Little-o properties in two variable functions

Question

I am studying Taylor polynomials with Peano remainder in my Multivariable course, and I am trying to understand how little-o works in two variables. However, I am having difficulty with a few concepts in this area, and would like to ask for not only clarification, but if possible some literature references. So, here is what I know:

As I wrote in this question, we introduced the Taylor-MacLaurin polynomial of degree 2 of a generic two-variable real function as follows: $$ f(x,y) = f(0, 0) + f_x (0, 0) x + f_y (0, 0) y + \frac{1}{2}\left[ f_{xx} (0,0)x^2 + 2f_{xy} (0, 0)xy + f_{yy} (0, 0) y^2 \right] + o \left(x^2 +y^2 \right) $$ where $o \left(x^2 +y^2 \right)$ is the Peano's form of remainder of this function. By definition, $f(x, y) = o(g(x, y))$ ["$f(x,y)$ is little-o of $g(x)$"] if $$ \lim_{(x, y) \to (0,0)} \frac{f(x, y)}{g(x, y)} = 0 $$ Now, for single-variable functions there are some properties concerning the little-o notation [Taken from: Marcellini, Paolo; Sbordone, Carlo, Analisi Matematica Uno. Napoli: Liguori, 1998, ISBN: 88-207-2819-2, p. 373] $$ \begin{align*} o(x^n) + o(x^n) &= o(x^n) \\ c \cdot o(x^n) &= o(cx^n) = o(x^n) \\ o(x^n) - o(x^n) &= o(x^n) \\ x^m \cdot o(x^n) &= o(x^{m+n}) \\ o(x^m) \cdot o(x^n) &= o(x^{m+n}) \\ o(o(x^n)) &= o(x^n) \\ o(x^n + o(x^n)) &= o(x^n) \end{align*} $$ where $m, n \in \mathbb{N}$ and $c$ is a non-negative constant.

My question is: are there any similar properties concerning $o((x^2 + y^2)^\alpha)$, where $\alpha > 0$? I tried to look it up on the net and only found this thread from an Italian mathematics forum which (from what I've interpreted) says that there is no such form due to the nature of limits of functions of two variables, so the definition must be used.

Now, here's the thing: our lecturer, to "better" explain how some things work, gave us a series of equations, such as: $$ x^3 y = o(x^2 + y^2) $$ She told us this holds true because, and I quote, "it's like saying $\rho^4 = o(\rho^2)$". Hence I thought that in some cases it is possible to use the properties involving little-os in one variable in combination with polar coordinates, where $\rho = \sqrt{x^2 + y^2}$. I've tried finding more information about it, with no avail.

I am therefore asking for your help in clarifying this issue. If you can, I would also be grateful if you could provide me with some useful, reliable sources from which I can draw this information and explore the topic further.

I thank you in advance for your attention.

Answer 1

Let $L=\frac{x^{2\alpha}+y^{2\alpha}}{(x^2+y^2)^\alpha}$. Trivially, ($x=0\ \text{or}\ y=0\ \text{or}\ \alpha=1)\implies L=1$; $x^2=y^2\implies L=2^{1-\alpha}$
Now for $(x\not=0) \wedge (y\not=0) \wedge (x^2\not=y^2)\wedge(\alpha\not=1)$ , wlog assume $x,y>0\ ;\ x^2<y^2$, let $\beta=\frac{x^2}{y^2}$ ($\beta\in(0,1)$): $$L=\frac{x^{2\alpha}+y^{2\alpha}}{(x^2+y^2)^\alpha}=\frac{1+\beta^\alpha}{(1+\beta)^{\alpha}}\in \begin{cases}(1,2^{1-\alpha}),\alpha\in(0,1)\\(2^{1-\alpha},1),\alpha\in(1,\infty)\end{cases}$$ In all cases: $0\ll L\ll\infty$ , this implies $o((x^2+y^2)^{\alpha})=o(x^{2\alpha}+y^{2\alpha})$
This can be generalized to: $o\left((\sum x_i^{k_i})^\alpha\right)=o\left(\sum x_i^{\alpha k_i}\right)$

And fyi, perhaps just recently, we do know a little more about the nature of limits of multivariable functions. See my question on MSE: here and Pinelis's question on MO: here .
There are some distinctive creatures like $o(x^3+xy+y^3)$