An Extension of Young's Inequality
AAN EXTENSION OF YOUNG’S INEQUALITY
FLAVIA-CORINA MITROI AND CONSTANTIN P. NICULESCU
Abstract.
Young’s inequality is extended to the context of absolutely con-tinuous measures. Several applications are included. Introduction
Young’s inequality [18] asserts that every strictly increasing continuous function f : [0 , ∞ ) −→ [0 , ∞ ) with f (0) = 0 and lim x →∞ f ( x ) = ∞ verifies an inequality of thefollowing form,(1.1) ab ≤ (cid:90) a f ( x ) dx + (cid:90) b f − ( y ) dy, whenever a and b are nonnegative real numbers. The equality occurs if and only if f ( a ) = b . See [4], [8], [9] and [14] 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?F. Cunningham Jr. and N. Grossman [2] noticed that the question (Q1) hasa positive answer (correcting the prevalent belief that Young’s inequality is thebusiness of strictly increasing continuous functions). The aim of the present paperis to extend the entire discussion to the framework of locally absolutely continuousmeasures and to prove several improvements.As well known, Young’s inequality is an illustration of the Legendre duality.Precisely, the functions F ( a ) = (cid:90) a f ( x ) dx and G ( b ) = (cid:90) b f − ( x ) dx, are both continuous and convex on [0 , ∞ ) and (1.1) can be restated as(1.2) ab ≤ F ( a ) + G ( b ) for all a, b ∈ [0 , ∞ ) , with equality if and only if f ( a ) = b. Because of the equality case, the formula (1.2)leads to the following connection between the functions F and G :(1.3) F ( a ) = sup { ab − G ( b ) : b ≥ } and G ( b ) = sup { ab − F ( a ) : a ≥ } . Date : June 2011.2000
Mathematics Subject Classification.
Primary 26A51, 26D15; Secondary 90B06.
Key words and phrases.
Young’s inequality, Legendre duality, convex function.Corresponding author: Constantin P. Niculescu. a r X i v : . [ m a t h . C A ] J un FLAVIA-CORINA MITROI AND CONSTANTIN P. NICULESCU
It turns out that each of these formulas produces a convex function (possibly ona different interval). Some details are in order.By definition, the conjugate of a convex function F defined on a nondegenerateinterval I is the function F ∗ : I ∗ → R , F ∗ ( y ) = sup { xy − F ( x ) : x ∈ I } , with domain I ∗ = { y ∈ R : F ∗ ( y ) < ∞} . Necessarily I ∗ is an non-empty intervaland 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 fordifferentiability.The subdifferential of a real function F defined on an interval I is a multivaluedfunction ∂F : I → P ( R ) defined by ∂F ( x ) = { λ ∈ R : F ( y ) ≥ F ( x ) + λ ( y − x ), for every y ∈ I } . Geometrically, the subdifferential gives us the slopes of the supporting lines forthe graph of F . The subdifferential at a point is always a convex set, possiblyempty, but the convex functions F : I → R have the remarkable property that ∂F ( x ) (cid:54) = ∅ at all interior points. It is worth noticing that ∂F ( x ) = { F (cid:48) ( x ) } at eachpoint where F is differentiable (so this formula works for all points of I except fora countable subset). See [9], page 30. Lemma 1. ( Legendre duality, [9] , page . Let F : I → R be a closed convexfunction. 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 ) − ( as graphs ); iv ) F ∗∗ = F. Recall that the inverse of a graph Γ is the set Γ − = { ( y, x ) : ( x, y ) ∈ Γ } . How far is Young’s inequality from the Legendre duality? Surprisingly, they arepretty closed in the sense that in most cases the Legendre duality can be convertedinto a Young like inequality. Indeed, every continuous convex function admits anintegral representation.
Lemma 2. ( See [9] , page . Let F be a continuous convex function defined onan 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 we have F ( b ) − F ( a ) = (cid:90) ba ϕ ( t ) dt. As a consequence, the heuristic meaning of the formula i ) in Lemma 1 is thefollowing Young like inequality, ab ≤ (cid:90) aa ϕ ( x ) dx + (cid:90) bb ψ ( y ) dy for all a ∈ I, b ∈ I ∗ , where ϕ and ψ are selection functions for ∂F and respectively ( ∂F ) − . Now itbecomes clear that Young’s inequality should work outside strict monotonicity (aswell 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 theframework of integrability to all positive measures ρ which are locally absolutely OUNG’S INEQUALITY 3 continuous with respect to the planar Lebesgue measure dxdy . See Theorem 1below.A special case of Young’s inequality is xy ≤ x p p + y q q , which works for all x, y ≥
0, and p, q > /p + 1 /q = 1. Theorem 1yields the following companion to this inequality in the case of Gaussian measure π e − x − y dxdy on [0 , ∞ ) × [0 , ∞ ) :erf( x ) erf( y ) ≤ √ π (cid:90) x erf (cid:0) s p − (cid:1) e − s ds + 2 √ π (cid:90) y erf (cid:0) t q − (cid:1) e − t dt, where(1.4) erf( x ) = 2 √ π (cid:90) x e − s ds is the Gauss error function (or the erf function).The precision of our generalization of Young’s inequality makes the objective ofSection 3.In Section 4 we discuss yet another extension of Young’s inequality, based onrecent work done by J. Jakˇseti´c and J. E. Peˇcari´c [13].The paper ends by noticing the connection of our result to the theory of c -convexity (that is, of convexity associated to a cost density function).Last but not the least, all results in this paper can be extended verbatim to theframework of nondecreasing functions f : [ a , a ) → [ A , A ) such that a < a ≤ ∞ and A < A ≤ ∞ , f ( a ) = A and lim x → a f ( x ) = A . 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 nonin-creasing functions, but this is outside the scope of the present paper.2. Young’s inequality for weighted measures
In what follows f : [0 , ∞ ) −→ [0 , ∞ ) will denote a nondecreasing function suchthat f (0) = 0 and lim x →∞ f ( x ) = ∞ . Since f is not necessarily injective we will attachto f a pseudo-inverse by the following formula: f − : [0 , ∞ ) −→ [0 , ∞ ) , f − ( y ) = inf { x ≥ f ( x ) > y } . Clearly, f − is nondecreasing and f − ( f ( x )) ≥ x for all x. Moreover, with theconvention f (0 − ) = 0 ,f − ( 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, f − ( y ) = max { x ≥ y = f ( x ) } . Remark 1. ( F. Cunningham Jr. and N. Grossman [2]) . Since pseudo-inverses willbe used as integrands, it is convenient to enlarge the concept of pseudo-inverse byreferring to any function g such that f − ≤ g ≤ f − , FLAVIA-CORINA MITROI AND CONSTANTIN P. NICULESCU where f − ( y ) = sup { x ≥ f ( x ) < y } . Necessarily, g is nondecreasing and anytwo pseudo-inverses agree except on a countable set (so their integrals will be thesame) . Given 0 ≤ a < b, we define the epigraph and the hypograph of f | [ a,b ] respectivelyby epi f | [ a,b ] = { ( x, y ) ∈ [ a, b ] × [ f ( a ) , f ( b )] : y ≥ f ( x ) } , and hyp f | [ a,b ] = { ( x, y ) ∈ [ a, b ] × [ f ( a ) , f ( b )] : y ≤ f ( x ) } . Their intersection is the graph of f | [ a,b ] , graph f | [ 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 measure ρ on [0 , ∞ ) × [0 , ∞ ) , which is locally absolutelycontinuous with respect to the Lebesgue measure dxdy, that is, ρ is of the form ρ ( A ) = (cid:90) A K ( x, y ) dxdy, where K : [0 , ∞ ) × [0 , ∞ ) −→ [0 , ∞ ) is a Lebesgue locally integrable function, and A is any compact subset of [0 , ∞ ) × [0 , ∞ ).Clearly, ρ (cid:0) hyp f | [ a,b ] (cid:1) + ρ (cid:0) epi f | [ a,b ] (cid:1) = ρ ([ a, b ] × [ f ( a ) , f ( b )])= (cid:90) ba (cid:90) f ( b ) f ( a ) K ( x, y ) dydx. Moreover, ρ (cid:0) hyp f | [ a,b ] (cid:1) = (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx. and ρ (cid:0) epi f | [ a,b ] (cid:1) = (cid:90) f ( b ) f ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy. The discussion above can be summarized as follows:
Lemma 3.
Let f : [0 , ∞ ) −→ [0 , ∞ ) be a nondecreasing function such that f (0) =0 and lim x →∞ f ( x ) = ∞ . Then for every Lebesgue locally integrable function K :[0 , ∞ ) × [0 , ∞ ) −→ [0 , ∞ ) and every pair of nonnegative numbers a < b, (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) f ( b ) f ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy = (cid:90) ba (cid:90) f ( b ) f ( a ) K ( x, y ) dydx. We can now state the main result of this section:
OUNG’S INEQUALITY 5
Theorem 1. ( Young’s inequality for nondecreasing functions ) . Under the assump-tions of Lemma , for every pair of nonnegative numbers a < b, and every number c ≥ f ( a ) we have (cid:90) ba (cid:90) cf ( a ) K ( x, y ) dydx ≤ (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy. If in addition K is strictly positive almost everywhere, then the equality occurs ifand only if c ∈ [ f ( b − ) , f ( b +)] . Proof.
We start with the case where f ( a ) ≤ c ≤ f ( b − ). See Figure 1. Figure 1.
The geometry of Young’s inequality when f ( a ) ≤ c ≤ f ( b − ) . In this case, (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy = (cid:90) f − ( c ) a (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy + (cid:90) bf − ( c ) (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx = (cid:90) f − ( c ) a (cid:90) cf ( a ) K ( x, y ) dydx + (cid:90) bf − ( c ) (cid:32)(cid:90) f ( x ) c K ( x, y ) dy (cid:33) dx + (cid:90) bf − ( c ) (cid:90) cf ( a ) K ( x, y ) dydx ≥ (cid:90) ba (cid:90) cf ( a ) K ( x, y ) dydx, FLAVIA-CORINA MITROI AND CONSTANTIN P. NICULESCU
Figure 2.
The case c ≥ f ( b +) . with equality if and only if (cid:82) bf − ( c ) (cid:16)(cid:82) f ( x ) c K ( x, y ) dy (cid:17) dx = 0 . When K is strictlypositive almost everywhere, this means that c = f ( b − ).If c ≥ f ( b +) , then (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy = (cid:90) f − ( c ) a (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy − (cid:90) f − ( c ) b (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx = (cid:90) f − ( c ) a (cid:90) cf ( a ) K ( x, y ) dydx − (cid:32)(cid:90) f − ( c ) b (cid:32)(cid:90) f ( c ) f ( a ) K ( x, y ) dy (cid:33) dx − (cid:90) cf ( b +) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy (cid:33) ≥ (cid:90) ba (cid:90) cf ( a ) K ( x, y ) dydx. Equality holds if and only if (cid:82) cf ( b +) (cid:16)(cid:82) f − ( y ) a K ( x, y ) dx (cid:17) dy , that is, when c = f ( b +)(provided that K is strictly positive almost everywhere). See Figure 2.If c ∈ ( f ( b − ) , f ( b +)) , then f − ( c ) = b and the inequality in the statement ofTheorem 1 is actually an equality. See Figure 3. (cid:3) OUNG’S INEQUALITY 7
Figure 3.
The equality case.
Corollary 1. ( Young’s inequality for continuous increasing functions ) . If f :[0 , ∞ ) −→ [0 , ∞ ) is also continuous and increasing, then (cid:90) ba (cid:90) cf ( a ) K ( x, y ) dydx ≤ (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy for every real number c ≥ f ( a ) . Assuming K strictly positive almost everywhere,the equality occurs if and only if c = f ( b ) . If K ( x, y ) = 1 for every x, y ∈ [0 , ∞ ), then Corollary 1 asserts that bc − af ( a ) < (cid:90) ba f ( x ) dx + (cid:90) cf ( a ) f − ( y ) dy for all 0 < a < b and c > f ( a );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. ( The probabilistic companion of Theorem 1 ) . Suppose there is givena nonnegative random variable X : [0 , ∞ ) → [0 , ∞ ) whose cumulative distributionfunction F X ( x ) = P ( X ≤ x ) admits a density, that is, a nonnegative Lebesgue-integrable function ρ X such that P ( x ≤ X ≤ y ) = (cid:90) yx ρ X ( u ) du for all x ≤ y. The quantile function of the distribution function F X (also known as the increasingrearrangement of the random variable X ) is defined by Q X ( x ) = inf { y : F X ( y ) ≥ x } . Thus, a quantile function is nothing but a pseudo-inverse of F X . Motivated byStatistics, a number of fast algorithms were developed for computing the quantile
FLAVIA-CORINA MITROI AND CONSTANTIN P. NICULESCU functions with high accuracy. See [1] . Without entering the details, we recall herethe remarkable formula (due to G. Steinbrecher) for the quantile function of thenormal distribution: erf − ( z ) = ∞ (cid:88) k =0 c k (cid:16) √ π z (cid:17) k +1 k + 1 , where c = 1 and c k = k − (cid:88) m =0 c m c k − m − ( m + 1) (2 m + 1) for all k ≥ . According to Theorem for every pair of continuous random variables Y, Z :[0 , ∞ ) → [0 , ∞ ) with density ρ Y,Z , and every positive numbers b and c, the followinginequality holds: P ( Y ≤ b ; Z ≤ c ) ≤ (cid:90) b (cid:32)(cid:90) F X ( x )0 ρ Y,Z ( x, y ) dy (cid:33) dx + (cid:90) c (cid:32)(cid:90) Q X ( y )0 ρ Y,Z ( x, y ) dx (cid:33) dy. This can be seen as a principle of uncertainty, since it shows that the functions x → (cid:90) F X ( x )0 ρ Y,Z ( x, y ) dy and y → (cid:90) Q X ( y )0 ρ Y,Z ( x, y ) dx cannot be made simultaneously small. Remark 3. ( The higher dimensional analogue of Theorem . Consider a lo-cally absolutely continuous kernel K : [0 , ∞ ) × ... × [0 , ∞ ) −→ [0 , ∞ ) , K = K ( s , s , ..., s n ) , and a family φ , ..., φ n : [ a i , b i ] → R of nondecreasing functionsdefined on subintervals of [0 , ∞ ) . Then (cid:90) φ ( b ) φ ( a ) (cid:90) φ ( b ) φ ( a ) · · · (cid:90) φ n ( b n ) φ n ( a n ) K ( s , s , ..., s n ) ds n ...ds ds ≤ n (cid:88) i =1 (cid:90) φ i ( b i ) φ i ( a i ) (cid:32)(cid:90) φ ( s ) φ ( a ) · · · (cid:90) φ n ( s ) φ n ( a n ) K ( s , ..., s n ) ds n ...ds i +1 ds i − ...ds (cid:33) 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 asobtained by Oppenheim [10] (as well as the main result in [12]).The following stronger version of Corollary 1 incorporates the Legendre duality.
Theorem 2.
Let f : [0 , ∞ ) −→ [0 , ∞ ) be a continuous nondecreasing function and Φ : [0 , ∞ ) → R a convex function whose conjugate is also defined on [0 , ∞ ) . Thenfor all b > a ≥ , c ≥ f ( a ) , and ε > we have (cid:90) ba Φ (cid:32) ε (cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) Φ ∗ (cid:32) ε (cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dx ≥ (cid:90) ba (cid:90) cf ( a ) K ( x, y ) dydx − ( c − f ( a ))Φ ( ε ) − ( b − a )Φ ∗ (1 /ε ) . OUNG’S INEQUALITY 9
Proof.
According to the Legendre duality,(2.1) Φ( εu ) + Φ ∗ ( v/ε ) ≥ uv for all u, v, ε ≥ . For u = (cid:82) f ( x ) f ( a ) K ( x, y ) dy and v = 1 we getΦ (cid:32) ε (cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) + Φ ∗ (1 /ε ) ≥ (cid:90) f ( x ) f ( a ) K ( x, y ) dy, and by integrating both sides from a to b we obtain the inequality (cid:90) ba Φ (cid:32) ε (cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + ( b − a )Φ ∗ (1 /ε ) ≥ (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx. In a similar manner, starting with u = 1 and v = (cid:82) f − ( y ) a K ( x, y ) dx, we arrive firstat the inequalityΦ ( ε ) + Φ ∗ (cid:32) ε (cid:90) f − ( y ) a K ( x, y ) dx (cid:33) ≥ (cid:90) f − ( y ) a K ( x, y ) dx, and then to( c − f ( a ))Φ ( ε ) + (cid:90) cf ( a ) Φ ∗ (cid:32) ε (cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dx ≥ (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy. Therefore, (cid:90) ba Φ (cid:32) ε (cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) Φ ∗ (cid:32) ε (cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dx ≥ (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy − ( b − a )Φ ∗ (1 /ε ) − ( c − f ( a ))Φ ( ε ) . According to Theorem 1, (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy ≥ (cid:90) ba (cid:90) cf ( a ) K ( x, y ) dydx, and the inequality in the statement of Theorem 2 is now clear. (cid:3) In the special case where K ( x, y ) = 1 , a = f ( a ) = 0 and Φ( x ) = x p /p (for some p > (cid:90) b f p ( x ) dx + (cid:90) c (cid:0) f − ( y ) (cid:1) p dy ≥ pbc − ( p −
1) ( b + c ) , for every b, c ≥ . This remark extends a result due to W. T. Sulaiman [15].We end this section by noticing the following result that complements Theorem1.
Proposition 1.
Under the assumptions of Lemma 3, (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy ≤ max (cid:40)(cid:90) ba (cid:90) f ( b ) f ( a ) K ( x, y ) dydx, (cid:90) f − ( c ) a (cid:90) cf ( a ) K ( x, y ) dydx (cid:41) . Assuming K strictly positive almost everywhere, the equality occurs if and only if c = f ( b ) . Proof. If c < f ( b ), then from Lemma 3 we infer that (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy = (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) f ( b ) f ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy − (cid:90) f ( b ) c (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy ≤ (cid:90) ba (cid:90) f ( b ) f ( a ) K ( x, y ) dydx The other case, c ≥ f ( b ), has a similar approach. (cid:3) Proposition 1 extends a result due to M. J. Merkle [6].3.
The precision in Young’s inequality
The main result of this section is as follows:
Theorem 3.
Under the assumptions of Lemma 3, for all b ≥ a ≥ and c ≥ f ( a ) , (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy − (cid:90) ba (cid:90) cf ( a ) K ( x, y ) dydx ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) bf − ( c ) (cid:90) f ( b ) c K ( x, y ) dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .Assuming K strictly positive almost everywhere, the equality occurs if and only if c = f ( b ) .Proof. The case where f ( a ) ≤ c ≤ f ( b − ) is illustrated in Figure 4. The left-handside of the inequality in the statement of Theorem 3 represents the measure of thecross-hatched curvilinear trapezium, while right-hand side is the measure of the ABCD rectangle.
OUNG’S INEQUALITY 11
Figure 4.
The geometry of the case f ( a ) ≤ c ≤ f ( b − ) . Therefore, (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy − (cid:90) ba (cid:90) cf ( a ) K ( x, y ) dydx = (cid:90) bf − ( c ) (cid:32)(cid:90) f ( x ) c K ( x, y ) dy (cid:33) dx ≤ (cid:90) bf − ( c ) (cid:90) f ( b ) c K ( x, y ) dydx. The equality holds if and only if (cid:82) bf − ( c ) (cid:16)(cid:82) f ( x ) c K ( x, y ) dy (cid:17) dx = 0 , that is, when f ( b − ) = c. The case where c ≥ f ( b +) is similar to the precedent one. The first term willbe: (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy − (cid:90) ba (cid:90) cf ( a ) K ( x, y ) dydx = (cid:90) f − ( c ) b (cid:32)(cid:90) f ( x ) f ( b ) K ( x, y ) dy (cid:33) dx ≤ (cid:90) f − ( c ) b (cid:90) cf ( b ) K ( x, y ) dydx. Equality holds if and only if (cid:82) f − ( c ) b (cid:82) cf ( b ) K ( x, y ) dydx = 0 , so we must have f ( b +) = c .The case where c ∈ [ f ( b − ) , f ( b +)] is trivial, both sides of our inequality beingequal to zero. (cid:3) Corollary 2. ( E. Minguzzi [7]) . If moreover K ( x, y ) = 1 on [0 , ∞ ) × [0 , ∞ ) , and f is continuous and increasing, then (cid:90) ba f ( x ) dx + (cid:90) cf ( a ) f − ( y ) dy − bc + af ( a ) ≤ (cid:0) f − ( c ) − b (cid:1) · ( 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.
Let f be a nondecreasing continuous function, which is convex onthe interval (cid:2) min (cid:8) f − ( c ) , b (cid:9) , max (cid:8) f − ( c ) , b (cid:9)(cid:3) . Then: i ) (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy − (cid:90) ba (cid:90) cf ( a ) K ( x, y ) dydx ≤ (cid:90) bf − ( c ) (cid:90) c + f ( b ) − cb − f − c ) ( x − f − ( c )) c K ( x, y ) dydx , for every c ≤ f ( b ) ; ii ) (cid:90) ba (cid:32)(cid:90) f ( x ) f ( a ) K ( x, y ) dy (cid:33) dx + (cid:90) cf ( a ) (cid:32)(cid:90) f − ( y ) a K ( x, y ) dx (cid:33) dy − (cid:90) ba (cid:90) cf ( a ) K ( x, y ) dydx ≥ (cid:90) f − ( c ) b (cid:90) f ( b )+ c − f ( b ) f − c ) − b ( x − b ) f ( b ) K ( x, y ) dydx , for every c ≥ f ( b ) . If f is concave on the aforementioned interval, then the inequalities above work inthe reverse way.Assuming K strictly positive almost everywhere, the equality occurs if and onlyif f is an affine function or f ( b ) = c .Proof. We will restrict here to the case of convex functions, the argument for theconcave functions being similar.The left-hand side term of each of the inequalities in our statement representsthe measure of the cross-hatched surface. See Figure 5 and Figure 6.
OUNG’S INEQUALITY 13
Figure 5. The geometry of the case c ≤ f ( b ) . Figure 6. The geometry of the case c ≥ f ( b ) . As the points of the graph of the convex function f (restricted to the interval ofendpoints b and f − ( c )) are under the chord joining ( b, f ( b )) and (cid:0) f − ( c ) , c (cid:1) , itfollows that this measure is less than the measure of the enveloping triangle M N Q when c ≤ f ( b ) . This yields i ). The assertion ii ) follows in a similar way. (cid:3) Corollary 3 extends a result due to J. Jakˇseti´c and J. E. Peˇcari´c [13] . Theyconsidered the special case were K ( x, y ) = 1 on [0 , ∞ ) × [0 , ∞ ) and f : [0 , ∞ ) → [0 , ∞ ) is increasing and differentiable, with an increasing derivative on the interval (cid:2) min (cid:8) f − ( c ) , b (cid:9) , max (cid:8) f − ( c ) , b (cid:9)(cid:3) and f (0) = 0 . In this case the conclusion ofCorollary 3 reads as follows: i ) (cid:90) b f ( x ) dx + (cid:90) c f − ( y ) dy − bc ≤ (cid:0) f − ( c ) − b (cid:1) ( c − f ( b )) for c < f ( b ) ; ii ) (cid:90) b f ( x ) dx + (cid:90) c f − ( y ) dy − bc ≥ (cid:0) f − ( c ) − b (cid:1) ( c − f ( b )) for c > f ( b ) . The equality holds if f ( b ) = c or f is an affine function. The inequality signshould be reversed if f has a decreasing derivative on the interval (cid:2) min (cid:8) f − ( c ) , b (cid:9) , max (cid:8) f − ( c ) , b (cid:9)(cid:3) . The connection with c -convexity Motivated by the mass transportation theory, several people [3], [5] drew a par-allel 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 followinggeneralization of the notion of convex function: Definition 1.
A function F : 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 ) } , for all x ∈ X. We abbreviate (4.1) by writing F = G c . A useful remark is the equality F cc = F, that is,(4.2) F ( x ) = sup y ∈ Y { c ( x, y ) − F c ( y ) } , for all x ∈ X. The classical notion of convex function corresponds to the case where X is acompact interval and c ( x, y ) = xy . The details can be found in [9], pp. 40-42.Theorem 1 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(4.3) c ( x, y ) = (cid:90) xa (cid:90) yf ( a ) K ( s, t ) dtds .In fact, under the hypotheses of this theorem, the functions F ( x ) = (cid:90) xa (cid:32)(cid:90) f ( s ) f ( a ) K ( s, t ) dt (cid:33) ds, x ≥ a, and G ( y ) = (cid:90) yf ( a ) (cid:32)(cid:90) f − ( t ) a K ( s, t ) ds (cid:33) dt, y ≥ f ( a ) , verify the relations F c = G and G c = F (due to the equality case as specified inthe statement of Theorem 1, so they are both c -convex.On the other hand, a simple argument shows that F and G are also convex inthe usual sense.Let us call the functions c that admits a representation of the form (4.3) with K ∈ L ( R × R ) , absolutely continuous in the hyperbolic sense . With this terminology,Theorem 1 can be rephrased as follows: Theorem 4.
Suppose that c : [ a, b ] × [ A, B ] → R is an absolutely continuousfunction in the hyperbolic sense with mixed derivative ∂ c∂x∂y ≥ , and f : [ a, b ] → [ A, B ] is a nondecreasing function such that f ( a ) = A. Then (4.4) c ( x, y ) − c ( a, f ( a )) ≤ (cid:90) xa ∂c∂t ( t, f ( t )) dt + (cid:90) yf ( a ) ∂c∂s ( f − ( s ) , s ) ds for all ( x, y ) ∈ [ a, A ] × [ b, B ] .If ∂ c∂x∂y > almost everywhere, then (4.4) 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 function c in the hyperbolic sense, is con-tinuous. It admits partial derivatives of the first order and a mixed derivative ∂ c∂x∂y almost everywhere. Besides, the functions y → ∂c∂x ( x, y ) and x → ∂c∂y ( x, y ) are de-fined everywhere in their interval of definition and represent absolutely continuousfunctions; they are also nondecreasing provided that ∂ c∂x∂y ≥ c : [ a, A ] × [ b, B ] → R a continuously differentiable function with nondecreasingderivatives y → ∂c∂x ( x, y ) and x → ∂c∂y ( x, y ) , and f : [ a, b ] → [ A, B ] an increasinghomeomorphism). An example which escapes his result but is covered by Theorem4 is offered by the function c ( x, y ) = (cid:90) x (cid:26) s (cid:27) ds (cid:90) y (cid:26) t (cid:27) dt, x, y ≥ , OUNG’S INEQUALITY 15 where (cid:8) s (cid:9) denotes the fractional part of s if s > , and (cid:8) s (cid:9) = 0 if s = 0. Accordingto Theorem 4, (cid:90) x (cid:26) s (cid:27) ds (cid:90) y (cid:26) t (cid:27) dt ≤ (cid:90) x (cid:32)(cid:26) s (cid:27) (cid:90) f ( s )0 (cid:26) t (cid:27) dt (cid:33) ds + (cid:90) y (cid:32)(cid:26) t (cid:27) (cid:90) f − ( t )0 (cid:26) s (cid:27) ds (cid:33) dt, for every nondecreasing function f : [0 , ∞ ) → [0 , ∞ ) such that f (0) = 0 . Acknowledgement.
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