Functional Logarithm in the Sense of Convex Analysis

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1 Journal of Convex Analysis Volume (3), No., 9 44 Funcional Logarihm in he Sense of Convex Analysis Musapha Raïssouli Groupe AFA - UFR AFACS, Universié Moulay Ismaïl, Faculé des Sciences, Déparemen de Mahémaiques e Informaique, BP 4, Zioune, Meknés, Maroc raissoul@fsmek.ac.ma Hanane Bouziane Groupe AFA - UFR AFACS, Universié Moulay Ismaïl, Faculé des Sciences, Déparemen de Mahémaiques e Informaique, BP 4, Zioune, Meknés, Maroc Received December 7, Revised manuscrip received December 7, Le Γ (H) denoe he usual cone of lower semiconinuous convex funcionals from a Hilber H ino IR + } no idenically equal o +. In his paper we inroduce a map L : Γ (H) IR H (IR = IR, + }) which exends he Logarihm from posiive inverible operaors o convex funcionals. Aferwards, we presen a funcional limied developmen of L(f σ + λf) where f σ :=., f Γ (H) and λ goes o +. The paper will be illusraed wih some examples which jusify he chosen erminology and he imporance of his work. Keywords: Convex analysis, conjugae funcion, subdifferenial, funcional logarihm, funcional limied developmen Mahemaics Subjec Classificaion: 49-XX, 5-XX. Inroducion Recenly, he exension of he means from posiive real numbers o posiive operaors has exensive several developmens and applicaions [4], [5], [9]. Since calculaions involving operaor-valued funcions are feasible wih large compuers, many auhors have used he operaor means in solving some scienific problems. Le a and b be wo posiive real numbers and define a map φ by φ(a, b) = ( a + b, ab). If φ k denoes he k-h ierae of φ, i is no hard o prove ha here is a posiive number M = M(a, b) such ha lim k + φk (a, b) = (M, M).The number M is called he "arihmeico-geomeric mean of a and b". An explici form of M(a, b) is given by he following ellipic inegral [9]: (M(a, b)) = π π ( a cos θ + b sin θ ) dθ. () The limi of he above ieraive process (φ k (a, b)) k can be used o compue he inegral () which is of mahemaical and physical ineres. The generalizaion of he preceding noions and resuls o posiive linear operaors is sufficienly sudied. More precisely, le σ = (σ, σ,..., σ m ) IR m be a probabiliy vecor, ISSN / $.5 c Heldermann Verlag

2 3 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis i.e. σ i >, i m and m i= σ i =, R. D. Nussbaum and J. E. Cohen [9] have suggesed ha a reasonable analogue of m i= aσ i i (a i posiive numbers) is ( m ) exp σ i Log(A i ), () i= where A i, i m, are posiive inverible operaors, exp is he exponenial of operaor, i.e. expa = A n n= and Log is he logarihm of operaor defined by ([9], page 53) n! LogA = I (( )I + A) d, (3) where I denoes he ideniy operaor. I will be an ineresing aemp o exend he previous noions and resuls from posiive inverible operaors o convex funcionals. In his sense, M.Aeia and M.Raïssouli [] have recenly inroduced he "Convex Geomeric Funcional Mean" from an ieraive process ha operaes recursively on pairs (f, g) Γ (H) Γ (H) where Γ (H) denoes he cone of proper lower semiconinuous convex funcionals defined on a real Hilber space H, wih values in IR + }. In paricular, he funcional "square roo" R : Γ (IR m ) Γ (IR m ) was inroduced in a way ha for f(x) = < Ax, x >, where A is a symmeric posiive marix, [R(f)](x) = < Ax, x > where A is he posiive square roo of A. Very recenly, M.Raïssouli and M.Chergui [] have consruced he arihmeico-geomeric funcional mean f + τg of wo funcions f, g Γ (H) as follows: f + τg = lim k + Φk (f, g) where Φ k denoes he k-h ierae of Φ defined by Φ(f, g) = ( f + g, fτg), fτg is he convex geomeric funcional mean of f and g consruced in []. An explici form of f τg + analogue o () is, for he momen, unknown. For his, he soluion of he reverse problem of he square roo funcional is very ineresing, i.e. wha should be he analogue of A for convex funcionals. The sandard definiion of A comes from he produc AB of wo operaors A and B. The exension of his produc from operaors o convex funcionals is no obvious and appears o be ineresing. For posiive inverible operaors, anoher (equivalen) definiion of A can be given by A A (λi + A ) = lim. (4) λ λ In [] he auhors suggesed ha an analogue of he second member of (4) o f Γ (H) is f (λf σ + f ) lim, (5) λ λ where f denoes he Fenchel-conjugae of f, i.e. f (x ) = sup< x, x > f(x)} and x H f σ :=. is he only self- conjugae funcion. They have esablished ha he limi (5) exiss and f(x) (λf σ + f ) (x) lim λ λ = f σ (p f (x)), (6)

3 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis 3 for all x in(dom f), where p f (x) = P f(x) () is he unique poin projecion from o he nonempy closed convex f(x) subdifferenial of f a x. Anoher ineresing problem is o exend formula () from operaors o funcionals. This exension is imporan in order, for example, o explici he geomeric mean fτg of f and g and secondly o define he geomeric mean of hree or more convex funcionals. More precisely, wha should be he regular analogue of Logarihm and Exponenial from operaors o funcionals. The fundamenal goal of his work is o describe a reasonable analogue exension of he Logarihm from he case ha he variable is posiive inverible operaor o he case ha he variable is Γ (H)-funcional.We sugges ha a reasonable analogue of LogA given by (3) o f Γ (H) is: x H [L(f)](x) = f σ (x) (( )f σ + f) (x) d. (7) Using (7), we presen some funcional-valued properies of he map L analogous o ha of operaor-valued Logarihm. Noe ha in paricular he mapping L : Γ (H) IR H is increasing, concave (wih respec o he poinwise order on IR H ) and saisfies ha: f Γ (H) L(f ) = L(f). In case f(x) = < Ax, x > where A is a symmeric posiive inverible linear operaor of H, we obain [L(f)](x) = < (LogA)x, x > wih LogA is he Logarihm of A defined by (3). To deermine how o obain he soluion of he reverse problem of he funcional Logarihm (i.e. he funcional Exponenial ) is no obvious and appears o be ineresing. This paper will be divided ino hree pars: We begin by recalling some basic resuls from convex analysis ha will be needed laer. Secondly, we inroduce he noion of funcional Logarihm and we sudy is elemenary properies. This secion will be compleed by illusraing he heoreical resuls wih some examples. In he fourh secion, we presen a second order funcional limied developmen of L(f σ + λf) where f σ :=., λ goes o (λ > ) and f Γ (H). Some consequences and examples are given.. Background maerial and preliminary resuls Le us recall some basic noions and resuls from convex analysis which are needed hroughou his paper. H will denoe a real Hilber space wih inner produc <.,. > and is associaed hilberian norm.. Le IR = IR, + }, we prolong he srucure of IR on IR by seing: x IR, < x < +, + + x = +, + x =, + + ( ) = +,.(+ ) = +. We can define a parial ordering on IR H by f g x H f(x) g(x).

4 3 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis Consider a funcion f : H IR, f denoes he Fenchel conjugae of f defined by he formula x H f (x ) := sup< x, x > f(x)}. x H If f, g : H IR are wo given funcionals, i is easy o see ha if f g hen g f. The following inequaliy holds for every f, g IR H. α ], [ (αf + ( α)g) αf + ( α)g, (8) Denoe dom f he effecive domain of f defined by dom f := x H ; f(x) IR}, and Γ (H) he cone of lower semiconinuous convex funcionals from H ino IR + } no idenically equal o +. We recall ha if f Γ (H) hen, f Γ (H) and f = f. Le S be a subse of H, we denoe by Ψ S he indicaor funcional of S defined as follows: Ψ S (x) = if x S and Ψ S (x) = + oherwise. Ψ S Γ (H) if and only if S is a nonempy closed convex subse of H. In paricular, if B = B(, ) is he closed uni ball of cener, hen Ψ B =. and. = Ψ B. We se below f σ :=. he only self-conjugae funcional defined on H. I is easy o see he following resuls: f IR H f f (f f σ ) (9) a > (af σ ) = a f σ. () (f σ + Ψ S ) = f σ d S, () where S is a nonempy closed convex subse of H and d S is defined by d S (x) := x y. inf y S We recall ha, ([], page 66) for all real λ > and f Γ (H), he funcional (λf σ + f) is finie everywhere and for all x H here holds (λf σ + f) (x) = inf y H f (y) + λ y x }, () moreover he applicaion x (λf σ + f) (x) from H ino IR is Freche-differeniable. Le f : H ĨR = IR + }, we say ha p is a subgradien of f a he poin x if x dom f and: f(y) f(x)+ < y x, p >, for all y H. The subdifferenial of f a x is defined by: f(x) := p H; y H f(y) f(x)+ < y x, p >}. Le us denoe by in(dom f) he opological inerior of dom f, we recall ha if f Γ (H) and in(dom f) is nonempy hen, for all x in(dom f), f is coninuous a x and f(x).

5 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis Funcional Logarihm in convex analysis This secion is devoed o inroduce he funcional Logarihm L : Γ (H) IR H ogeher wih is elemenary properies. Firs, we sae he following heorem which will be needed in he sequel. Theorem 3.. For all f Γ (H) and all λ >, he following formula holds λ(λf σ + f ) + (f σ + λf) = f σ Remark 3.. Theorem 3. is proved in ([8], page 84) for λ = and ([7], page XIII-6), ([], Theorem 3.) for λ >. This resul can be wrien under he following "convex" form (ake λ = ): f Γ (H) ], [ ( ) (( )f σ + f ) + (f σ + ( )f) = f σ (3) Le f Γ (H) be a fixed funcional, he mapping: (( )f σ + f) is derivable on ], [, (cf. [], Corollary 3.) hence coninuous on ], [ and consequenly we inroduce he following definiion: Definiion 3.3. The mapping L : Γ (H) IR H defined by x H [L(f)](x) = f σ (x) (( )f σ + f) (x) d, (4) is called he Funcional Logarihm in he sense of Convex Analysis. Remark 3.4. The denominaion "Funcional Logarihm in he sense of Convex Analysis" will be jusified by Examples 3.3 and 4.4 below and he properies of L discussed hroughou he paper. Proposiion 3.5. Relaion (4) is equivalen o he following one x H [L(f)](x) = Proof. According o relaion (4), we have for all x H [L(f)](x) = (( )f σ + f ) (( )f σ + f) } (x)d (5) ( f σ (( )f σ + f) ) + ( f σ (( )f σ + f) } ) (x)d By a simple variable change (s = ), i is easy o see ha [L(f)](x) = = ( f σ and wih (3), he desired resul follows. (f σ + ( )f) ) + ( f σ (( )f σ + f) fσ (f σ + ( )f) } (( )f σ + f) (x)d, } ) (x)d

6 34 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis Proposiion 3.6. The map L : Γ (H) IR H saisfies he following properies: (i) L() = Ψ } and f Γ (H) α IR L(f + α) = L(f) + α. (ii) L(f σ ) = and a > L(af σ ) = (Loga)f σ. (iii) f Γ (H), L(f ) = L(f). Proof. (i) I is immediae from he definiion of L. (ii) Relaion L(f σ ) = is obvious. Le us show ha L(af σ ) = (Loga)f σ, for all a >. To simplify he wriing below, we omi he x in relaions (4) and (5), so by (4) and (), we have successively L(af σ ) = = f σ (( )f σ + af σ ) d = ( )} f σ d = + a = [Log( + (a ))] f σ = (Loga)f σ. (iii) Le f Γ (H), hen f = f and by (5) one has L(f ) = f σ (( + a)f σ ) d } a f σ d + (a ) (( )f σ + f) (( )f σ + f ) d, since, for all f Γ (H), he mapping (( )f σ + f) is wih finie values hen L(f ) = (( )f σ + f ) (( )f σ + f) d = L(f). Proposiion 3.7. Wih respec o he poinwise ordering on IR H, he map L is: (i) Increasing: f, g Γ (H); if f g hen L(f) L(g). (ii) Concave: f, g Γ (H), α ], [, L (αf + ( α)g) αl(f) + ( α)l(g). Proof. (i) Le f, g Γ (H) such ha f g, we derive for all ], [ (( )f σ + f) (( )f σ + g). Using (4) we deduce L(f) L(g), so L is increasing. (ii) Show ha L is concave: Le f, g Γ (H), and α ], [, we can wrie ( )f σ + (αf + ( α)g) = α[( )f σ + f] + ( α)[( )f σ + g]. Then according o inequaliy (8), we deduce ha (( )f σ + (αf + ( α)g)) α (( )f σ + f) + ( α) (( )f σ + g), from which we observe ha f σ (( )f σ + (αf + ( α)g)) α [f σ (( )f σ + f) ] + ( α) [f σ (( )f σ + g) ], Noe ha if f, g : H IR he equaliy f g = (g f) is no always rue.

7 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis 35 and by relaion (4), finally which complees he proof. L(αf + ( α)g) αl(f) + ( α)l(g), Proposiion 3.8. For all f in Γ (H), he following formula holds Proof. Using (4), combined wih (8), we ge ha L(f) = f σ f L(f) f f σ (6) f σ (( )f σ + f) d f σ (( )f σ + f ) d, and simplifying we obain L(f) f σ f, for all f Γ (H). Replace f by f in he laer inequaliy o obain, wih Proposiion., (iii), f f σ. The proof is complee. L(f) From he above proposiion, i is easy o see he following remarks. Remark 3.9. (i) If dom f = dom f = H hen doml(f) = H. (ii) For all f Γ (H), L(f) is no idenically equal o + (resp. ). f σ (x) (( )f σ + f) (x) Remark 3.. The inegral d converges o a finie real number for every x dom f dom f. In paricular, if dom f = dom f = H hen he above inegral converges for all x H. Corollary 3.. Le f Γ (H), hen one has (i) L(f) = f = f σ. (ii) L(f) f f σ ( resp. L(f) f f σ ). Proof. I follows immediaely from (6). Corollary 3.. Le (f n ) n IN be defined recursively by n f n+ = f n + f n, (7) wih f Γ (H). Then (f n ) n IN converges poinwise o f σ on dom f dom f. Proof. Since f Γ (H) hen, by inducion, we deduce ha f n Γ (H) for all n IN. On he oher hand, L is concave so n IN L(f n+ ) L(f n) + L(f n), and according o Proposiion 3.6, (iii) and he srucure of IR, we ge n IN L(f n+ ) L(f n) L(f n),

8 36 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis and by Corollary 3., we deduce ha n IN f n+ f σ and n IN f n+ f σ. Using hese previous inequaliies, relaion (7) yields which implies ha n IN f n+ f σ = (f n f σ ) + (f n f σ ) (f n f σ ), n IN f n f σ n (f f σ ). If x dom f = dom f dom f hen lim n n (f f σ )(x) =, from which he desired resul follows. We now end his secion by illusraing our heoreical resuls wih hree examples. We begin by an example which explains ha he funcional Logarihm inroduced above conains ha of posiive inverible operaors. Example Le H = IR and x IR f(x) = ax (a > ), due o Proposiion 3.6, (ii) we have x IR [L(f)](x) = (Loga)x.. More generally, le H = IR n be he usual n-dimensional Euclidean space and f given by: x IR n, f(x) = < Ax, x > where A is a symmeric posiive definie marix of order n. In his case, we recall ha ([], page 38) f is given by x IR n f (x) = < A x, x >. Using he fac ha, for all ], [, he marix ( )I + A is also symmeric posiive definie, hen we obain for all x IR n, and relaion (4) implies ha (( )f σ + f) (x) = < (( )I + A) x, x >, x IR n [L(f)](x) = By (3) and he bilineariy of <.,. >, we deduce ha I (( )I + A) < x, x > d. x IR n [L(f)](x) = < (LogA)x, x >.

9 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis 37 If P is an inverible n n marix such ha λ A = P λ P, wih λ i > ( i =,,...p) he eigenvalues of A, λ p hen x IR n [L(f)](x) = < (LogA)x, x >, Logλ where LogA = P Logλ P. Logλ p 3. The above example for finie-dimensional Hilber spaces works, by he same argumen, for general Hilber spaces: Le f A (x) = < Ax, x >, for all x H, be he quadraic form associaed o he symmeric posiive inverible operaor A hen L(f A ) = f LogA where LogA is he Logarihm of A defined by (3). Relaion (iii) of Proposiion 3.6, L(f ) = L(f), can be ineresing in order, for example, o compue L(f) when f has a simple expression (as f). The nex example explains his siuaion. Example 3.4. Le us consider he space H = M n of symmeric square marices of order n equipped wih is usual inner produc < A, B >= T raceab. The variance of a given marix A M n is defined by: (var)(a) = ( ) T racea n < A, A >. n I is easy o see ha he funcional var : M n IR is convex and hence var Γ (M n ). The conjugae (var) : M n IR + } of var is given by ([3], page 6): n (var) (B) = n f σ(b) + Ψ T (B) = f σ(b) if T raceb = + oherwise where T := A M n ; T racea = } is a closed subspace of M n. So,we begin by compuing L((var) ). By virue of () and an elemenary manipulaion, we ge ha (( )f σ + (var) ) = ( ( + n )f σ + Ψ T ) = + n ( f σ ) d T. Using formula (4) combined wih a simple compuaion, we obain L(var)(A) = L((var) )(A) = (Log n ).f (Log n σ(a) Ψ T (A) = ).f σ(a) if A T oherwise

10 38 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis Example Le S be a nonempy closed convex subse of H and f = Ψ S he indicaor funcion of S. By () and a rouine ransformaion, we have (( )f σ + Ψ S ) = (( )f σ + Ψ S ) = (f σ ) (d ( )S), which, wih relaion (4), yields L(Ψ S ) = f σ + or equivalenly (by a variable change s = ) L(Ψ S ) = } ( ) (d ( )S) d, ( ) (ds ) f σ d. (8) ( ) In paricular, if S = B = B(, ), he unied ball, Proposiion 3.6, (iii) and relaion (8) give (since Ψ B =. ): L(. ) = ( fσ (d B ) ) d. ( ). Wih he above noaions, assume ha S is a closed subspace of H and f = Ψ S. I is known ([], pages 4, 5) ha f = Ψ S where S = x H; < x, x >= } is he orhogonal of S. Taking λ = and f = Ψ S, Theorem 3. combined wih () yields he celebrae ideniy d S + d =., and relaion (8) gives (since S = S) S L(Ψ S ) = 4 ( ) (d S d S )d, which becomes in a differen way if d S (x) > d S (x) [L(Ψ S )](x) = if d S (x) = d S (x) + if d S (x) < d S (x) or equivalenly where L(Ψ S ) = Ψ S + Ψ S, S + = x H; d S (x) d S (x)} and S = x H; d S (x) d S (x)}. 4. Funcional limied developmens (F.L.D) of L(f σ + λf) We preserve he same noaions as previous. In his secion, we shall give a second order developmen of L(f σ + λf), λ +, which exends ha of operaors: Log(I + λa) = λa λ A + λ θ λ (A), wih θ λ (A) ends o as λ. We begin by recalling he following lemma.

11 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis 39 Lemma 4.. ([]) Le f Γ (H), hen for all x dom f one has (f σ + λf) (x) = f σ (x) λf(x) + λ(θ λ (f))(x), (9) where θ λ (f) ends poinwise o when λ goes o +. If moreover in(dom f) is nonempy, he funcional limied developmen (FLD): (f σ + λf) (x) = f σ (x) λf(x) + λ f σ (p f (x)) + λ (θ λ (f))(x), () holds for all x in(dom f), where lim θ λ (f) = for he poinwise convergence and λ p f (x) = P f(x) () is he unique poin projecion from o he nonempy closed convex f(x). By applying he above lemma, we will prove he following heorem: Theorem 4.. Le f Γ (H), for each x dom f we have [L(f σ + λf)](x) = λf(x) + λ(θ λ (f))(x), () where θ λ (f) ends poinwise o when λ +. If moreover in(dom f) is nonempy, he following developmen [L(f σ + λf)](x) = λf(x) λ f σ (p f (x)) + λ (θ λ (f))(x), () holds for every x in(dom f), where lim θ λ (f) = in he poinwise convergence and λ p f (x) = P f(x) () is he unique poin projecion from o he nonempy closed convex f(x). Proof. Noe ha, firs, he θ λ (f) in (9) and () (resp. () and ()) is no he same bu only o simplify he wriing below. For he same reason we omi he x in all relaion and we wrie θ insead of θ λ (f). Le us prove (). According o Proposiion 3.8, one has and using (9), we ge which implies (). f σ (f σ + λf) L(f σ + λf) λf, λf + λθ λ (f) L(f σ + λf) λf, We now prove (). From relaion (4), i follows fσ L(f σ + λf) = (f } σ + λf) d. By virue of (), we obain L(f σ + λf) = = fσ f } σ λf + λ f σ (p f ) + λ θ() d λf λ f σ (p f ) λ θ() } d = λf λ f σ(p f ) λ θ()d. (3)

12 4 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis On he oher hand, by combining Theorem 3. and relaion (), we infer ha from which we deduce x in(dom f) λ(λf σ + f ) = λf λ f σ (p f ) λ θ(), [θ λ (f)()](x) := [θ()](x) = f(x) (λf σ + f ) (x) λ To complee he proof we need he nex lemma. f σ (p f (x)). (4) Lemma 4.3. (θ λ (f)) λ converges uniformly (wih respec o ) o when λ ends o +, i.e. x in(dom f) sup [θ λ (f)()](x) =. lim λ ],[ Proof of Lemma 4.3. For each x in(dom f), recall ha we have f(x). Le x f(x) hen, by definiion, f(x) f(y) < x, x y > for all y H. Cauchy-Schwarz s inequaliy implies ha, for all y H and < <, and by Young s inequaliy f(x) f(y) λ λx x y, which obviously yields f(x) f(y) λ x + λ y x, f(x) f(y) λ y x λ x. If we ake he supremum for all y H, his previous inequaliy gives or equivalenly Thanks o relaion (), we conclude ha supf(x) f(y) y H λ y x } λ x, f(x) inf y H f(y) + λ y x } λ x. x f(x) Furher, i is well known ha, ([], Lemma 3.) f(x) (λf σ + f ) (x) λ x. (5) lim f λ (x) = lim (λf σ + f ) (x) = p f (x) f(x). (6) λ λ If we ake x = p f (x) in (5) we obain f(x) (λf σ + f ) (x) λ f σ (p f (x)). (7)

13 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis 4 Moreover, we know ha ([], Theorem 3.) which, wih (7), gives f σ (( f λ (x))) f(x) (λf σ + f ) (x), λ f σ (( f λ (x))) f σ (p f (x)) f(x) (λf σ + f ) (x) λ f σ (p f (x)), and herefore f(x) (λf σ + f ) (x) f σ (p f (x)) λ f σ(p f (x)) f σ ( f λ (x)). (8) Recalling ha ([3], page II.) ( f λ ) λ is decreasing, i becomes for all ], [, and λ > f λ f λ and hus f σ ( f λ ) f σ ( f λ ). I follows ha f(x) (λf σ + f ) (x) f σ (p f (x)) λ f σ(p f (x)) f σ ( f λ (x)), and sup ],[ In (9) use (4) o wrie f(x) (λf σ + f ) (x) λ f σ (p f (x)) f σ(p f (x)) f σ ( f λ (x)). (9) x in(dom f) sup [θ λ (f)()](x) f σ (p f (x)) f σ ( f λ (x)). (3) ],[ By virue of relaion (6) and he coninuiy of f σ, he second member of (3) ends o as λ. The proof of Lemma 4.3 is complee. To end he proof of Theorem 4., we can wrie by virue of Lemma 4.3: lim λ (θ λ (f))()d = which, wih (3), gives he desired resul. lim (θ λ (f))()d = λ We now illusrae Theorem 4. wih he nex example. lim λ (θ λ (f))()d =, Example Le H = IR and x IR f(x) = ax (a > ). Theorem 4. and Example 3.3, - give he following classical resul: Log( + λa) = λa λ a + λ θ λ (a), where θ λ (a) ends o when λ.. More generally, le A be a symmeric posiive marix of order n, and consider he funcional x IR n f(x) := f A (x) = < Ax, x >.

14 4 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis Since f is convex and (G aeaux) differeniable hen f(x) = f(x)}, A is symmeric gives f(x) = Ax} and f σ (p f (x)) = Ax = < A x, x >. According o Theorem 4. and Example 3.3,., we can wrie f Log(I+λA) = f λa λ A + λ θ λ (f), (3) from which we deduce ha θ λ (f) is a quadraic form, i.e. θ λ (f) = f θλ (A) where θ λ (A) is ( a symmeric marix because θ λ (A) = Log(I + λa) λa + λ A ). Then, relaion (3) is equivalen o he following one λ f Log(I+λA) = f λa λ A +λ θ λ (A) (3) From he relaion θ λ (f) = f θλ (A) and he fac ha [θ λ (f)](x) λ for all x in(dom f) = IR n, we deduce < (θ λ (A))x, x > λ for all x IR n, and by symmery of θ λ (A) we have < (θ λ (A))x, y > λ for all x, y IR n. I follows ha x IR n (θ λ (A))x λ in IR n. Using he fac ha he marices associaed o he quadraic forms of (3) are symmeric we find he known developmen: Log(I + λa) = λa λ A + λ θ λ (A). Corollary 4.5. Le f Γ (H) such ha in(dom f) is nonempy. Then, for each s in a neighbourhood of, here holds x in(dom f) d ds L(f σ + sf)(x) = (sf σ + f ) (x) + s(θ s (f))(x), where θ s (f) ends (poinwise) o when s +. Proof. I follows by combining Theorem 3., Lemma 4. and Theorem 4.. Proposiion 4.6. Le f Γ (H) be a fixed funcional such ha in(dom f) is nonempy. Then he mapping s L(sf σ + f) defined from IR + :=], + [ ino IR H is differeniable and x in(dom f) d ds L(sf σ + f)(x) = f σ ( (( + s)f σ + f) ) (x)d (33) Proof. Here also, we omi he x and we wrie θ insead of θ λ (f). Le s IR + and λ +, we have fσ L((λ + s)f σ + f) = (( )f σ + (λ + s)f σ + f) } d fσ = (λf σ + ( + s)f σ + f) } d. (34)

15 M. Raïssouli, H. Bouziane / Funcional Logarihm in he Sense of Convex Analysis 43 Seing f = ( + s)f σ + f, i is clear ha f Γ (H) and dom f = dom f. Using relaion () wih Theorem 3., we obain which, wih (34), yields L((λ + s)f σ + f) = (λf σ + f) = f λf σ (p f ) λ θ(), fσ (( )f σ + sf σ + f) = L(sf σ + f) + λ } + λf σ (p f ) + λ θ() d f σ ( (( + s)f σ + f) ) d + λ θ()d, where [ θ λ (f)]() := θ() = f (λf σ + f) λ Lemma 4.3 implies ha f σ (p f ). lim λ [ θ λ (f)]()d = lim[ θ λ (f)]()d =. λ Le us observe ha seing [ θ λ (f)]()d = θ λ (f), i follows L((λ + s)f σ + f) = L(sf σ + f) + λ and consequenly d ds L(sf σ + f) = f σ ( (( + s)f σ + f) ) d + λθ λ (f), f σ ( (( + s)f σ + f) ) d. Remark 4.7. Le A be a symmeric posiive operaor from H ino H and ake f(x) = < Ax, x > for all x H. In his case, relaion (33) yields, afer a simple compuaion, he known classical resul: s IR + d ds Log(sI + A) = (si + A), and he "Logarihm" erminology is again jusified. Acknowledgemens. The auhors are graeful o he anonymous referee for his helpful suggesions and commens which have been included in he revised version of his paper. References [] M. Aeia, M. Raïssouli: Self dual operaors on convex funcionals; geomeric mean and square roo of convex funcionals, J. Convex Anal. 8 () 3 4. [] J. P. Aubin: Analyse non Linéaire e ses Moivaions Économiques, Masson (98).

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