Is there a name for the function $(ax^2+by^2)/(ax+by)$

Question

Assume $x,y$ are non-negative reals and $a,b$ are positive real constants. Now define: $$g(x,y)=\frac{ax^2+by^2}{ax+by}$$ Does this have a common name? It acts somehow as a weighted average of $x$ and $y$ with $a$ and $b$ being the weights. For example

$g(x,y)=g(y,x)$(assuming $a=b$), $g(x,0)=x$, $g(kx,ky)=k g(x,y)$, $\min(x,y)\leq g(x,y) \leq \max(x,y)$, $g(x,x)=x$

This showed up as a rule to obtain an equivalent parameter in a dynamical system when merging two similar components, much like when you replace two resistors in parallel with one equivalent resistor.

Answer 1

You may consider this a Multivariate Rational Function [1].

Wolfram MathWorld:

A quotient of two polynomials P(z) and Q(z),

$$R(z)=\frac{P(z)}{Q(z)}$$

is called a rational function, or sometimes a rational polynomial function. More generally, if P and Q are polynomials in multiple variables, their quotient is called a (multivariate) rational function

Answer 2

$g(x,y)=m$ is equivalent to the equation for an ellipse centered at $(c,c)$ with vertical axis of length $\sqrt{c^2(1+R)}$ and horizontal axis of length $\sqrt{c^2(1+1/R)}$ where $R \equiv a/b$ and $c \equiv m/2$. Demonstrating with SymPy,

>>> from sympy import *
>>> from sympy.abc import a, b, m, x, y
>>> c = m/2
>>> R = a/b
>>> h = sqrt(c**2*(1+1/R))
>>> v = sqrt(c**2*(1+R))
>>> ell = Ellipse((c,c),h, v).equation(x,y)
>>> eq = (a*x**2+b*y**2)/(a*x+b*y)-m
>>> set(solve(eq, y)) == set(solve(ell, y))
True

The ellipse equation defines the same values for $y$ as those defined by $g(x,y) = m$.

This mean, for a given $(x,y)$ pair, is different than the arithmetic mean, $A(a,y) = (x+y)/2$. For a given arithmetic mean, the range of $x$ and $y$ are unbounded so (14,16) and (10,20) both have an arithmetic mean of 15 -- as do (-85, 115): there is no bound to values of $x$ and $y$ that have an average of 15. For g(x,y), the infinite line of possible values is wrapped up into an ellipse. If $a = b = 1$ then the the values of $(x,y)$ giving $g(x,y) = 15$ are on a circle centered at $(7.5,7.5)$ with a radius of $\sqrt{2(7.5)^2}$, e.g. 14 and either of $15/2 \pm sqrt(281)/2 \approx -0.88$ or $15.88$ have an "average" or 15. As indicated in the link given by @jjagmath, if $x$ and $y$ are positive, then the value of $m$ will be between $x$ and $y$ (and the average can be extended for multiple values, preserving the same property that the $m$ will be within the interval containing the max and min of the values being added together). But this property of the mean being withing the interval of values being "averaged" which is always true for the arithmetic mean does not hold when negative are included (as was demonstrated above).

Answer 3

This is the weighted version of the contraharmonic mean. The (unweighted) contraharmonic mean of two numbers $x$ and $y$ is $$C(x,y) =\frac{x^2+y^2}{x+y}$$ and as with other means (arithmetic, geometric, harmonic) one can assign weights to each term. In this case, assigning the weights $a$ and $b$ to $x$ and $y$ respectively you get $$\frac{a x^2+b y^2}{a x+b y}$$