# An Extension of Young's Inequality

Authors: Flavia-Corina Mitroi; Constantin P. Niculescu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

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


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## Body

## 1. Introduction

Young's inequality [1] asserts that every strictly increasing continuous function f:[0,∞)→[0,∞) with f(0)=0 and limx→∞f(x)=∞ verifies an inequality of the following form:(1.1)ab≤∫0af(x)dx+∫0bf-1(y)dy,
whenever a and bare nonnegative real numbers. The equality occurs if and only if f(a)=b; 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(1.2)F(a)=∫0af(x)dx,G(b)=∫0bf-1(x)dx
are both continuous and convex on [0,∞), and (1.1) can be restated as(1.3)ab≤F(a)+G(b)∀a,b∈[0,∞),
with equality if and only if f(a)=b. Because of the equality case, formula (1.3) leads to the following connection between the functions F and G:(1.4)F(a)=sup⁡{ab-G(b):b≥0},G(b)=sup⁡{ab-F(a):a≥0}.It turns out that each of these formulas produces a convex function (possibly on a different interval). Some details are in order.By definition, theconjugate of a convex function F defined on a nondegenerate interval I is the function
(1.5)F*:I*⟶R,F*(y)=sup⁡{xy-F(x):x∈I},
with domain I*={y∈ℝ:F*(y)<∞}. Necessarily I* is a nonempty interval, and F* is a convex function whose level sets {y:F*(y)≤λ} are closed subsets of ℝ for each λ∈ℝ (usually such functions are called closed convex functions).A convex function may not be differentiable, but it admits a good substitute for differentiability.Thesubdifferentialof a real function F defined on an interval I is a multivalued function ∂F:I→𝒫(ℝ) defined by(1.6)∂F(x)={λ∈R:F(y)≥F(x)+λ(y-x),foreveryy∈I}.Geometrically, the subdifferential gives us the slopes of the supporting lines for the graph ofF. The subdifferential at a point is always a convex set, possibly empty, but the convex functions F:I→ℝ have the remarkable property that ∂F(x)≠∅ at all interior points. It is worth noticing that ∂F(x)={F′(x)} at each point where F is differentiable (so this formula works for all points of I except for a countable subset), see [4, page 30].Lemma 1.1 (Legendre duality, [4, page 41]).
LetF:I→ℝ be a closed convex function. Then its conjugate F*:I*→ℝ is also convex and closed and(i)
xy≤F(x)+F*(y) for all x∈I, y∈I*;(ii)
xy=F(x)+F*(y) if and only if y∈∂F(x);(iii)
∂F*=(∂F)-1 (as graphs);(iv)
F**=F.Recall that the inverse of a graphΓ is the set Γ-1={(y,x):(x,y)∈Γ}.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]).
LetF be a continuous convex function defined on an interval I, and let φ:I→ℝ be a function such that φ(x)∈∂F(x) for every x∈I. Then for every a<b in I one has
(1.7)F(b)-F(a)=∫abφ(t)dt.As a consequence, the heuristic meaning of formula (i) in Lemma1.1 is the following Young-like inequality:(1.8)ab≤∫a0aφ(x)dx+∫b0bψ(y)dy∀a∈I,b∈I*,
where φ and ψ are selection functions for ∂F and,  (∂F)-1respectively. 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 ρ which are locally absolutely continuous with respect to the planar Lebesgue measure dxdy, see Theorem 2.3.A special case of Young's inequality is(1.9)xy≤xpp+yqq,
which works for all x,y≥0, and p,q>1 with 1/p+1/q=1. Theorem 2.3 yields the following companion to this inequality in the case of Gaussian measure (4/2π)e-x2-y2dxdy on [0,∞)×[0,∞):(1.10)erf⁡(x)erf⁡(y)≤2π∫0xerf⁡(sp-1)e-s2ds+2π∫0yerf⁡(tq-1)e-t2dt,
where(1.11)erf⁡⁡(x)=2π∫0xe-s2ds
is the Gauss error function (or the erf function).The precision of our generalization of Young's inequality makes the objective of Section3.In Section4 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 ofc-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 functionsf:[a0,a1)→[A0,A1) such that a0<a1≤∞ and A0<A1≤∞,f(a0)=A0 and limx→a1f(x)=A1. In other words, the interval [0,∞) 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 followsf:[0,∞)→[0,∞) will denote a nondecreasing function such that f(0)=0 and limx→∞f(x)=∞. Since f is not necessarily injective we will attach to a  pseudoinversef by the following formula:(2.1)fsup⁡-1:[0,∞)⟶[0,∞),fsup⁡-1(y)=inf⁡{x≥0:f(x)>y}.Clearly,fsup-1 is nondecreasing and fsup-1(f(x))≥x for all x. Moreover, with the convention f(0-)=0,(2.2)fsup⁡-1(y)=sup⁡{x:y∈[f(x-),f(x+)]}.
Here f(x-) and f(x+) represent the lateral limits at x. When f is also continuous,(2.3)fsup⁡-1(y)=max⁡⁡{x≥0:y=f(x)}.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 functiong such that
(2.4)finf⁡-1≤g≤fsup⁡-1,
where finf-1(y)=sup{x≥0:f(x)<y}. Necessarily, g is nondecreasing, and any two pseudoinverses agree except on a countable set (so their integrals will be the same).
Given0≤a<b, we define the epigraph and the hypograph of f|[a,b], respectively, by
(2.5)epi⁡f|[a,b]={(x,y)∈[a,b]×[f(a),f(b)]:y≥f(x)},hypf|[a,b]={(x,y)∈[a,b]×[f(a),f(b)]:y≤f(x)}.
Their intersection is thegraph of f|[a,b],
(2.6)graphf|[a,b]={(x,y)∈[a,b]×[f(a),f(b)]:y=f(x)}.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 measureρ on [0,∞)×[0,∞), which is locally absolutely continuous with respect to the Lebesgue measure dxdy that is, ρ is of the form(2.7)ρ(A)=∫AK(x,y)dxdy,
where K:[0,∞)×[0,∞)→[0,∞)is a Lebesgue locally integrable function, and A is any compact subset of [0,∞)×[0,∞).Clearly,(2.8)ρ(hypf|[a,b])+ρ(epi⁡f|[a,b])=ρ([a,b]×[f(a),f(b)])=∫ab∫f(a)f(b)K(x,y)dydx.Moreover,(2.9)ρ(hypf|[a,b])=∫ab(∫f(a)f(x)K(x,y)dy)dx,ρ(epi⁡f|[a,b])=∫f(a)f(b)(∫afsup⁡-1(y)K(x,y)dx)dy.The discussion above can be summarized as follows.Lemma 2.2.
Letf:[0,∞)→[0,∞) be a nondecreasing function such that f(0)=0 and limx→∞f(x)=∞. Then for every Lebesgue locally integrable function K:[0,∞)×[0,∞)→[0,∞) and every pair of nonnegative numbers a<b,
(2.10)∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)f(b)(∫afsup⁡-1(y)K(x,y)dx)dy=∫ab∫f(a)f(b)K(x,y)dydx.We can now state the main result of this section.Theorem 2.3 (Young's inequality for nondecreasing functions).
Under the assumptions of Lemma2.2, for every pair of nonnegative numbers a<b and every number c≥f(a), one has
(2.11)∫ab∫f(a)cK(x,y)dydx≤∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy.
If in addition K is strictly positive almost everywhere, then the equality occurs if and only if c∈[f(b-),f(b+)].Proof.
We start with the case wheref(a)≤c≤f(b-), see Figure 1. In this case,
(2.12)∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy=∫afsup⁡-1(c)(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy+∫fsup⁡-1(c)b(∫f(a)f(x)K(x,y)dy)dx=∫afsup⁡-1(c)∫f(a)cK(x,y)dydx+∫fsup⁡-1(c)b(∫cf(x)K(x,y)dy)dx+∫fsup⁡-1(c)b∫f(a)cK(x,y)dydx≥∫ab∫f(a)cK(x,y)dydx,
with equality if and only if ∫fsup-1(c)b(∫cf(x)K(x,y)dy)dx=0. When K is strictly positive almost everywhere, this means that c=f(b-).
Ifc≥f(b+), then
(2.13)∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy=∫afsup⁡-1(c)(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy-∫bfsup⁡-1(c)(∫f(a)f(x)K(x,y)dy)dx=∫afsup⁡-1(c)∫f(a)cK(x,y)dydx-(∫bfsup⁡-1(c)(∫f(a)cK(x,y)dy)dx-∫f(b+)c(∫afsup⁡-1(y)K(x,y)dx)dy)≥∫ab∫f(a)cK(x,y)dydx.
The equality holds if and only if∫f(b+)c(∫afsup-1(y)K(x,y)dx)dy=0, that is, when c=f(b+) (provided that K is strictly positive almost everywhere), see Figure 2.
Ifc∈(f(b-),f(b+)), then fsup-1(c)=b 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 whenf(a)≤c≤f(b-).Figure 2
The casec≥f(b+).Figure 3
The equality case.Corollary 2.4.
(Young's inequality for continuous increasing functions). Iff:[0,∞)→[0,∞) is also continuous and increasing, then
(2.14)∫ab∫f(a)cK(x,y)dydx≤∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫af-1(y)K(x,y)dx)dy
for every real number c≥f(a). Assuming K is strictly positive almost everywhere, the equality occurs if and only if c=f(b).IfK(x,y)=1 for every x,y∈[0,∞), then Corollary 2.4 asserts that(2.15)bc-af(a)<∫abf(x)dx+∫f(a)cf-1(y)dy∀0<a<b,c>f(a).
The equality occurs if and only if c=f(b). In the special case where a=f(a)=0, this reduces to the classical inequality of Young.Remark 2.5 (the probabilistic companion of Theorem2.3).
Suppose there is a given nonnegative random variableX:[0,∞)→[0,∞)  whose cumulative distribution function FX(x)=P(X≤x) admits a density, that is, a nonnegative Lebesgue integrable function ρX,  such that
(2.16)P(x≤X≤y)=∫xyρX(u)du∀x≤y.
The quantile function of the distribution function FX (also known as the increasing rearrangement of the random variable X) is defined by
(2.17)QX(x)=inf⁡⁡{y:FX(y)≥x}.
Thus, a quantile function is nothing but a pseudoinverse of FX. 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:
(2.18)erf-1(z)=∑k=0∞ck((π/2)z)2k+12k+1,
where c0=1  and
(2.19)ck=∑m=0k-1cmck-m-1(m+1)(2m+1)∀k≥1.
According to Theorem2.3, for every pair of continuous random variables Y,Z:[0,∞)→[0,∞) with density ρY,Z  and all positive numbers b  and c,  the following inequality holds:
(2.20)P(Y≤b;Z≤c)≤∫0b(∫0FX(x)ρY,Z(x,y)dy)dx+∫0c(∫0QX(y)ρY,Z(x,y)dx)dy.
This can be seen as a principle of uncertainty, since it shows that the functions
(2.21)x⟶∫0FX(x)ρY,Z(x,y)dy,y⟶∫0QX(y)ρY,Z(x,y)dx
cannot be made simultaneously small.Remark 2.6 (the higher dimensional analogue of Theorem2.3).
Consider a locally absolutely continuous kernelK:[0,∞)×⋯×[0,∞)→[0,∞),K=K(s1,s2,…,sn),  and a family ϕ1,…,ϕn:[ai,bi]→ℝof nondecreasing functions defined on subintervals of [0,∞). Then
(2.22)∫ϕ1(a1)ϕ1(b1)∫ϕ2(a2)ϕ2(b2)⋯∫ϕn(an)ϕn(bn)K(s1,s2,…,sn)dsn⋯ds2ds1≤∑i=1n∫ϕi(ai)ϕi(bi)(∫ϕ1(a1)ϕ1(s)⋯∫ϕn(an)ϕn(s)K(s1,…,sn)dsn…dsi+1dsi-1⋯ds1)ds.
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 Corollary2.4 incorporates the Legendre duality.Theorem 2.7.
Letf:[0,∞)→[0,∞) be a continuous nondecreasing function, and Φ:[0,∞)→ℝ be a convex function whose conjugate is also defined on [0,∞). Then for all b>a≥0,c≥f(a), and ε>0 one has
(2.23)∫abΦ(ε∫f(a)f(x)K(x,y)dy)dx+∫f(a)cΦ*(1ε∫afsup⁡-1(y)K(x,y)dx)dy≥∫ab∫f(a)cK(x,y)dydx-(c-f(a))Φ(ε)-(b-a)Φ*(1ε).Proof.
According to the Legendre duality,(2.24)Φ(εu)+Φ*(vε)≥uv∀u,v,ε≥0.
For u=∫f(a)f(x)K(x,y)dy and v=1, we get
(2.25)Φ(ε∫f(a)f(x)K(x,y)dy)+Φ*(1ε)≥∫f(a)f(x)K(x,y)dy,
and by integrating both sides from a to b, we obtain the inequality
(2.26)∫abΦ(ε∫f(a)f(x)K(x,y)dy)dx+(b-a)Φ*(1ε)≥∫ab(∫f(a)f(x)K(x,y)dy)dx.
In a similar manner, starting with u=1 and v=∫afsup-1(y)K(x,y)dx, we arrive first at the inequality
(2.27)Φ(ε)+Φ*(1ε∫afsup⁡-1(y)K(x,y)dx)≥∫afsup⁡-1(y)K(x,y)dx,
and then to
(2.28)(c-f(a))Φ(ε)+∫f(a)cΦ*(1ε∫afsup⁡-1(y)K(x,y)dx)dy≥∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy.
Therefore,
(2.29)∫abΦ(ε∫f(a)f(x)K(x,y)dy)dx+∫f(a)cΦ*(1ε∫afsup⁡-1(y)K(x,y)dx)dy≥∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy-(b-a)Φ*(1ε)-(c-f(a))Φ(ε).
According to Theorem 2.3,
(2.30)∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy≥∫ab∫f(a)cK(x,y)dydx,
and the inequality in the statement of Theorem 2.7 is now clear.In the special case whereK(x,y)=1,a=f(a)=0, and Φ(x)=xp/p (for some p>1), Theorem 2.7 yields the following inequality:(2.31)∫0bfp(x)dx+∫0c(fsup⁡-1(y))pdy≥pbc-(p-1)(b+c),foreveryb,c≥0.This remark extends a result due to Sulaiman [11].We end this section by noticing the following result that complements Theorem2.3.Proposition 2.8.
Under the assumptions of Lemma2.2,
(2.32)∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy≤max⁡{∫ab∫f(a)f(b)K(x,y)dydx,∫afsup⁡-1(c)∫f(a)cK(x,y)dydx}.
Assuming K is strictly positive almost everywhere, the equality occurs if and only if c=f(b).Proof.
Ifc<f(b), then from Lemma 2.2, we infer that
(2.33)∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy=∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)f(b)(∫afsup⁡-1(y)K(x,y)dx)dy-∫cf(b)(∫afsup⁡-1(y)K(x,y)dx)dy≤∫ab∫f(a)f(b)K(x,y)dydx.
The other case, c≥f(b), has a similar approach.Proposition2.8 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 Lemma2.2,   for all b≥a≥0 and c≥f(a),
(3.1)∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy-∫ab∫f(a)cK(x,y)dydx≤|∫fsup⁡-1(c)b∫cf(b)K(x,y)dydx|.
Assuming K is strictly positive almost everywhere, the equality occurs if and only if c=f(b).Proof.
The case wheref(a)≤c≤f(b-) 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 ABCD rectangle.
Therefore,(3.2)∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy-∫ab∫f(a)cK(x,y)dydx=∫fsup⁡-1(c)b(∫cf(x)K(x,y)dy)dx≤∫fsup⁡-1(c)b∫cf(b)K(x,y)dydx.
The equality holds if and only if ∫fsup-1(c)b(∫cf(x)K(x,y)dy)dx=0, that is, when f(b-)=c.
The case wherec≥f(b+) is similar to the precedent one. The first term will be
(3.3)∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy-∫ab∫f(a)cK(x,y)dydx=∫bfsup⁡-1(c)(∫f(b)f(x)K(x,y)dy)dx≤∫bfsup⁡-1(c)∫f(b)cK(x,y)dydx.
The equality holds if and only if ∫bfsup-1(c)∫f(b)cK(x,y)dydx=0,so we must have f(b+)=c.
The case wherec∈[f(b-),f(b+)] is trivial, both sides of our inequality being equal to zero.Figure 4
The geometry of the casef(a)≤c≤f(b-).Corollary 3.2 (Minguzzi [13]).
If, moreover,K(x,y)=1 on [0,∞)×[0,∞), and f is continuous and increasing, then
(3.4)∫abf(x)dx+∫f(a)cf-1(y)dy-bc+af(a)≤(f-1(c)-b)⋅(c-f(b)).
The equality occurs if and only if c=f(b).More accurate bounds can be indicated under the presence of convexity.Corollary 3.3.
Letf be a nondecreasing continuous function, which is convex on the interval [min{fsup-1(c),b},max{fsup-1(c),b}]. Then (i)
(3.5)∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy-∫ab∫f(a)cK(x,y)dydx≤∫fsup⁡-1(c)b∫cc+[(f(b)-c)/(b-fsup⁡-1(c))](x-fsup⁡-1(c))K(x,y)dydx,foreveryc≤f(b),(ii)
(3.6)∫ab(∫f(a)f(x)K(x,y)dy)dx+∫f(a)c(∫afsup⁡-1(y)K(x,y)dx)dy-∫ab∫f(a)cK(x,y)dydx≥∫bfsup⁡-1(c)∫f(b)f(b)+[(c-f(b))/(fsup⁡-1(c)-b)](x-b)K(x,y)dydx,foreveryc≥f(b).If f is concave on the aforementioned interval, then the inequalities above work in the reverse way.
AssumingK is strictly positive almost everywhere, the equality occurs if and only if f is an affine function or f(b)=c.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 Figures5 and 6.
As the points of the graph of the convex functionf (restricted to the interval of endpoints b and fsup-1(c)) are under the chord joining (b,f(b)) and (fsup-1(c),c), it follows that this measure is less than the measure of the enveloping triangle MNQ when c≤f(b). This yields (i). The assertion (ii) follows in a similar way.Figure 5
The geometry of the casec≤f(b).Figure 6
The geometry of the casec≥f(b).Corollary3.3 extends a result due to Pečarić and Jakšetić [7]. They considered the special case where K(x,y)=1 on [0,∞)×[0,∞) and f:[0,∞)→[0,∞) is increasing and differentiable, with an increasing derivative on the interval [min{f-1(c),b},max{f-1(c),b}] and f(0)=0. In this case the conclusion of Corollary 3.3 reads as follows:(i)
(3.7)∫0bf(x)dx+∫0cf-1(y)dy-bc≤12(f-1(c)-b)(c-f(b))forc<f(b),(ii)
(3.8)∫0bf(x)dx+∫0cf-1(y)dy-bc≥12(f-1(c)-b)(c-f(b))forc>f(b).The equality holds iff(b)=c or f is an affine function. The inequality sign should be reversed if f has a decreasing derivative on the interval(3.9)[min⁡{f-1(c),b},max⁡{f-1(c),b}].


## 4.   The Connection withc-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 X and Y and a cost density function c:X×Y→ℝ (which is supposed to be continuous), we may consider the following generalization of the notion of convex function.Definition 4.1.
A functionF:X→ℝ is c-convex if there exists a function G:Y→ℝ such that
(4.1)F(x)=sup⁡y∈Y⁡{c(x,y)-G(y)},∀x∈X.
We abbreviate (4.1) by writing F=Gc. A useful remark is the equality
(4.2)Fcc=F,
that is,
(4.3)F(x)=sup⁡y∈Y⁡{c(x,y)-Fc(y)},∀x∈X.
The classical notion of convex function corresponds to the case whereX is a compact interval and c(x,y)=xy. The details can be found in [4, pages 40–42].
Theorem2.3 illustrates the theory of c-convex functions for the spaces X=[a,∞], Y=[f(a),∞] (the Alexandrov one point compactification of [a,∞) and, respectively, [f(a),∞)), and the cost function
(4.4)c(x,y)=∫ax∫f(a)yK(s,t)dtds.
In fact, under the hypotheses of this theorem, the functions
(4.5)F(x)=∫ax(∫f(a)f(s)K(s,t)dt)ds,x≥a,G(y)=∫f(a)y(∫afsup⁡⁡(t)-1K(s,t)ds)dt,y≥f(a),
verify the relations Fc=G and Gc=F (due to the equality case as specified in the statement of Theorem 2.3), so they are both c-convex.
On the other hand, a simple argument shows thatF and G are also convex in the usual sense.
Let us call the functionc that admits a representation of form (4.4) with K∈L1(ℝ×ℝ),  absolutely  continuous in the hyperbolic sense. With this terminology, Theorem 2.3 can be rephrased as follows.Theorem 4.2.
Suppose thatc:[a,b]×[A,B]→ℝ is an absolutely continuous function in the hyperbolic sense with mixed derivative ∂2c/∂x∂y≥0, and f:[a,b]→[A,B] is a nondecreasing function such that f(a)=A. Then
(4.6)c(x,y)-c(a,f(a))≤∫ax∂c∂t(t,f(t))dt+∫f(a)y∂c∂s(fsup⁡-1(s),s)ds,
for all (x,y)∈[a,A]×[b,B].
If∂2c/∂x∂y>0  almost everywhere, then (4.6) becomes an equality if and only if y∈[f(x-),f(x+)]; here we made the convention f(a-)=f(a) and f(b+)=f(b).Necessarily, an absolutely continuous functionc in the hyperbolic sense is continuous. It admits partial derivatives of the first order and a mixed derivative∂2c/∂x∂yalmost everywhere. Besides, the functions y→(∂c/∂x)(x,y)and x→(∂c/∂y)(x,y) are defined everywhere in their interval of definition and represent absolutely continuous functions; they are also nondecreasing provided that ∂2c/∂x∂y≥0 almost everywhere.A special case of Theorem4.2 was proved by Páles [10, 16] (assuming c:[a,A]×[b,B]→ℝ is a continuously differentiable function with nondecreasing derivatives y→(∂c/∂x)(x,y) and x→(∂c/∂y)(x,y), and f:[a,b]→[A,B]is an increasing homeomorphism). An example which escapes his result but is covered by Theorem 4.2 is offered by the function (4.7)c(x,y)=∫0x{1s}ds∫0y{1t}dt,x,y≥0,
where {1/s} denotes the fractional part of 1/s if s>0, and {1/s}=0 if s=0. According to Theorem 4.2,(4.8)∫0x{1s}ds∫0y{1t}dt≤∫0x({1s}∫0f(s){1t}dt)ds+∫0y({1t}∫0fsup⁡-1(t){1s}ds)dt,
for every nondecreasing function f:[0,∞)→[0,∞) such that f(0)=0.



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*Source: 162049-2011-09-26.xml*