Lower Bounds on Expressions Depending on the Functions φ(n), ψ(n) and σ(n): Part 4

In this article, we establish four lower-bound inequalities for rational expressions involving the Euler totient function φ(n), the Dedekind psi function ψ(n), and the sum-of-divisors function σ(n). The inequalities concern rational expressions involving sums of cubes, products of these arithmetic functions, and sums of pairs of these functions. The lower bounds are given explicitly as rational functions of n and are attained when n is prime. Furthermore, we formulate four conjectures that extend the established inequalities to arbitrary positive integer powers k.