Instead of expanding to the class of all analytic functions (where the asymptotics can be hard to get control over, due to oscillatory behavior), my inclination would be to focus on large classes of functions with well-behaved asymptotics, including all the functions that arise in ordinary asymptotic analysis. The usual buzz phrase for this is "Hardy field" (mentioned for example in my answer herehere), which by definition is an ordered field of germs at infinity of $C^\infty$ functions.
Instead of expanding to the class of all analytic functions (where the asymptotics can be hard to get control over, due to oscillatory behavior), my inclination would be to focus on large classes of functions with well-behaved asymptotics, including all the functions that arise in ordinary asymptotic analysis. The usual buzz phrase for this is "Hardy field" (mentioned for example in my answer here), which by definition is an ordered field of germs at infinity of $C^\infty$ functions.
Instead of expanding to the class of all analytic functions (where the asymptotics can be hard to get control over, due to oscillatory behavior), my inclination would be to focus on large classes of functions with well-behaved asymptotics, including all the functions that arise in ordinary asymptotic analysis. The usual buzz phrase for this is "Hardy field" (mentioned for example in my answer here), which by definition is an ordered field of germs at infinity of $C^\infty$ functions.
Note: This is a major rewrite of my earlier answer, to include necessary and sufficient conditions applicable to an even wider class of functions.
Instead of expanding to the class of all analytic functions (where the asymptotics can be hard to get control over, due to oscillatory behavior), my inclination would be to focus on large classes of functions with well-behaved asymptotics, including all the functions that arise in ordinary asymptotic analysis. The usual buzz phrase for this is "Hardy field" (mentioned for example in my answer here), which by definition is an ordered field of germs at infinity of $C^\infty$ functions.
Let us considerI will describe several classes of such functions. The first is the class of all functions which are first-order definable in the structure $(\mathbb{R}, 0, 1, +, \cdot, <, \exp)$ (if you like, add in$(\mathbb{R}, +, \cdot, <, \exp)$ together with all real numbers adjoined as constants). This type of structure arisesclass contains all functions that are constructible from polynomials, $\exp$, $\log$ and closed under the usual arithmetic operations and composition. It thus contains all the functions that usually arise in asymptotic analysis, and many more besides. This class enjoys the following strong model-theoretic property (as developed more fully in the theory of o-minimal structures, and has some rather remarkable properties. For example,):
(O) The zero set of any function $F: [a, \infty) \to \mathbb{R}$ in this class is a finite union of points and intervals (finite or infinite in extent).
Condition O ensures that every such function in this class$F$ is either eventually positive ($F(x) > 0$ for all sufficiently large $x$), eventually zero, or eventually negative. As a result, the ring of germs at infinity of the definable functions in this class forms an ordered field, i.e., a Hardy field. This Hardy field is in fact a real closedHardy field.
Also, if $F$ is definable, then $F'$ is also first-order definable (and its domain can be shown to be the domain of definition contains all sufficiently large reals$F$ save for finitely many points). Finally, these functions include all functions constructible from polynomials,Applying condition O to $\exp$$F'$, every definable $\log$ using the usual arithmetic operations and composition, and so includes all functions that arise$F$ in ordinary asymptotic analysisthis class is either eventually increasing, eventually decreasing, or eventually constant.
Now let us apply Willie Wong's necessary condition (that $x^n = O(F(x))$ and $F(x) = O(x^N)$ for someProposition: A function $n, N$) to$F$ in this class of functions. I claim this is a sufficient condition forsatisfies (*) provided that$F(x) = O(xF'(x))$ and $xF'(x) = O(F(x))$ if and only if there exist $n, N$ are either, both positive or both negative, for which $x^n < |F(x)| < x^N$ for all sufficiently large $x$.
Proof: WLOG we may assume $F$ is eventually positive, and is not eventually constant. Suppose thatThus $F$ is either eventually increasing or eventually decreasing, say eventually increasing. If $F$ is eventually bounded above by some $x^n \leq F(x) \leq x^N$ where$x^N$, $n, N$ are both positive. By adjusting the exponents$N > 0$, then by increasing $N$ if necessary, we may assume that $F(x)/x^n$ and $x^N/F(x)$ tend$F(x)/x^N$ tends to infinity. Thuszero, these functions are eventually increasing (their derivatives arewhence it is eventually positive)decreasing. By a simpleTaking the derivative calculation one finds, we conclude that eventually
$$0 < nF(x) < xF'(x) \qquad 0 < xF'(x) < NF(x)$$$$x^N F'(x) - Nx^{N-1}F(x) < 0$$
andwhence $xF'(x) < NF(x)$, i.e., $|xF'(x)| < N|F(x)|$ or $xF'(x) = O(F(x))$. However, if $F(x)$ is eventually bounded above by every positive-power function $x^N$ (think $N$ small!), this also shows $xF'(x) = o(F(x))$, so that $F(x)$ is not $O(xF'(x))$.
By a similar argument, if $F(x)$ is eventually bounded below by some positive-power function $x^n$, we get condition$nF(x) < xF'(x)$ eventually, so that $F(x) = O(xF'(x))$. This also shows that if $F(x)$ is bounded below by every positive-power function (*think $n$ large!), then $F(x) = o(xF'(x))$, so $xF'(x)$ is not $O(F(x))$. The case where
Thus, if $n, N$ are both negative$F(x)$ is positive and eventually increasing, a necessary and sufficient condition that $F'$ satisfy condition ($\ast$) in the question is that there exist two positive-power functions that $F$ is eventually squeezed between. An entirely similar exceptanalysis shows that we have insteadif $F(x)$ is positive and eventually decreasing, a necessary and sufficient condition that $F'$ satisfy condition ($\ast$) is that there exist two negative-power functions that $F$ is eventually squeezed between. Thus the proposition is proved.
$$nF(x) < xF'(x) < 0 \qquad xF'(x) < NF(x) < 0$$
which also leadsThe almost freshman-level triviality of this proof testifies to the great power of condition O (*which is a special case of the o-minimality axiom), from which all flows. Thus it is of interest to know of classes of functions which satisfy it. I will mention an extraordinary result in this regard, due largely to Patrick Speissegger (The Pfaffian closure of an o-minimal structure, J. Reine Angew. Math. 508 (1999), 189--211):
Edit: There are knownis an o-minimal structures much larger thanexpansion of the structure I mentioned above; for instance you can expandordered exponential field $\mathbb{R}$ (thus, including the structure by adjoining all analyticclass of functions whose domaindescribed above) so large that
The structure includes the restriction of any analytic function to a compact box,
If $f: [a, \infty) \to \mathbb{R}$ is first-order definable within this structure, then so is any antiderivative $F$ (even though general antiderivatives are not definable by a first-order construction),
This may fit better with Adam Hughes's formulation in terms of antiderivatives. Since condition O is a compact boxsatisfied (extendedaccording to the exterior of the box by setting the function to be identically zero theremore general o-minimality condition), the same analysis as above applies.
Instead of expanding to the class of all analytic functions (where the asymptotics can be hard to get control over, due to oscillatory behavior), my inclination would be to focus on large classes of functions with well-behaved asymptotics, including all the functions that arise in ordinary asymptotic analysis. The usual buzz phrase for this is "Hardy field" (mentioned for example in my answer here), which by definition is an ordered field of germs at infinity of $C^\infty$ functions.
Let us consider all functions which are first-order definable in the structure $(\mathbb{R}, 0, 1, +, \cdot, <, \exp)$ (if you like, add in all real numbers as constants). This type of structure arises in the theory of o-minimal structures, and has some rather remarkable properties. For example, every function in this class is either eventually positive, eventually zero, or eventually negative. As a result, the ring of germs at infinity forms an ordered field, i.e., a Hardy field. This Hardy field is in fact a real closed field. Also, if $F$ is definable, then $F'$ is also first-order definable (and its domain of definition contains all sufficiently large reals). Finally, these functions include all functions constructible from polynomials, $\exp$, $\log$ using the usual arithmetic operations and composition, and so includes all functions that arise in ordinary asymptotic analysis.
Now let us apply Willie Wong's necessary condition (that $x^n = O(F(x))$ and $F(x) = O(x^N)$ for some $n, N$) to this class of functions. I claim this is a sufficient condition for (*) provided that $n, N$ are either both positive or both negative. WLOG we may assume $F$ is eventually positive. Suppose that eventually $x^n \leq F(x) \leq x^N$ where $n, N$ are both positive. By adjusting the exponents if necessary, we may assume that $F(x)/x^n$ and $x^N/F(x)$ tend to infinity. Thus, these functions are eventually increasing (their derivatives are eventually positive). By a simple derivative calculation one finds that eventually
$$0 < nF(x) < xF'(x) \qquad 0 < xF'(x) < NF(x)$$
and so we get condition (*). The case where $n, N$ are both negative is similar except that we have instead
$$nF(x) < xF'(x) < 0 \qquad xF'(x) < NF(x) < 0$$
which also leads to condition (*).
Edit: There are known o-minimal structures much larger than the structure I mentioned above; for instance you can expand the structure by adjoining all analytic functions whose domain is a compact box (extended to the exterior of the box by setting the function to be identically zero there).
Note: This is a major rewrite of my earlier answer, to include necessary and sufficient conditions applicable to an even wider class of functions.
Instead of expanding to the class of all analytic functions (where the asymptotics can be hard to get control over, due to oscillatory behavior), my inclination would be to focus on large classes of functions with well-behaved asymptotics, including all the functions that arise in ordinary asymptotic analysis. The usual buzz phrase for this is "Hardy field" (mentioned for example in my answer here), which by definition is an ordered field of germs at infinity of $C^\infty$ functions.
I will describe several classes of such functions. The first is the class of all functions which are first-order definable in the structure $(\mathbb{R}, +, \cdot, <, \exp)$ together with all real numbers adjoined as constants. This class contains all functions that are constructible from polynomials, $\exp$, $\log$ and closed under the usual arithmetic operations and composition. It thus contains all the functions that usually arise in asymptotic analysis, and many more besides. This class enjoys the following strong model-theoretic property (as developed more fully in the theory of o-minimal structures):
(O) The zero set of any function $F: [a, \infty) \to \mathbb{R}$ in this class is a finite union of points and intervals (finite or infinite in extent).
Condition O ensures that every such function $F$ is either eventually positive ($F(x) > 0$ for all sufficiently large $x$), eventually zero, or eventually negative. As a result, the ring of germs at infinity of the definable functions in this class forms an ordered field, i.e., is a Hardy field.
Also, if $F$ is definable, then $F'$ is also first-order definable (and its domain can be shown to be the domain of $F$ save for finitely many points). Applying condition O to $F'$, every definable $F$ in this class is either eventually increasing, eventually decreasing, or eventually constant.
Proposition: A function $F$ in this class satisfies $F(x) = O(xF'(x))$ and $xF'(x) = O(F(x))$ if and only if there exist $n, N$, both positive or both negative, for which $x^n < |F(x)| < x^N$ for all sufficiently large $x$.
Proof: WLOG we may assume $F$ is eventually positive, and is not eventually constant. Thus $F$ is either eventually increasing or eventually decreasing, say eventually increasing. If $F$ is eventually bounded above by some $x^N$, $N > 0$, then by increasing $N$ if necessary we may assume $F(x)/x^N$ tends to zero, whence it is eventually decreasing. Taking the derivative, we conclude that eventually
$$x^N F'(x) - Nx^{N-1}F(x) < 0$$
whence $xF'(x) < NF(x)$, i.e., $|xF'(x)| < N|F(x)|$ or $xF'(x) = O(F(x))$. However, if $F(x)$ is eventually bounded above by every positive-power function $x^N$ (think $N$ small!), this also shows $xF'(x) = o(F(x))$, so that $F(x)$ is not $O(xF'(x))$.
By a similar argument, if $F(x)$ is eventually bounded below by some positive-power function $x^n$, we get $nF(x) < xF'(x)$ eventually, so that $F(x) = O(xF'(x))$. This also shows that if $F(x)$ is bounded below by every positive-power function (think $n$ large!), then $F(x) = o(xF'(x))$, so $xF'(x)$ is not $O(F(x))$.
Thus, if $F(x)$ is positive and eventually increasing, a necessary and sufficient condition that $F'$ satisfy condition ($\ast$) in the question is that there exist two positive-power functions that $F$ is eventually squeezed between. An entirely similar analysis shows that if $F(x)$ is positive and eventually decreasing, a necessary and sufficient condition that $F'$ satisfy condition ($\ast$) is that there exist two negative-power functions that $F$ is eventually squeezed between. Thus the proposition is proved.
The almost freshman-level triviality of this proof testifies to the great power of condition O (which is a special case of the o-minimality axiom), from which all flows. Thus it is of interest to know of classes of functions which satisfy it. I will mention an extraordinary result in this regard, due largely to Patrick Speissegger (The Pfaffian closure of an o-minimal structure, J. Reine Angew. Math. 508 (1999), 189--211):
There is an o-minimal expansion of the ordered exponential field $\mathbb{R}$ (thus, including the class of functions described above) so large that
The structure includes the restriction of any analytic function to a compact box,
If $f: [a, \infty) \to \mathbb{R}$ is first-order definable within this structure, then so is any antiderivative $F$ (even though general antiderivatives are not definable by a first-order construction),
This may fit better with Adam Hughes's formulation in terms of antiderivatives. Since condition O is satisfied (according to the more general o-minimality condition), the same analysis as above applies.
Instead of expanding to the class of all analytic functions (where the asymptotics can be hard to get control over, due to oscillatory behavior), my inclination would be to focus on large classes of functions with well-behaved asymptotics, including all the functions that arise in ordinary asymptotic analysis. The usual buzz phrase for this is "Hardy field" (mentioned for example in my answer here), which by definition is an ordered field of germs at infinity of $C^\infty$ functions.
Let us consider all functions which are first-order definable in the structure $(\mathbb{R}, 0, 1, +, \cdot, <, \exp)$ (if you like, add in all real numbers as constants). This type of structure arises in the theory of o-minimal structures, and has some rather remarkable properties. For example, every function in this class is either eventually positive, eventually zero, or eventually negative. As a result, the ring of germs at infinity forms an ordered field, i.e., a Hardy field. This Hardy field is in fact a real closed field. Also, if $F$ is definable, then $F'$ is also first-order definable (and its domain of definition contains all sufficiently large reals). Finally, these functions include all functions constructible from polynomials, $\exp$, $\log$ using the usual arithmetic operations and composition, and so includes all functions that arise in ordinary asymptotic analysis.
Now let us considerapply Willie Wong's necessary condition (that $x^n = O(F(x))$ and $F(x) = O(x^N)$ for some $n, N$) to this class of functions. I claim this is a sufficient condition for (*) provided that $n, N$ are either both positive or both negative. WLOG we may assume $F$ is eventually positive. Suppose that eventually $x^n \leq F(x) \leq x^N$ where $n, N$ are both positive. By adjusting the exponents if necessary, we may assume that $F(x)/x^n$ and $x^N/F(x)$ tend to infinity. Thus, these functions are eventually increasing (their derivatives are eventually positive). By a simple derivative calculation one finds that eventually
$$0 < nF(x) < xF'(x) \qquad 0 < xF'(x) < NF(x)$$
and so we get condition (*). The case where $n, N$ are both negative is similar except that we have instead
$$nF(x) < xF'(x) < 0 \qquad xF'(x) < NF(x) < 0$$
which also leads to condition (*).
Edit: There are known o-minimal structures much larger than the structure I mentioned above; for instance you can expand the structure by adjoining all analytic functions whose domain is a compact box (extended to the exterior of the box by setting the function to be identically zero there).
Instead of expanding to the class of all analytic functions (where the asymptotics can be hard to get control over, due to oscillatory behavior), my inclination would be to focus on large classes of functions with well-behaved asymptotics, including all the functions that arise in ordinary asymptotic analysis. The usual buzz phrase for this is "Hardy field" (mentioned for example in my answer here), which by definition is an ordered field of germs at infinity of $C^\infty$ functions.
Let us consider all functions which are first-order definable in the structure $(\mathbb{R}, 0, 1, +, \cdot, <, \exp)$ (if you like, add in all real numbers as constants). This type of structure arises in the theory of o-minimal structures, and has some rather remarkable properties. For example, every function in this class is either eventually positive, eventually zero, or eventually negative. As a result, the ring of germs at infinity forms an ordered field, i.e., a Hardy field. This Hardy field is in fact a real closed field. Also, if $F$ is definable, then $F'$ is also first-order definable (and its domain of definition contains all sufficiently large reals). Finally, these functions include all functions constructible from polynomials, $\exp$, $\log$ using the usual arithmetic operations and composition, and so includes all functions that arise in ordinary asymptotic analysis.
Now let us consider Willie Wong's necessary condition (that $x^n = O(F(x))$ and $F(x) = O(x^N)$ for some $n, N$) to this class of functions. I claim this is a sufficient condition for (*) provided that $n, N$ are either both positive or both negative. WLOG we may assume $F$ is eventually positive. Suppose that eventually $x^n \leq F(x) \leq x^N$ where $n, N$ are both positive. By adjusting the exponents if necessary, we may assume that $F(x)/x^n$ and $x^N/F(x)$ tend to infinity. Thus, these functions are eventually increasing (their derivatives are eventually positive). By a simple derivative calculation one finds that eventually
$$0 < nF(x) < xF'(x) \qquad 0 < xF'(x) < NF(x)$$
and so we get condition (*). The case where $n, N$ are both negative is similar except that we have instead
$$nF(x) < xF'(x) < 0 \qquad xF'(x) < NF(x) < 0$$
which also leads to condition (*).
Instead of expanding to the class of all analytic functions (where the asymptotics can be hard to get control over, due to oscillatory behavior), my inclination would be to focus on large classes of functions with well-behaved asymptotics, including all the functions that arise in ordinary asymptotic analysis. The usual buzz phrase for this is "Hardy field" (mentioned for example in my answer here), which by definition is an ordered field of germs at infinity of $C^\infty$ functions.
Let us consider all functions which are first-order definable in the structure $(\mathbb{R}, 0, 1, +, \cdot, <, \exp)$ (if you like, add in all real numbers as constants). This type of structure arises in the theory of o-minimal structures, and has some rather remarkable properties. For example, every function in this class is either eventually positive, eventually zero, or eventually negative. As a result, the ring of germs at infinity forms an ordered field, i.e., a Hardy field. This Hardy field is in fact a real closed field. Also, if $F$ is definable, then $F'$ is also first-order definable (and its domain of definition contains all sufficiently large reals). Finally, these functions include all functions constructible from polynomials, $\exp$, $\log$ using the usual arithmetic operations and composition, and so includes all functions that arise in ordinary asymptotic analysis.
Now let us apply Willie Wong's necessary condition (that $x^n = O(F(x))$ and $F(x) = O(x^N)$ for some $n, N$) to this class of functions. I claim this is a sufficient condition for (*) provided that $n, N$ are either both positive or both negative. WLOG we may assume $F$ is eventually positive. Suppose that eventually $x^n \leq F(x) \leq x^N$ where $n, N$ are both positive. By adjusting the exponents if necessary, we may assume that $F(x)/x^n$ and $x^N/F(x)$ tend to infinity. Thus, these functions are eventually increasing (their derivatives are eventually positive). By a simple derivative calculation one finds that eventually
$$0 < nF(x) < xF'(x) \qquad 0 < xF'(x) < NF(x)$$
and so we get condition (*). The case where $n, N$ are both negative is similar except that we have instead
$$nF(x) < xF'(x) < 0 \qquad xF'(x) < NF(x) < 0$$
which also leads to condition (*).
Edit: There are known o-minimal structures much larger than the structure I mentioned above; for instance you can expand the structure by adjoining all analytic functions whose domain is a compact box (extended to the exterior of the box by setting the function to be identically zero there).
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