Of course, the classical inequality between means contains arbitrarily long chains of inequalities. Accordingly, my proposed question WOULD BE CONCERNED TO THE CLASSES OF CHAINS OF ANALYTIC INEQUALITIES WHICH ARE NOT KNOWN IN RELATED REFFERED LITERATURE!!!

For two distinct positive real numbers $x$ and $y$, let $H$, $G$, $A$ and $Q$ be the harmonic mean, the geometric mean, the arithmetic mean and the quadratic mean of $x$ and $y$, respectively. By using routine calculations, it can be proved the following curious chain containing 9 inequalities:

$$

\sqrt{AH}

Dear Romeo:

The Question is challenging and technical for anyone other than experts in the specialized field.

A further Question rather than Answer: Can this be extended to higher moments?

Dear Chia-Lin,

Thank you very much for your interesting, intriguing and useful observations on my question and related possible answers. I'm sure that your idea on considerations of higher moments (i.e., means of higher order) would lead to many very long (nontrivial) chains of inequalities. (There is an interesting inequality between the first four moments of any random variable given by Theorem 1 in the paper available at https://arxiv.org/pdf/1111.6220.pdf; Iosif Pinelis “Relations between the first four moments”, American Mathematical Monthly, Vol. 122, No. 5 (2015), 479-481, https://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.479?seq=1#page_scan_tab_contents).

Best regards

To Romeo:

Please excuse an attempt at a serious Answer by a non-expert to your intriguing and challenging Question that is intended for experts.

Your refer to 4 means, namely the first moments.

Are these analysed in a univariate setting of individual means?

Can the analysis be extended to a multivariate setting of any 2, 3 or all 4 means simultaneously?

Can the median and/or the mode be included in the analysis, or is the non-trivial chain of inequalities intended only for analytic means and not for non-parametric moments?

Dear Romeo:

Are there any possible or likely practical applications of a non-trivial chain of inequalities?

Dear Chia-Lin and Michael John,

My question is entirely theoretical and it is addressed to mathematical experts on Inequalities. Formally, every (not necessary analytic) inequality can be considered as a chain of containing one inequality. Some interesting (and I believe useful in Statistics with applications) inequalities involving the median and/or mode can be found at links given below.

http://www.mathnet.ru/links/92f154e92fb08ba0a475abdbcfe6707a/tvp2942.pdf

https://www.researchgate.net/publication/23564935_The_mean-median-mode_inequality_Counterexamples

http://www.larrylisblog.net/WebContents/Interview%20Preparations/medianMeanInequality.pdf

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.596.6346&rep=rep1&type=pdf

http://www.fao.org/docs/up/easypol/448/simple_inequality_mesures_080en.pdf

https://www.researchgate.net/publication/254330359_The_Mode_Median_and_Mean_Inequality (fultext unavailable at RG)

https://books.google.me/books?id=xaH8AQAAQBAJ&pg=PA50&lpg=PA50&dq=%22inequality+for+median+and+mode&source=bl&ots=eubT_e2g4O&sig=ACfU3U2bBo9MQhd7IrS1bSRU3cSmithBAA&hl=en&sa=X&ved=2ahUKEwj8psDujqXiAhUhx4sKHc9iD7k4HhDoATAIegQICRAB#v=onepage&q=%22inequality%20for%20median%20and%20mode&f=false

https://www.tib.eu/en/search/id/BLSE%3ARN080294876/The-mean-median-mode-inequality-for-discrete-unimodal/

https://experts.umn.edu/en/publications/the-mode-median-and-mean-inequality

https://math.stackexchange.com/questions/517929/concentration-inequality-for-the-median

https://www.amazon.com/mean-median-mode-inequality-distributions-Institute-Statistics/dp/B000713DZ6

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-842X.1981.tb00784.x

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/59636/1/0916-6.pdf

https://www.tandfonline.com/doi/abs/10.1080/00031305.1977.10479215

https://www.jstor.org/stable/3533476?seq=1#page_scan_tab_contents

https://epubs.siam.org/doi/abs/10.1137/S0040585X97975447?journalCode=tprbau

https://www.stat.berkeley.edu/~aldous/Unpub/13-00698.pdf

To Romeo:

Many thanks for your incredibly detailed Answer, that I would not be able to decipher if I had several lifetimes.

I wonder if you might be willing to write a brief review of this well-defined literature for journal publication.

I have several respectable outlets in mind in financial economics and risk management.