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).

What is the abc conjecture?by Keith Conrad, lecture notes from the official channelUConn Mathematics(date March, 23th 2014) I tried to study the case $p=2$, thecontraharmonic mean, with the script written in Pari/GP that you can evaluate from the web Sage Cell Server (choosing as languageGP)`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))))`

$\endgroup$PROGof the article dedicated to the sequenceA007947from theOn-Line Encyclopedia of Integer Sequences. $\endgroup$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(MSE3580506, Mar 14' 20) andA weak form of the abc conjecture involving the definition of Hölder mean(MSE3648776, 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). $\endgroup$