Smith Numbers are Infinite in Number/Lemma

Theorem
Let $\map S m$ denote the sum of the digits of a positive integer $m$.

Let $\map {S_p} m$ denote the sum of the digits of the prime decomposition of $m$.

Let $\map N m$ denote the number of digits in $m$.

Suppose $m = p_1 p_2 \dots p_r$, where $p_i$ are prime numbers, not necessarily distinct.

Then $\map {S_p} m < 9 \map N m - 0.54 r$.