Generalized fraction rules for monotonicity with higher antiderivatives and derivatives

We first introduce the generic versions of the fraction rules for monotonicity, i.e. the one that involves integrals known as the Gromov theorem and the other that involves derivatives known as L'H\^opital rule for monotonicity, which we then extend to high order antiderivatives and derivatives, respectively.


Introduction
Roughly speaking, the application of either the integral or the differential operation to both the numerator and the denominator of a fraction, preserves the monotonicity of the fraction.The integral case of such fact is known as the Gromov theorem (see, e.g., [5,10]), while the differential case is called the L'Hôpital rule for monotonicity (see, e.g., [14,12,2,18,10]).The Gromov theorem first appeared in [6], i.e. about a decade before the introduction of the L'Hôpital rule for monotonicity in [1].
Below follow the most generic versions of these fraction rules for monotonicity, for the statement of which we remind that a real function, defined in an interval of the extended real line, [−∞, ∞], is locally characterized by a property when it is characterized by that property in every compact subinterval of its domain (we remind that an unbounded interval of the form [−∞, ∞], [−∞, a] or [a, ∞], for some a ∈ R, is compact).
Theorem 1 (the Gromov theorem).Consider i. f and g are both locally Lebesgue integrable and ii.g preserves Lebesgue-almost everywhere a non zero sign.
It can be shown (see §3.2) that Theorem 1 is stronger than Theorem 2, a fact that has already been observed in [10].However, the latter one is an independent result of differential calculus, for the proof of which no tools of the integration theory are needed (see §A).
The goal of the present manuscript is not only the proof of Theorem 1 and Theorem 2, but also the introduction of their generalizations to higher antiderivatives and derivatives, respectively.Our analysis is organized as follows.In §2 we review some necessary notions used for the compact statement of the aforementioned generalizations.In §3, after the statement of the main results, we examine the relation between them and we proceed to their proof.In §4 we employ our findings in some novel applications.In §A we provide an alternative proof of the generalized L'Hôpital rule for monotonicity with the exclusive utilization of the differential calculus toolbox.

Basic notions
For the statement of our results we make a short, necessary note on the notation used.

For every
A n,f,c stands for the antiderivative of order n for f at c, i.e.
The name of this function is nothing but random.It comes from the Cauchy formula of repeated integration, This formula is introduced in [ If, in addition, f is continuous, then A n,f,c I∩R is n-times differentiable, with 2. For every i. n ∈ N, ii.interval I ⊆ R, iii.c ∈ I and iv.locally Lebesgue integrable f ∶ , I → R, M n,f,c stands for the mean of order n for f at c, i.e.
The concept behind the above definition lies in the observation that which confirms the expected equality T n,f,c and R n,f,c stand for the Taylor polynomial and remainder, respectively, of order n for f at c, i.e. and If, in addition, n ∈ N and f (n) ∶ I ∩ R → R is locally Lebesgue integrable, then the integral form of the remainder (see, e.g., [3, §1.6 in page 62]) implies that 3 Generalized fraction rules for monotonicity

Statement
For the proper statement of the main results, we need the following result. Proof.
1. To begin with, we have that Since f preserves Lebesgue-almost everywhere a non zero sign, we deduce that for every x ∈ I ∖{c} the function (x − id) n−1 f ∶ (min {c, x}, max {c, x}) → R also preserves Lebesgue-almost everywhere a non zero sign, where id stands for the identity function.Thus A n,f,c (x) ≠ 0 and the result then follows.

We have that R
Since f (n) (x) ≠ 0, for all x ∈ I ∩ R, from the Darboux theorem (see, e.g., [8, Theorem 8.3.2 in page 228]) we have that f (n) preserves a non zero sign, that is f (n−1) ∶ I ∩ R → R is strictly monotonic, hence f (n) is locally Lebesgue integrable (see, e.g., [8,Theorem B.2.5 in 490]).Now, we first apply point 1. for the function f (n) and we then employ (3), in order to get the desired result.
With Proposition 1 at hand, we can now state the generalizations of Theorem 1 and Theorem 2 to higher antiderivatives and derivatives, respectively.
Proof.Under the hypothesis of Theorem 4, we first deduce that both f (n) , g (n) ∶ I ∩ R → R are locally Lebesgue integrable.Indeed, we can argue as in the proof of point 2. of Proposition 1, in order to show that g (n) is locally Lebesgue integrable.Moreover, is locally bounded since it is (strictly) monotonic, hence we write and we conclude that f (n) is also locally Lebesgue integrable as a product of a locally bounded function and a locally Lebesgue integrable one.Now, we first apply Theorem 3 for the functions f (n) and g (n) and we then employ (3).
We can weaken Theorem 3 in a specific manner, in order to get the reverse implication of Proposition 2. Proposition 3. Theorem 4 implies Theorem 3, when the latter one is equipped with the hypothesis that f and g are both continuous instead of being just locally Lebesgue integrable.
Proof.Under the hypothesis of the weakened Theorem 3, (2) implies that A n,f,c I∩R and A n,g,c I∩R are both n-times differentiable.
Now, all we have to do is to apply Theorem 4 for the functions A n,f,c and A n,g,c .

Proof
In view of Proposition 2, we only need to prove the stronger of the main results, in particular, Theorem 3.
Proof of Theorem 5.It suffices to show the result only for the case where g preserves Lebesguealmost everywhere the positive sign.Indeed, we can employ such a result for −f and −g instead of f and g, respectively, in order to get the corresponding one for g that preserves Lebesgue-almost everywhere the negative sign.Moreover, it suffices to show Theorem 3 only for the case where f g is Lebesgue-almost everywhere (strictly) increasing.Indeed, we can employ such a result for −f instead of f , in order to get the corresponding one for f g that is Lebesgue-almost everywhere (strictly) decreasing.Hence, we assume, without loss of generality, that g preserves Lebesgue-almost everywhere the positive sign and that f is Lebesgue-almost everywhere (strictly) increasing.
We will show the desired result by induction on n.
1.The base case.The case where n = 1 is nothing but Theorem 1 itself.
Since g preserves Lebesgue-almost everywhere the positive sign, the function A 1,g,c is strictly increasing, which implies that its inverse A 1,g,c −1 ∶ A 1,g,c (I) → I is not only well defined but also strictly increasing.In addition, the continuity of A 1,g,c guarantees that A 1,g,c (I) is an interval.
We then consider the function h ∶= A 1,f,c ○ A 1,g,c −1 ∶ A 1,g,c (I) → R and we claim that Indeed, observing that we get the desired equality by the use of the change of variable formula (see, e.g., [17, point (i) of Corollary 6.97 in page 326]).
is Lebesgue-almost everywhere (strictly) increasing as a composition of a strictly increasing function and an Lebesgue-almost everywhere (strictly) increasing function.
The combination of the above two facts implies that h is (strictly) convex (see, e.g., [16, Theorem A in page 9 and Remark B in page 13] or [19, Theorem 14.14 in page 334]).Hence, from the equality h(0) = 0 along with the Galvani lemma (see, e.g., [11, Theorem 1.3.1 in page 20]) we deduce that the function is (strictly) increasing and so is since A 1,g,c is strictly increasing.The result then follows from the fact that h ○ A 1,g,c = A 1,f,c .
We assume that A k,g,c is (strictly) increasing and we will show that A k+1,g,c is (strictly) increasing.
We consider the functions which are both locally Lebesgue integrable.
We claim that g preserves Lebesgue-almost everywhere the positive sign.Indeed, we have that In addition, With the above facts at hand, all we have to do is first to apply Theorem 1 for the functions f and g and second to employ (1), in order to obtain the desired result.

Corollaries and examples
Below follow some applications of the generalized fraction rules for monotonicity.If f is Lebesgue-almost everywhere (strictly) monotonic, then from Theorem 3 for g ≡ 1 we deduce that M n,f,c is (strictly) monotonic of the same (strict) monotonicity.From the Galvani lemma we have that the function f −f (c) id−c ∶ I ∖{c} → R is (strictly) increasing.Extending the above function as and remembering that every convex function is locally Lebesgue integrable, we employ Theorem 3, in order to obtain that the function is also (strictly) increasing.Hence, again from the Galvani lemma we deduce that M n,f,c is (strictly) convex.
Hence, S ′ (t) < 0, for all t ∈ [0, ∞), which implies that S is strictly decreasing.By the use of Theorem 2 we deduce that These facts imply that i.e. a useful a priori estimate when c = 0.
Moreover, by the use of Theorem 3 we deduce that which can also be deduced directly from the previous one.We then set where B(0 n , r) stands for the n-dimensional ball of radius r > 0 centered at the origin 0 n ∈ R n and ⋅ stands for the standard euclidean norm in R n .
Employing the change of variables formula, we can deduce that, for every fixed r > 0, the functions φ(r, ⋅), ψ(r, ⋅)∶ B(0 n , r) → R are both Lebesgue integrable.Indeed, we have where for the first equality we employed the polar coordinates change of variables formula for the radial functions (see, e.g., [19,Theorem 26.20 in page 695]).Similarly follows the result for the other function, ψ, for which we also note that, in view of Proposition 1, we have B(0n,r) ψ(r, x)dx ≠ 0, ∀r > 0.
We now claim that if is Lebesgue-almost everywhere (strictly) monotonic, then the well defined function ψ(⋅,x)dx ∶ (0, ∞) → R is (strictly) monotonic of the same (strict) monotonicity.Indeed, from Theorem 3 we have that An,g,0 ∶ (0, ∞) → R n is (strictly) monotonic of the same (strict) monotonicity as of f g and the result then follows since With Proposition 4 at hand, Theorem 4 is properly stated in the context of differential calculus.We now proceed to its proof.
Proof of Theorem 2. Since g ′ (x) ≠ 0, for all x ∈ I ∩ R, from the Darboux theorem we have that g ′ preserves a non zero sign, hence g is strictly monotonous.Hence inverse function of g, g −1 ∶ g(I) → I is well defined.Additionally, g −1 is differentiable with Arguing as in the proof of Theorem 3, it suffices to show the result for g being strictly increasing and f being (strictly) increasing.Therefore, we make such assumptions.From the strict monotonicity of g, the function g −1 is also strictly increasing.Now, we consider the function h ∶= f ○ g −1 ∶ g(I) → R, which is differentiable, due to the chain rule, with thus h ′ is (strictly) increasing as a composition of a strictly increasing function and a (strictly) increasing function.Hence h is (strictly) convex.
We then consider two arbitrary x 1 , x 2 ∈ I ∖ {c}, such that x 1 < x 2 .Since g(x 1 ) < g(x 2 ), from the Galvani lemma we deduce that Proof of Theorem 4. It is only left to show the result for n > 1 (with I ⊆ R), thus we make such an assumption.
To begin with, in view of Proposition 4 we have the following sequence of equalities Additionally, the following is true.Now, we inductively apply Theorem 2 n times, in order to get that both are (strict) monotonic of the same (strict) monotonicity as of f (n) g (n) .If c ∈ ∂I, then the proof is complete.
Next, we deal with the case where c ∈ I ○ .From the above (strict) monotonicity we deduce that the one sided limits to c of these functions exist in [−∞, ∞], i.e.

∶
I ∖ {c} → R is (strictly) monotonic of the same (strict) monotonicity.Theorem 2 (the L'Hôpital rule for monotonicity).Consider 1. an interval I ⊆ [−∞, ∞], 2. a point c ∈ I and 3. two functions f, g ∶ I → R, such that i. f I∩R and g I∩R are both differentiable and

Proposition 1 .
Consider i. a natural number n ∈ N, ii. an interval I ⊆ [−∞, ∞] when n = 1 or I ⊆ R when n ≠ 1, iii. a point c ∈ I and iv. a function f ∶ I → R. 1.If f a. is locally Lebesgue integrable and b. preserves Lebesgue-almost everywhere a non zero sign,

Theorem 3 (
generalization to higher antiderivatives).Consider 1. a natural number n ∈ N,2. an intervalI ⊆ [−∞, ∞] when n = 1 or I ⊆ R when n ≠ 1,3. a point c ∈ I and 4. two functions f, g ∶ I → R, such that i. f and g are both locally Lebesgue integrable and ii.g preserves Lebesgue-almost everywhere a non zero sign.

Theorem 4 (
generalization to higher derivatives).Consider 1. a natural number n ∈ N, 2. an interval I ⊆ [−∞, ∞] when n = 1 or I ⊆ R when n ≠ 1, 3. a point c ∈ I and 4. two functions f, g ∶ I → R, such that i. f I∩R and g I∩R are both n-times differentiable and ii.g since sgn(x − c) = sgn(x − t), ∀t ∈ (min {c, x}, max {c, x}) , ∀x ∈ I ∖ {c} , therefore g I∖{c} preserves the positive sign.

1 .
Monotonicity of high order mean: We consider i. a natural number n ∈ N, ii. an interval I ⊆ R, iii. a point c ∈ I and iv. a locally Lebesgue integrable function f ∶ I → R.

2 .
Convexity of high order mean: We consider i. a natural number n ∈ N, ii. an interval I ⊆ R, iii. a point c ∈ I and iv. a convex function f ∶ I → R.

4 .
Multidimensional analogue for specific radial functions: We consider i. a natural number n ∈ N and ii.two functions f, g ∶ [0, ∞) → R, such that a. f and g are both locally Lebesgue integrable and b. g preserves Lebesgue-almost everywhere a non zero sign.