A\in\mathcal{B}}\} $$
and the concentration function of the decomposition $\widehat P$:
\begin{align*} \widehat P({[{-z,0})})&=\widehat P({({0,z}]})=(\widehat Q(P,2z)-\widehat Q(P,0))/2, \qquad z > 0, \\ \widehat P({\{0\}})&=\widehat Q({P,0}). \end{align*}
are introduced.It is proved that if the finite measures $P_k $ and $T_k $ satisfy $\widehat Q(P_k ,z) \le \widehat Q(T_k ,z), k = 1, \ldots ,n$, then $\widehat Q(P_1 * \cdots * P_n ,z) \le Q(\widehat P_1 * \cdots * \widehat P_n ,z) \le Q(\widehat T_1 * \cdots * \widehat T_n ,z)$.