GLAESER’S INEQUALITY ON AN INTERVAL

ADAM COFFMAN AND YIFEI PAN

Abstract. We use elementary methods to find pointwise bounds for the first derivative of a real valued function with a continuous, bounded second derivative on an interval.

1. Introduction

By “Glaeser’s inequality,” we refer to the elementary calculus theorem that for a function f which is nonnegative on ℝ and has continuous second derivative bounded above by M > 0 on ℝ, the first derivative satisfies the inequalities

(1)  −\sqrt{2Mf(x)} ≤ f′(x) ≤ \sqrt{2Mf(x)}

at every point x ∈ ℝ. The surprising part of the statement, and possibly the reason it is not more well-known, is that even though the conclusion is a pointwise inequality, the assumptions are global: f must satisfy the stated conditions on the entire number line ℝ. The inequality |f′(x)| ≤ \sqrt{2Mf(x)} is clearly false for nonnegative functions f defined only on an interval (a, b) or ray (a, ∞); the function f(x) = x on the interval (0, ∞) is an easy counterexample.

We refer to the literature ([G], [LN]) for the history and various proofs of Glaeser’s inequality, and we note that it and its generalizations to several variables are topics of recent research interest ([CV], [NS]) in the field of partial differential equations. The related problem of estimating the maximum value of |f′| on ℝ or a subinterval goes back at least to Landau ([CS]).

Our goal here is to find “local” versions of Glaeser’s pointwise inequality for f′: to replace the domain ℝ in the theorem with a subinterval of ℝ, possibly at the expense of imposing boundary conditions or getting a weaker pointwise bound.

Some further examples are instructive regarding the need for all the hypotheses in Glaeser’s theorem. Given f(x) ≥ 0 on an interval (a, b), a first attempted approach might be to extend f to a nonnegative function on ℝ, and then apply Glaeser’s inequality to the extension. The example f(x) = x on (0, ∞) already shows what goes wrong: f does not extend to a nonnegative differentiable function on ℝ; if it did, then f′(0) would have to be 1 and f would be increasing near 0 and negative for x → 0^{−}, a contradiction. To avoid this issue, we could try assuming that f′ approaches 0 near the left endpoint, but this boundary condition by itself is still not enough for the extension to satisfy either the assumptions or the conclusion of Glaeser’s inequality. Considering the example f(x) = x^{3/2} on (0, ∞), f extends to the nonnegative, differentiable function |x|^{3/2} on ℝ, but f′′ is unbounded and f′ is not bounded pointwise on (0, ∞) by μ\sqrt{f} for any scalar μ.

MSC 2010: 26D10. Keywords: Calculus, pointwise inequality.

If we consider an extra assumption: that f is increasing on an interval (a, b), in addition to satisfying f(x) ≥ 0, lim_{x→a^{+}} f′(x) = 0, and f′′(x) ≤ M, then we get the inequalities (1) for

x ∈ (a, b) via a one-line proof:

(2)  \frac{1}{2}(f′(x))^{2} = lim_{w→a^{+}} ∫_{w}^{x} f′(t)f′′(t)dt ≤ ∫_{a}^{x} f′(t)Mdt ≤ Mf(x).

However, the “increasing” assumption seems too strong and it would be more satisfactory to find a result that holds even if f′ changes sign infinitely many times on an interval.

2. The inequality with boundary conditions

Our main result, Theorem 2.1, does not assume that f is increasing; the Proof uses a method different from (2), but which is still elementary. One might expect a limit condition on f′ at one endpoint, a, to lead to estimates for f′ that holds only near a, but it turns out that this condition (and f′′(x) ≤ M for a < x < b) is enough to guarantee the same upper bound as (1) on the whole interval (a, b). Under the further condition that f′′ is bounded below by B, we obtain the lower bound f′(x) ≥ −\sqrt{2Mf(x)} on a certain subinterval (a, c], where c depends on M and B but not on f.

For simplicity, we consider the interval I = (0, C); the result clearly can be translated to an interval (a, b).

Theorem 2.1. Given f : I → ℝ, suppose:

• f(x) ≥ 0 for all x ∈ I;

• f′′(x) is continuous on I [we denote this by f ∈ C^{2}(I)];

• there is a constant M > 0 so that f′′(x) ≤ M for all x ∈ I;

• lim_{x→0^{+}} f′(x) = 0.

Let

s(x) = max \{1, \frac{\sqrt{2f(x)}}{\sqrt{M}(C − x)}\} .

Then, for all x ∈ I,

(3)  f′(x) ≤ \sqrt{2Mf(x)}, and
(4)  f′(x) ≥ −\sqrt{2Mf(x)} (\frac{1}{2s(x)} + \frac{s(x)}{2}) .

If, in addition, there is a constant B < M so that B ≤ f′′(x) for all x ∈ I, let

ŝ(x) = max \{1, \frac{\sqrt{2f(x)}}{\sqrt{M}(C − x)}χ_{( \frac{M}{M−B} C,∞)}(x)\};

then f′ satisfies:

f′(x) ≥ max \{Bx, −\sqrt{2Mf(x)} (\frac{1}{2ŝ(x)} + \frac{ŝ(x)}{2})\} .

Proof. Extending f′ to a continuous function on [0, C) with f′(0) = 0, the Mean Value Theorem applies for 0 < t < C:

(5)  \frac{f′(t) − f′(0)}{t − 0} = f′′(c_{1}) ⟹ f′(t) = f′′(c_{1})t ⟹ f′(t) ≤ Mt.

Similarly, if there is a lower bound f′′ ≥ B, then

(6)  Bt ≤ f′(t).

We refer to (5) and (6) as the linear bounds on f′.

Fix x ∈ I; then, for any y ∈ I, integration gives:

∫_{y}^{x} ∫_{x}^{t} f′′(s)dsdt = ∫_{y}^{x}(f′(t) − f′(x))dt
= [f(t) − tf′(x)]_{t=y}^{t=x}
= (f(x) − xf′(x)) − (f(y) − yf′(x))
(7)  ⟹ f(y) = f(x) − (x − y)f′(x) + ∫_{y}^{x} ∫_{t}^{x} f′′(s)dsdt
(8)  ⟹ 0 ≤ f(x) − (x − y)f′(x) + \frac{1}{2}(x − y)^{2}M.

The first line is where the assumption of the continuity of f′′ is used, in the Fundamental Theorem of Calculus (so, the C^{2} hypothesis may be weakened as in [CS], [NS]). Line (7) is the well-known first-order Taylor polynomial with integral remainder term. The last line uses the upper bound f′′ ≤ M and the assumption f(y) ≥ 0. Let Δx = x − y; then 0 < y < C is equivalent to x − C < Δx < x, and the non-negativity of the quadratic (8) for all y ∈ I is equivalent to

(9)  \frac{1}{2}M(Δx)^{2} − f′(x)Δx + f(x) ≥ 0

for all Δx ∈ (x − C, x), which (using M > 0) leads to three cases.

Case 1. The quadratic (9) is always positive or has a double root:

(−f′(x))^{2} − 4 · \frac{1}{2}Mf(x) ≤ 0 ⟹ |f′(x)| ≤ \sqrt{2Mf(x)}.

Case 2. The quadratic (9) has distinct real zeros, the lesser one is to the right of the interval (x − C, x), so (f′(x))^{2} − 2Mf(x) > 0 and

x ≤ \frac{f′(x) − \sqrt{(f′(x))^{2} − 2Mf(x)}}{M}
⟹ f′(x) ≥ Mx + \sqrt{(f′(x))^{2} − 2Mf(x)},

however, this contradicts the linear bound f′(x) ≤ Mx.

Case 3. The quadratic (9) has distinct real zeros, the greater one is to the left of the interval (x − C, x), so (f′(x))^{2} − 2Mf(x) > 0 and

x − C ≥ \frac{f′(x) + \sqrt{(f′(x))^{2} − 2Mf(x)}}{M}
(10)  ⟹ f′(x) ≤ M(x − C) − \sqrt{(f′(x))^{2} − 2Mf(x)},

which implies f′(x) < 0 and establishes (3).

f′(x) + M(C − x) ≤ −\sqrt{(f′(x))^{2} − 2Mf(x)} < 0
⟹ (f′(x) + M(C − x))^{2} ≥ (f′(x))^{2} − 2Mf(x)
⟹ 2f′(x)M(C − x) + M^{2}(C − x)^{2} ≥ −2Mf(x)
(11)  ⟹ f′(x) ≥ −\frac{f(x)}{C − x} − \frac{M(C − x)}{2}.

The inequalities (10) and (11) are consistent only if:

−\frac{f(x)}{C − x} − \frac{M(C − x)}{2} < M(x − C)
⟺ \frac{2f(x)}{(C − x)^{2}} > M.

The lower bounds from Cases 1 and 3 are:

(12)  f′(x) ≥ −\sqrt{2Mf(x)}, if M ≥ \frac{2f(x)}{(C − x)^{2}} ,
(13)  f′(x) ≥ − (\frac{f(x)}{C − x} + \frac{M(C − x)}{2}), if M < \frac{2f(x)}{(C − x)^{2}} .

It follows from the arithmetic-geometric mean inequality that \sqrt{2Mf(x)} ≤ \frac{f(x)}{C−x} + \frac{M(C−x)}{2}, so the lower bound from (13) is always less than or equal to the lower bound from (12), and they coincide when M = \frac{2f(x)}{(C−x)^{2}} . It is elementary to check that (12) and (13) together imply (4).

In the case where f′′ ≥ B, from (10) and the linear bound (6) we can conclude

Bx ≤ f′(x) < M(x − C),

and it follows that x > \frac{M}{M−B} C. We include the characteristic function χ_{( \frac{M}{M−B} C,∞)}(x) in the

expression for ŝ(x) to show that Case 3 does not occur for 0 < x ≤ \frac{M}{M−B} C; in particular, if B ≥ 0, Case 3 does not occur for any x ∈ I.

Example 2.2. Considering f(x) = \frac{1}{2}Mx^{2} on an interval (0, C), the upper bound (3) in the Theorem is sharp, in the sense that for some f(x), equality is achieved at every point x. However, for some other f(x) such as \frac{1}{2}Mx^{2} + 1, the linear upper bound (5) is better.

Example 2.3. This example shows that the lower bound can also be reached, in this case by a function that vanishes at the endpoint C.

Let f(x) = x^{2}(1 − x) = x^{2} − x^{3}, on the interval (0, 1). Then f′(0) = 0 and f′′(x) = 2 − 6x is bounded above by M = 2 and below by B = −4. For x in the interval (0, \frac{1}{2}(\sqrt{5} − 1)], \frac{2f(x)}{(1−x)^{2}} ≤ M = 2, and the Theorem’s bounds (3) and (12) are:

(14)  −\sqrt{4(x^{2} − x^{3})} ≤ f′(x) = 2x − 3x^{2} ≤ \sqrt{4(x^{2} − x^{3})}.

In fact, (14) holds on the larger interval (0, \frac{8}{9}] and the upper bound holds on the whole interval (0, 1), but the lower bound from (14) is false for x close to 1. For x in the interval (\frac{1}{2}(\sqrt{5}−1), 1), \frac{2f(x)}{(1−x)^{2}} > M = 2, x > \frac{M}{M−B} = \frac{1}{3}, and the Theorem’s bounds (3), (6), and (13) are:

max \{−4x, − (\frac{f(x)}{1 − x} + \frac{2(1 − x)}{2})\} = −x^{2} + x − 1 ≤ 2x − 3x^{2} ≤ \sqrt{4(x^{2} − x^{3})}.

As x → 1^{−}, the inequalities approach −1 ≤ −1 ≤ 0.

From the Proof of Theorem 2.1, we see that Glaeser’s lower bound (1) fails only for x that are close to the endpoint without the boundary condition, and that fall into Case 3. The inequality (1) can be recovered by avoiding Case 3 in two ways: first, we could replace (0, C) by (0, ∞) in Theorem 2.1; an examination of the Proof shows that Case 3 is excluded. Second, we could add another limit condition at the other endpoint of I = (0, C), lim_{x→C^{−}} f′(x) = 0. Then there

is another linear bound f′(x) ≥ M(x − C), which contradicts (10), so Case 3 in the Proof of Theorem 2.1 is excluded again. Proposition 2.4. Let I denote the interval (0, C) or the ray (0, ∞). Given f : I → ℝ, suppose:

• f(x) ≥ 0 for all x ∈ I;

• f ∈ C^{2}(I);

• there is a constant M > 0 so that f′′(x) ≤ M for all x ∈ I;

• lim_{x→0^{+}} f′(x) = 0;

• either I = (0, ∞) or lim_{x→C^{−}} f′(x) = 0.

Then, for all x ∈ I, −\sqrt{2Mf(x)} ≤ f′(x) ≤ \sqrt{2Mf(x)}.

3. The inequality without boundary conditions

Removing all the boundary conditions on f′ leads to this version of the inequality, which we state for any interval (a, b). It can be proved by a straightforward adaptation of the Proof of Theorem 2.1. Proposition 3.1. Given f : (a, b) → ℝ, suppose:

• f(x) ≥ 0 for all x ∈ (a, b);

• f ∈ C^{2}((a, b));

• there is a constant M > 0 so that f′′(x) ≤ M for all x ∈ (a, b).

Let s_{ℓ}(x) = max \{1, ^{\frac{\sqrt{2f(x)}}{\sqrt{M}(x−a)}}\} and s_{r}(x) = max \{1, ^{\frac{\sqrt{2f(x)}}{\sqrt{M}(b−x)}}\}. Then, for all x ∈ (a, b),

−\sqrt{2Mf(x)} (\frac{1}{2s_{r}(x)} + \frac{s_{r}(x)}{2}) ≤ f′(x) ≤ \sqrt{2Mf(x)} (\frac{1}{2s_{ℓ}(x)} + \frac{s_{ℓ}(x)}{2}) .

If, as in Proposition 2.4, there is no right endpoint b, then the lower bound −\sqrt{2Mf(x)} holds on (a, ∞). For (a, b) = (−R, R) and x at the midpoint 0, this is the same as the result stated in [LN]. Proposition 3.1 has Glaeser’s inequality for the domain ℝ as a limiting case, as a → −∞, b → +∞. Example 3.2. The function f(x) = \sqrt{1 − x^{2}} on the interval (−1, 1) gives an example where f′ is unbounded at both endpoints and f is concave down everywhere. For any x, allowing M to decrease to 0 makes both expressions s_{ℓ}(x) and s_{r}(x) greater than 1.

Acknowledgments. The authors thank anonymous referees for their comments which improved the statements of the results.

References

[CV] I. Capuzzo Dolcetta and A. Vitolo, C^{1,α} and Glaeser type estimates, Rendiconti di Matematica, Serie VII, 29 (2009), 17–27. MR 2548484 (2010i:35090).

[CS] C. Chui and P. Smith, A note on Landau’s problem for bounded intervals, American Math. Monthly (9) 82 (1975), 927–929. MR 0457654 (56 #15859).

[G] G. Glaeser, Racine carrée d’une fonction différentiable, Ann. Inst. Fourier (Grenoble) (2) 13 (1963), 203– 210. MR 0163995 (29 #1294).

[LN] Y. Y. Li and L. Nirenberg, Generalization of a well-known inequality, in Contributions to Nonlinear Analysis, 365–370, Progr. Nonlinear Differential Equations Appl. 66, Birkhäuser, Basel, 2006. MR 2187814 (2006g:26030).

[NS] T. Nishitani and S. Spagnolo, An extension of Glaeser inequality and its applications, Osaka J. Math. (1) 41 (2004), 145–157. MR 2040070 (2004m:26027).

Department of Mathematical Sciences, Indiana University - Purdue University Fort Wayne, 2101 E. Coliseum Blvd., Fort Wayne, IN, USA 46805-1499

   E-mail address: CoffmanA@ipfw.edu
   URL: http://ipfw.edu/math/

College of Mathematics and Information Sciences, Jiangxi Normal University, Nanchang, P.R.China E-mail address: Pan@ipfw.edu