Composition of Arithmetical functions with generalization of perfect and related numbers

In this paper we have studied the deficient and abundent numbers connected with the composition of $\phi$, ${\phi }^{\ast }$, $\sigma$, ${\sigma }^{\ast }$ and $\psi$ arithmetical functions, where $\phi$ is Euler totient, ${\phi }^{\ast }$ is unitary totient, $\sigma$ is sum of divisor, ${\sigma }^{\ast }$ is unitary sum of divisor and $\psi$ is Dedekind's function. In 1988, J. Sandor conjectured that $\psi (\phi (m))\ge m$, for all $m$, all odd $m$ and proved that this conjecture is equivalent to $\psi (\phi (m))\ge \frac{m}{2}$, we have studied this equivalent conjecture. Further, a necessary and sufficient conditions of primitivity for unitary r-deficient numbers and unitary totient r-deficient numbers have been obtained. We have discussed the generalization of perfect numbers for an arithmetical function $E_{\alpha }$.