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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).

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  • $\begingroup$ The scripts that I refer are similar than for(a=1, 1000, for(b=1, 1000, if(a<b&&gcd(a,b)==1&&2*(b-a)/log(b/a)>factorback(factorint(a*b*(a+b))[, 1]),print(a)))) that you can evaluate in the web Sage Cell Server choosing as Language GP. The code for the radical of an integer (see also Wikipedia Radical of an integer) is due to Andrew Lelechenko that posted it in the section PROG (May, 9th 2014) of the sequence A007947 in The On-Line Encyclopedia of Integer Sequences. $\endgroup$
    – user759001
    Mar 14 '20 at 8:56
  • $\begingroup$ I am waiting your feedback about my question, many thanks. $\endgroup$
    – user759001
    Mar 14 '20 at 8:58
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$a+b\ge 2 (b-a)/log(b/a)$ 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.

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  • $\begingroup$ Many thanks much for your answer. $\endgroup$
    – user759001
    Apr 6 '20 at 6:53
  • $\begingroup$ Again, thanks. I mark as favorite the post, and I am going to try to continue to study in the future the post. My knowledges about different formulations of the abc conjecture aren't the best. Feel free to study variants in the same way that I tried if isn't in the literature. What I evoke is try to build weak versions of the abc conjecture invoking inequatilies between (different) generalized means. Isn't required a response, and good day. $\endgroup$
    – user759001
    Apr 6 '20 at 10:50

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