On inequalities involving counts of the prime factors of an odd perfect number

Let \(N\) be an odd perfect number. Let \(\omega(N)\) be the number of distinct prime factors of \(N\) and let \(\Omega(N)\) be the total number (counting multiplicity) of prime factors of \(N\). We prove that \(\frac{99}{37}\omega(N) - \frac{187}{37} \leq \Omega(N)\) and that if \(3\nmid N\), then \(\frac{51}{19}\omega(N)-\frac{46}{19} \leq \Omega(N)\).