# Notation for bounds on derivative

I am working on a problem where the assumptions are that some derivatives are bounded. I want to refer to the individual bounds in the proof but there are about 7 of them in total. I am wondering if there is some standard notation that would prevent me from having to write something like $L_{1}, L_{2}, ... L_{7}$. For instance if they all happened to be functions of a single variable, something like: $\eta_{m}{f}$ for indicating the bound on $f^{(m)}$ seems reasonable, but I want to indicate the bounds on individual mixed partials, like $\frac{d^{2}f}{dx dw}$. Is there a standard, concise notation for this?

• If all you need is them being bounded by some bound, you can just take the maximum of your finite number of bounds, call it $L$ and use only that. Does that help? – Christoph Oct 11 '14 at 9:25
• I would use $L_x, L_{xx}, L_{xw}...$. Possibly also $L_{x^m}$. – Yves Daoust Oct 11 '14 at 9:51
• @Christoph that would be OK but my end result is an expression involving some of these quantities and I would like to give a more precise answer. – ttb Oct 11 '14 at 16:20
• @Yves This would be OK for one function, and I was thinking of perhaps for more functions using $L_{x}f, L_{w}g$ – ttb Oct 11 '14 at 16:22
• $f\rceil_x$, $f\rfloor_x$, $f>_x$, $f<_x$ ? – Yves Daoust Oct 11 '14 at 16:23

## 1 Answer

AFAIK there is no standard notation for this, but it is always a good idea to make notations as far as possible self-explaining, so instead of $L_1,\ L_2$,... I think that $L_x, L_y, L_{xw}$, etc, would be good idea. And yes, adding the concerned function as an additional "index" is a good idea, but take care to clarify whether, e.g., $L_x(f)$ is a bound for $f$ given somewhere explicitly, or whether it is a function of $f$. Often you would have different formulas or values for bounds for derivatives of one function and of another function, depending on what you know about them. (I.e., you would have, e.g., $|f_x| < L_x(f)$, $|g_x| < L_x(g)$, but $L_x$ would not be a function of $f$ in the strict sense, but just a notation for a bound given somewhere). It might be clearer, however, to use a different symbol in that case, e.g., $|g_x| < M_x(g)$, if $M_x(g)$ is given through a formula different from $L_x(f)$. Also make a distinction between $\sup|f_x|$ (which could in some cases also merit introduction of a dedicated notation = "abbreviation") and an ("arbitrary", not necessarily tight) upper bound $L_x(f)$.

• Is it clear what I mean by "function of $f$"? E.g., you may find $\sup|\partial_x f| < L_x(f) := 5 \,|f(0)|$ for a function $f$, but the formula may not hold for another function $g$ having other properties, with $5 g(0)$ on the r.h.s.. You would e.g. find, for the different function $g$, $\sup|\partial_x g|<L_x(g):=2\,|g'(0)|^2$. That's even more obvious if you get just a numerical bound, e.g., $L_x(f):=0.5$ for some $f$. There is no dependency on $f$ in the r.h.s. so you must expect that for another function it will be a different value, and the r.h.s. doesn't define a function $L_x$ of $f$. – Max Nov 6 '17 at 13:46