This post is cross posted from Mathematics Stack Exchange, due that there was a mistake from my part (see the excellent partial answer and my thread of edits of my question on MSE) this post on MathOverflow is slightly different of On variations of a claim due to Kaneko in terms of Lehmer means asked 30 days ago (May 13) as the question with identificator 3672588 of Mathematics Stack Exchange.
For a tuple of positive real numbers $\mathbb{x}=(x_1,x_2,\ldots,x_n)$ we denote its corresponding Lehmer mean as $L_q(\mathbb{x})$, where $-1\lt q\lt 0$, I know the definition of these means from the corresponding Wikipedia with title Lehmer mean..
Also we denote the sum of divisors function as $\sigma(n)=\sum_{1\leq d\mid n}d$ for integers $n\geq 1$.
The idea of the post was to combine this definition of Lehmer mean with an equivalent formulation of the Riemann hypothesis, I refer the last paragraph of [1] (Kaneko's claim for a suitable choice of the integer $n_0$ from which the inequality in Kaneko's claim is true $\forall n\geq n_0$).
From here, my belief that there should be an integer $n_0>1$ such that $\forall n\geq n_0$ the following inequality holds $$\sigma(n)<\exp\left(\frac{n}{L_q(1,\ldots,n)}\right)\log\left(\frac{n}{L_q(1,\ldots,n)}\right)\tag{1}$$ specifically for real numbers $-1\lt q\lt 0$.
Fact. We've from the theory of Lehmer mean that we recover the inequality of Kaneko as $q$ tends to $0^{-}$.
Sketch of proof. From the section of Properties and Special cases of the linked Wikipedia, and the continuity of a product of continuous functions (our exponential and logarithm).$\square$
Question. I would like to know what work can be done to prove an inequality of the form $(1)$ for a specific value of $-1<q<0$, the least/smallest value of $|q|$ for which it is possible to prove that your inequality is true (I mean very close to $0$), and that holds $\forall n\geq n_0$ for a suitable choice of $n_0>1$ for your corresponding choice of a specific value of $-1<q<0$.
Many thanks.
I emphasize that I'm asking what work can be done to prove an example for one of those inequalities $(1)$ for a very small quantity $|q|>0$ with $-1<q<0$ and that is true, our inequality, as a relaxation of Kaneko's claim for a segment of integers $n_0<n$. Feel free to add your feedback about if this type of inequalities (my interpretation of a relaxation or variation of Kaneko's claim) and combinations can be potentially interesting.
References:
[1] Jeffrey C. Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis, The American Mathematical Monthly, 109, No. 6 (2002), pp. 534-543.
[2] P. S. Bullen, Handbook of Means and Their Inequalities, Springer, (1987).
