Lower bounds on expressions dependent on functions φ(n), ψ(n) and σ(n)

Abstract

The inequalities \varphi^2(n)+\psi^2(n)+\sigma^2(n) \geq 3n^2+2n+3 , \varphi(n)\psi(n)+\varphi(n)\sigma(n)+\sigma(n)\psi(n) \geq 3n^2+2n-1 connecting \varphi(n), \psi(n) and \sigma(n)-functions are formulated and proved.