I wondered about a weak form of the abc conjecture, see the Wikipedia abc conjecture using the theory of generalized means, I mean this Wikipedia Generalized mean. We get the following claim, where $\operatorname{rad}(n)$ denotes the product of distinct primes dividing an integer $n>1$ with the definition $\operatorname{rad}(1)=1$.
Claim (Edited). On assumption of the abc conjecture $\forall \varepsilon>0$ there exists a constant $\mu(\epsilon)>0$ such that for triples of positive integers $a,b,c\geq 1$ satisfying $$\gcd(a,b)=\gcd(a,c)=\gcd(b,c)=1\tag{1}$$ and $$a+b=c\tag{2}$$ ones has $$c<\mu(\varepsilon)\left(\frac{\operatorname{rad}(a)^q+\operatorname{rad}(b)^q+\operatorname{rad}(c)^q}{3}\right)^{3(1+\varepsilon)/q},\tag{3}$$ where $q>0$.
Then from our construction we get/recover the formulation ABC conjecture II when $q\to0^{+}$ as a consequence of the theory of generalized means or Hölder mean.
Question. I would like to know what work can be done about the veracity (thus unconditionally) of the inequality $(3)$ for the given conditions $(1)$ and $(2)$, $\forall\varepsilon>0$ as in the formulation of the abc conjecture, for the smallest real number $q>0$ that we can evoke. Many thanks.
To emphasize we are interesting in the smallest real number $q>0$ for which our weak form of the abc conjecture $(3)$ is true. I don't know if this kind of inequalities are in the literature.
Please if you want, add also concise feedback in a paragraph as companion of your answer for my Question or in comments about if previous claim can be potentially interesting.
I've added below references, on other hand now this post is cross-posted on MathOverflow On weaker forms of the abc conjecture from the theory of Hölder and logarithmic means as the MO 359706 post.
References:
[1] P. S. Bullen, Handbook of Means and Their Inequalities, Dordrecht, Netherlands: Kluwer (2003).
[2] Andrew Granville and Thomas J. Tucker, It’s As Easy As abc, Notices of the AMS, Volume 49, Number 10 (November 2002).