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What's a simple, natural function that is "halfway" between addition and multiplication, in some sense? Can you think of a way to make that precise?

$$ x + y \quad\longrightarrow\quad ? \quad\longrightarrow\quad x \cdot y$$

Alternatively, what's a very natural curve in $\mathbb R^2$ between $xy=1$ and $x+y=1$?

In polar coordinates, these are $$r=\frac{1}{\cos\theta+\sin\theta}\implies r^2=\frac1{2\cos\theta\sin\theta+1}$$ and $r^2=\frac1{\cos\theta\sin\theta}$, so maybe: $r^2=\frac1{(3/2)\cos\theta\sin\theta+1/4}$ or $$\frac32 xy+\frac14(x^2+y^2)=1,\quad 6xy+x^2+y^2=4$$ $$(x+y)^2 + 4xy=4,\quad (x+y)^2 = 4(1-xy)$$

Blue
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Bjørn Kjos-Hanssen
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    There probably isn't one. Adding stuff is repetition of counting (or repeat use of the successor function); multiplying is repeated adding; exponentiating is repeated multiplication; tetrating is repeat exponentiation; and so on. Not that it necessarily prohibits one from making some sort of intermediary step through some form of generalization, though it's hard for me to guess what exactly we want the generalization to preserve... – PrincessEev Jan 03 '21 at 23:28
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    I think this question needs a bit more detail before it can be answered well. – Noah Schweber Jan 03 '21 at 23:30
  • There is something intermediate between $\frac{x+y}{2}$ and $\sqrt{xy}$ though which is $\operatorname{AGM}(x,y)$, but it seems uneasy to implicit draw level lines for this... – zwim Jan 03 '21 at 23:55
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    Alternately there is this discussion https://math.stackexchange.com/q/1269643/399263 – zwim Jan 04 '21 at 00:00
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    Also related. https://math.stackexchange.com/questions/1227761/example-x-y-and-z-values-for-x-uparrow-alpha-y-z-where-alpha-in-bbb –  Jan 04 '21 at 00:09
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    The "simplest" among your proposals would be $(1-t)(x+y)+txy=1$ with $t\in[0,1]$, but it artificially rejects the "negative" part of the curve at infinity. Approaches with tetration are most subtle. – zwim Jan 04 '21 at 00:30

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