To ask this question I was inspired in some words, if I understand well, from the authors of a preprint on arXiv in section 4.1, that I believe that is [1], to ask next question.
We consider the sequence of functions defined on $\mathbb{R}$ as $$Q_N(x)=\frac{\sum_{n=1}^{N+1}(a_n)^{f(x)}}{\sum_{n=1}^{N}(a_n)^{f(x)}},\tag{1}$$ where the terms the sequence $(a_n)_{n=1}^\infty\neq\text{ a constant}$ or if you prefer work for the sequence, also under previous requirements $$\hat{Q}_N(x)=\frac{\sum_{n=1}^{N+1}b_n\cdot(a_n)^{f(x)}}{\sum_{n=1}^{N}b_n\cdot(a_n)^{f(x)}},\tag{2}$$ for an available sequence $(b_n)_{n=1}^\infty$ used in the purpose to define our sequence $\hat{Q}_N(x)$.
Question. Can you show an example or set a conjecture of a sequence $Q_N(x)$, or if your choice is $\hat{Q}_N(x)$, such that $\forall N\geq 1$ our functions $Q_N(x)$, or $\hat{Q}_N(x)$ if you prefer, is log-convex (in $x$) on $\mathbb{R}$? Many thanks.
Also you can take the function $f(x)$ as you need it. I am looking an example different than was conjectured/evoked by the authors of [1]. If these examples that I evoke are in the literature, refere it and I try to search and read those statements from the literature. I am asking about sequences with good mathematical content and and not trivial from the choice of sequences $(a_n)_{n=1}^{\infty}$ or $(b_n)_{n=1}^\infty$, of the function $f(x)$. Thus a remarkable example or conjecture in the spirit that show the authors of [1].
I know the definition of what is a log-convex, and equivalent conditions, for example from the Wikipedia Logarithmically convex function.
References:
[1] J. Arias de Reyna and J. van de Lune, A proof of a trigonometric inequality. A glimpse inside the mathematical kitchen, J. Math. Inequal., Vol 5(3), 2011.
