In this post we denote the radical of an integer $n>1$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p$$ with the definition $\operatorname{rad}(1)=1$. The abc conjecture is an important problem in mathematics as you can see from the Wikipedia abc conjecture. In this post I mean the formulation ABC conjecture II stated in previous link.
Then it is easy to check using the inequality between the arithmetic and logarithmic means, that the abc conjecture implies that next Conjecture is true. I refer the Wikipedia Logarithmic mean, in case that you don't know this mean and the inequality that I refer.
Conjecture. For every real number $\varepsilon>0$, there exists a positive constant $\mu(\varepsilon)$ such that for all pairs $(a,b)$ of coprime positive integers $1\leq a<b$ $$2\,\frac{b-a}{\log\left(\frac{b}{a}\right)}\leq \mu(\varepsilon)\operatorname{rad}(ab(a+b))^{1+\varepsilon}\tag{1}$$ holds.
Question. I wondered what work can be done to prove/discuss unconditionally the veracity of previous Conjecture, since I evoke that this statement that involves the inequality $(1)$ is much weaker than the abc conjecture. Many thanks.
I don't know if this is obvious (if one can to get an answer easily for my question), I tried to get some idea from few and very simple experiments using a Pari/GP script. I don't know if this Conjecture is in the literature, if you know it from the literature feel free to answer this question as a reference request (or, please add a comment with your reference) and I try to search and read it from the literature.
Now this post is cross-posted on MathOverflow as On weaker forms of the abc conjecture from the theory of Hölder and logarithmic means, the MO 359706 post.
References:
[1] Andrew Granville and Thomas J. Tucker, It’s As Easy As abc, Notices of the AMS, Volume 49, Number 10 (November 2002).
[2] B. C. Carlson, Some inequalities for hypergeometric functions, Proc. Amer. Math. Soc., 17: in page 36 (1966).
$a+b\ge 2 (b-a)/log(b/a)$ does nothing for the right hand side of your new conjecture.
So, it is easy to show that the resulting quality analogue, q', in the new conjecture suffers the same fate as the original. Namely, that the $limsup(q') =1$ not $1+\epsilon$.
See abc Triples as a reference.