Abstract and Applied AnalysisVolume 2011 (2011), Article ID 162049, 18 pagesdoi:10.1155/2011/162049

Research Article

An Extension of Young's Inequality

Flavia-Corina Mitroi and Constantin P. Niculescu

Department of Mathematics, University of Craiova, Street A. I. Cuza 13, 200585 Craiova, Romania

Received 3 February 2011; Accepted 27 June 2011

Academic Editor: Irena Lasiecka

Copyright © 2011 Flavia-Corina Mitroi and Constantin P. Niculescu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Young's inequality is extended to the context of absolutely continuous measures. Several applications are included.

1. Introduction

Young's inequality [1] asserts that every strictly increasing continuous function with and verifies an inequality of the following form: whenever and are nonnegative real numbers. The equality occurs if and only if ; see [2–5] for details and significant applications.

Several questions arise naturally in connection with this classical result.(Q1): Is the restriction on strict monotonicity (or on continuity) really necessary?(Q2): Is there any weighted analogue of Young's inequality?(Q3): Can Young's inequality be improved?

Cunningham and Grossman [6] noticed that question (Q1) has a positive answer (correcting the prevalent belief that Young's inequality is the business of strictly increasing continuous functions). The aim of the present paper is to extend the entire discussion to the framework of locally absolutely continuous measures and to prove several improvements.

As is well known, Young's inequality is an illustration of the Legendre duality. Precisely, the functions are both continuous and convex on , and (1.1) can be restated as with equality if and only if . Because of the equality case, formula (1.3) leads to the following connection between the functions F and G:

It turns out that each of these formulas produces a convex function (possibly on a different interval). Some details are in order.

By definition, the conjugate of a convex function defined on a nondegenerate interval is the function with domain . Necessarily is a nonempty interval, and is a convex function whose level sets are closed subsets of for each (usually such functions are called closed convex functions).

A convex function may not be differentiable, but it admits a good substitute for differentiability.

The subdifferential of a real function defined on an interval is a multivalued function defined by

Geometrically, the subdifferential gives us the slopes of the supporting lines for the graph of . The subdifferential at a point is always a convex set, possibly empty, but the convex functions have the remarkable property that at all interior points. It is worth noticing that at each point where is differentiable (so this formula works for all points of except for a countable subset), see [4, page 30].

Lemma 1.1 (Legendre duality, [4, page 41]). Let be a closed convex function. Then its conjugate is also convex and closed and (i) for all , ;(ii) if and only if ;(iii) (as graphs);(iv).

Recall that the inverse of a graph is the set .

How far is Young's inequality from the Legendre duality? Surprisingly, they are pretty closed in the sense that in most cases the Legendre duality can be converted into a Young-like inequality. Indeed, every continuous convex function admits an integral representation.

Lemma 1.2 (see [4, page 37]). Let be a continuous convex function defined on an interval , and let be a function such that for every . Then for every in one has

As a consequence, the heuristic meaning of formula (i) in Lemma 1.1 is the following Young-like inequality: where and are selection functions for and, respectively. Now it becomes clear that Young's inequality should work outside strict monotonicity (as well as outside continuity). The details are presented in Section 2. Our approach (based on the geometric meaning of integrals as areas) allows us to extend the framework of integrability to all positive measures which are locally absolutely continuous with respect to the planar Lebesgue measure , see Theorem 2.3.

A special case of Young's inequality is which works for all , and with . Theorem 2.3 yields the following companion to this inequality in the case of Gaussian measure on : where is the Gauss error function (or the erf function).

The precision of our generalization of Young's inequality makes the objective of Section 3.

In Section 4 we discuss yet another extension of Young's inequality, based on recent work done by Pečarić and Jakšetić [7].

The paper ends by noticing the connection of our result to the theory of -convexity (i.e., of convexity associated to a cost density function).

Last but not least, all results in this paper can be extended verbatim to the framework of nondecreasing functions such that and and . In other words, the interval plays no special role in Young's inequality.

Besides, there is a straightforward companion of Young's inequality for nonincreasing functions, but this is outside the scope of the present paper.

2. Young's Inequality for Weighted Measures

In what follows will denote a nondecreasing function such that and . Since is not necessarily injective we will attach to a pseudoinverse by the following formula:

Clearly, is nondecreasing and for all . Moreover, with the convention , Here and represent the lateral limits at . When is also continuous,

Remark 2.1 (Cunningham and Grossman [6]). Since pseudoinverses will be used as integrands, it is convenient to enlarge the concept of pseudoinverse by referring to any function such that where . Necessarily, g is nondecreasing, and any two pseudoinverses agree except on a countable set (so their integrals will be the same).Given , we define the epigraph and the hypograph of , respectively, by Their intersection is the graph of ,

Notice that our definitions of epigraph and hypograph are not the standard ones, but agree with them in the context of monotone functions.

We will next consider a measure on , which is locally absolutely continuous with respect to the Lebesgue measure that is, is of the form where is a Lebesgue locally integrable function, and is any compact subset of .

Clearly,

Moreover,

The discussion above can be summarized as follows.

Lemma 2.2. Let be a nondecreasing function such that and . Then for every Lebesgue locally integrable function and every pair of nonnegative numbers ,

We can now state the main result of this section.

Theorem 2.3 (Young's inequality for nondecreasing functions). Under the assumptions of Lemma 2.2, for every pair of nonnegative numbers and every number , one has If in addition is strictly positive almost everywhere, then the equality occurs if and only if .

Proof. We start with the case where , see Figure 1. In this case, with equality if and only if . When is strictly positive almost everywhere, this means that .If , then The equality holds if and only if, that is, when (provided that is strictly positive almost everywhere), see Figure 2.If , then and the inequality in the statement of Theorem 2.3 is actually an equality, see Figure 3.

Figure 1: The geometry of Young's inequality when .

Figure 2: The case .

Figure 3: The equality case.

Corollary 2.4. (Young's inequality for continuous increasing functions). If is also continuous and increasing, then for every real number . Assuming is strictly positive almost everywhere, the equality occurs if and only if .

If for every , then Corollary 2.4 asserts that The equality occurs if and only if . In the special case where , this reduces to the classical inequality of Young.

Remark 2.5 (the probabilistic companion of Theorem 2.3). Suppose there is a given nonnegative random variable whose cumulative distribution function admits a density, that is, a nonnegative Lebesgue integrable function such that The quantile function of the distribution function (also known as the increasing rearrangement of the random variable ) is defined by Thus, a quantile function is nothing but a pseudoinverse of . Motivated by statistics, a number of fast algorithms were developed for computing the quantile functions with high accuracy; see [8]. Without going through the details, we recall here the remarkable formula (due to G. Steinbrecher) for the quantile function of the normal distribution: where and According to Theorem 2.3, for every pair of continuous random variables with density and all positive numbers and , the following inequality holds: This can be seen as a principle of uncertainty, since it shows that the functions cannot be made simultaneously small.

Remark 2.6 (the higher dimensional analogue of Theorem 2.3). Consider a locally absolutely continuous kernel , and a family of nondecreasing functions defined on subintervals of . Then The proof is based on mathematical induction (which is left to the reader). The above inequality cover the n-variable generalization of Young's inequality as obtained by Oppenheim [9] (as well as the main result in [10]).

The following stronger version of Corollary 2.4 incorporates the Legendre duality.

Theorem 2.7. Let be a continuous nondecreasing function, and be a convex function whose conjugate is also defined on . Then for all , and one has

Proof. According to the Legendre duality, For and , we get and by integrating both sides from to , we obtain the inequality In a similar manner, starting with and , we arrive first at the inequality and then to Therefore, According to Theorem 2.3, and the inequality in the statement of Theorem 2.7 is now clear.

In the special case where , and (for some ), Theorem 2.7 yields the following inequality:

This remark extends a result due to Sulaiman [11].

We end this section by noticing the following result that complements Theorem 2.3.

Proposition 2.8. Under the assumptions of Lemma 2.2, Assuming is strictly positive almost everywhere, the equality occurs if and only if .

Proof. If , then from Lemma 2.2, we infer that The other case, , has a similar approach.

Proposition 2.8 extends a result due to Merkle [12].

3. The Precision in Young's Inequality

The main result of this section is as follows.

Theorem 3.1. Under the assumptions of Lemma 2.2,   for all and , Assuming is strictly positive almost everywhere, the equality occurs if and only if .

Proof. The case where is illustrated in Figure 4. The left-hand side of the inequality in the statement of Theorem 3.1 represents the measure of the cross-hatched curvilinear trapezium, while right-hand side is the measure of the rectangle.Therefore, The equality holds if and only if , that is, when .The case where is similar to the precedent one. The first term will be The equality holds if and only if so we must have .The case where is trivial, both sides of our inequality being equal to zero.

Figure 4: The geometry of the case.

Corollary 3.2 (Minguzzi [13]). If, moreover, on , and is continuous and increasing, then The equality occurs if and only if .

More accurate bounds can be indicated under the presence of convexity.

Corollary 3.3. Let be a nondecreasing continuous function, which is convex on the interval . Then (i)(ii)If is concave on the aforementioned interval, then the inequalities above work in the reverse way. Assuming is strictly positive almost everywhere, the equality occurs if and only if is an affine function or .

Proof. We will restrict here to the case of convex functions, the argument for the concave functions being similar.The left-hand side term of each of the inequalities in our statement represents the measure of the cross-hatched surface, see Figures 5 and 6.As the points of the graph of the convex function (restricted to the interval of endpoints and are under the chord joining and , it follows that this measure is less than the measure of the enveloping triangle when . This yields (i). The assertion (ii) follows in a similar way.

Figure 5: The geometry of the case.

Figure 6: The geometry of the case .

Corollary 3.3 extends a result due to Pečarić and Jakšetić [7]. They considered the special case where on and is increasing and differentiable, with an increasing derivative on the interval and . In this case the conclusion of Corollary 3.3 reads as follows:(i)(ii)

The equality holds if or is an affine function. The inequality sign should be reversed if has a decreasing derivative on the interval

4. The Connection with -Convexity

Motivated by the mass transportation theory, several people [14, 15] drew a parallel to the classical theory of convex functions by extending the Legendre duality. Technically, given two compact metric spaces and and a cost density function (which is supposed to be continuous), we may consider the following generalization of the notion of convex function.

Definition 4.1. A function is -convex if there exists a function such that We abbreviate (4.1) by writing . A useful remark is the equality that is, The classical notion of convex function corresponds to the case where is a compact interval and . The details can be found in [4, pages 40–42].Theorem 2.3 illustrates the theory of -convex functions for the spaces , (the Alexandrov one point compactification of and, respectively, ), and the cost function In fact, under the hypotheses of this theorem, the functions verify the relations and (due to the equality case as specified in the statement of Theorem 2.3), so they are both -convex.On the other hand, a simple argument shows that and are also convex in the usual sense.Let us call the function that admits a representation of form (4.4) with ,  absolutely continuous in the hyperbolic sense. With this terminology, Theorem 2.3 can be rephrased as follows.

Theorem 4.2. Suppose that is an absolutely continuous function in the hyperbolic sense with mixed derivative , and is a nondecreasing function such that . Then for all . almost everywhere, then (4.6) becomes an equality if and only if ; here we made the convention and .

Necessarily, an absolutely continuous function in the hyperbolic sense is continuous. It admits partial derivatives of the first order and a mixed derivativealmost everywhere. Besides, the functions and are defined everywhere in their interval of definition and represent absolutely continuous functions; they are also nondecreasing provided that almost everywhere.

A special case of Theorem 4.2 was proved by Páles [10, 16] (assuming is a continuously differentiable function with nondecreasing derivatives and and is an increasing homeomorphism). An example which escapes his result but is covered by Theorem 4.2 is offered by the function where denotes the fractional part of if , and if . According to Theorem 4.2, for every nondecreasing function such that .

Acknowledgment

The authors were supported by CNCSIS Grant PN2 ID_420.

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