We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ with the definition $\operatorname{rad}(1)=1$. You can see this definition and the properties of this arithmetic function for example from this Wikipedia.

I know that there is literature about inequalities concerning particular values of the Euler's totient function $\varphi(m)$, say us comparisons of the type $$\varphi(an+b)>\varphi(cn+d).$$

Question.I'm curious about what can be interesting comparisons of particular values of the square-free kernel, say us $$\operatorname{rad}(an+b),$$ where $a,b$ are simple cases of integers, and the quantity $$\varphi(cn+d),$$ with $c,d$ also simple integers, by means of an inequality $>$ or well $<$.Many thanks

Thus I am asking about the inequalities that I've evoked: I am asking about an inequality/comparison $$\operatorname{rad}(an+b)<\varphi(cn+d)$$ or well $$\varphi(cn+d)<\operatorname{rad}(an+b),$$ for a simple choice of interegers $a,b,c$ and $d$, with a good mathematical content.

Of course if to make an asymptotic comparison for sufficiently large $n'$s you need to introduce in RHS terms like $$\text{the first arithmetic function}<\underbrace{A(n)\cdot\text{the second arithmetic function}+\text{error term}}_{\text{this RHS involves particular values of the second arithmetic function}},$$ being $A(n)$ a function (or a constant), feel free to do such comparison.