Deducing a similar derivation which is explained in the Wikipedia Stolarsky mean I define for real numbers $1\leq x<y$ the mean $$M(x,y)=\frac{1}{e}\left(\frac{x-y}{e^{W(x)}-e^{W(y)}}-1\right)\exp\left(\frac{x-y}{e^{W(x)}-e^{W(y)}}\right)\tag{1}$$ where $W(x)$ denotes the main/principal branch of the Lambert $W$ function. As usual the definition for $x=y$ will be $M(x,y)=x$.
I've used the function $f(x)=e^{W(x)}$ in the derivation, and calculating the inverse with the help of Wolfram Alpha online calculator.
My following question arises (I am inspired in this information) in a comment that added a professor in my own (now deleted post) MathOverflow 357442 with title A mean built on the principal branch of the Lambert $W$ function: remarkable relationships with other means (asked yesterday Apr 15 '20). I don't know if previous mean is in the literature, if you know it I would like to know about it, in this case please add a comment.
Question. I would like to know if it is possible to deduce a bound $\text{bound}(x,y)$ satisfying for any real numbers $1\leq x<y$ that $$0<\text{bound}(x,y)< M(x,y)-M_{\text{lm}}(x,y),\tag{2}$$ where $M_{\text{lm}}(x,y)$ denotes the logarithmic mean. Many thanks.
If you need it you've the Wikipedia articles Lambert $W$ function and Logarithmic mean.
Remarks as motivation. I think that this exercise can be interesting because I can to learn this technique to get refinements for inequalities. If you are not able to do it for the segment of real numbers $x$ and $y$ with $1\leq x<y$ but you are able to find a more sharp inequality $(2)$ that holds for the segment $B<x<y$, for a fixed and positive real constant $B$ (I evoke a suitable choice of $B>1$), feel free to add your post as an answer. In recent past years in the literature are arising, in my opinion, articles that explore the importance of the function $f(x)=e^{W(x)}$, I wanted to create a mean inspired in some branch of the Lambert $W$ function.
References:
[1] Kenneth B. Stolarsky, Generalizations of the logarithmic mean, Mathematics Magazine 48: pp. 87–92, (1975).
[2] The article Lambert W-Function from the online encyclopedia Wolfram MathWorld and its related bibliography.