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In this post we denote the Lehmer mean of a tuple $\text{x}$ of positive real numbers as $$L_p(\text{x})={\sum_{k=1}^nx_k^p\over\sum_{k=1}^nx_k^{p-1}},$$ see the reference Wikipedia Lehmer mean.

The idea of this post arises when I've consider the inequality $L_p(a,b,a+b)<a+b$ whenever $p$ is enough large, and the limit $\lim_{p\to\infty}L_p(a,b,a+b)=a+b$, where $L_p(\text{x})$ denotes the Lehmer mean for the tuple of positive integers $\text{x}=(a,b,a+b)$ in the context of variants of the abc conjecture. We use in this post the formulation ABC conjecture II from the Wikipedia abc conjecture, thus as usual we denote $$\operatorname{rad}(n)=\prod_{\substack{p\text{ prime}\\p\mid n}}p$$ for integers $n>1$ with the definition $\operatorname{rad}(1)=1$. From this scenario we've the following obvious claim.

Claim. On assumption of the abc conjecture, for all $\varepsilon>0$ and pairwise coprime integers $1\leq a$, $1\leq b$, we assume $a<b$, and such that $c=a+b$, there exists a positive constant $\mu(\varepsilon)$ such that $$L_p(a,b,a+b)<c<\mu(\varepsilon)\operatorname{rad}(ab(a+b))^{1+\varepsilon}\tag{1}$$ where we consider here $p\geq 2$ integer. Additionally we know $\lim_{p\to\infty} L_p(a,b,a+b)=c$.

Question. I would like to know if for each large enough integer $p\geq 2$ and, thus after this choice of a $p$ sufficiently large, for all $\varepsilon>0$ there exists a positive constant $\mu(\varepsilon,p)$ (thus that depends on our fixed integer $p$) such that $$\frac{a^p+b^p+(a+b)^p}{a^{p-1}+b^{p-1}+(a+b)^{p-1}}<\mu(\varepsilon,p)\operatorname{rad}(ab(a+b))^{1+\varepsilon}\tag{2}$$ for all those positive integers $1\leq a<b$ such that $\gcd(a,b)=1$. Many thanks.

I'm asking about what work can be done about it unconditionally, computations or reasonings with the intention to accept a suitable answer as soon is possible. I don't know how approach this interesting variant of the abc conjecture, I don't know if there is a relationship between the $\mu(\varepsilon,p)$ in $(2)$ and the $\mu(\varepsilon)$, in $(1)$, of the abc conjecture as $p\to\infty$.

I add the reference [2] where is added also the Oesterlé and Masser’s abc-conjecture, and subsequent references.

References:

[1] P. S. Bullen, Handbook of means and their inequalities, Springer (1987).

[2] Andrew Granville and Thomas J. Tucker, It’s As Easy As abc, Notices of the AMS, Volume 49, Number 10 (2002).

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  • $\begingroup$ (1/2) In the spirit to explore inequalities involving $\operatorname{rad}(n)$ explained in the slide at minute 21' from the video of YouTube What is the abc conjecture? by Keith Conrad, lecture notes from the official channel UConn Mathematics (date March, 23th 2014) I tried to study the case $p=2$, the contraharmonic mean, with the script written in Pari/GP that you can evaluate from the web Sage Cell Server (choosing as language GP) for(a=1, 100, for(b=1,100,if(a<b&&gcd(a,b)==1&&(a^2+b^2+a*b)/(a+b)>factorback(factorint(a*b*(a+b))[, 1])^(1),print(a," ",b)))) $\endgroup$
    – user142929
    Jan 23, 2020 at 12:21
  • $\begingroup$ (2/2) As you see I've encoded explicitly the exponent $1$ in the expression $\operatorname{rad}(ab(a+b))^1$. The code for the radical of an integer $\operatorname{rad}(n)$ is due to Andrew Lelechenko (May, 09th 2014) as you can see from the section PROG of the article dedicated to the sequence A007947 from the On-Line Encyclopedia of Integer Sequences. $\endgroup$
    – user142929
    Jan 23, 2020 at 12:21
  • $\begingroup$ Many thanks for your help @GerryMyerson $\endgroup$
    – user142929
    Jan 23, 2020 at 21:18
  • $\begingroup$ All, I've edited two posts on Mathematics Stack Exchange with titles and identificators, respectively, Weaker than abc conjecture invoking the inequality between the arithmetic and logarithmic means (MSE 3580506, Mar 14' 20) and A weak form of the abc conjecture involving the definition of Hölder mean (MSE 3648776, asked yesterday Apr 29' 20). Please, feel free if you or some of yours colleagues (professors studying the abc conjecture) want provide feedback about if this kind of inequalities of this MO or MSE posts are interesting in comments or as companion of your answer(s). $\endgroup$
    – user142929
    Apr 30, 2020 at 18:25

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