In 1994, Talagrand showed a generalization of the celebrated KKL theorem. In this work, we prove that the converse of this generalization also holds. Namely, for any sequence of numbers $0<a_1,a_2,ldots,a_nle 1$ such that $sum_{j=1}^n a_j/(1-log a_j)ge C$ for some constant $C>0$, it is possible to find a roughly balanced Boolean function $f$ such that $textrm{Inf}_j[f] < a_j$ for every $1 le j le n$.
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