Let ϕ(n) be the Euler totient function of n, defined as the number of positive integers less than or equal to n that are co-prime with n. In this paper, we consider the function ϕ_k, a generalization of ϕ, and establish some inequalities related to Lucas sequences of the first kind (U_n)_n≥1 with characteristic equation having real roots. As an application to these inequalities, we further establish inequalities related to Fibonacci, Pell, and balancing sequences.