Preliminaries Chapter1 1.1 Spaces of integrable, continuous and absolutely continuous functions

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1 Preliminries Chpter1 1.1 Spces of integrble, continuous nd bsolutely continuous functions In this section we listed definitions nd properties of integrble functions, continuous functions, bsolutely continuous functions nd bsic properties of the Lplce trnsform. Also we give required nottion, terms nd overview of some importnt results (more detils could be found in monogrphs [57, 59, 7, 74]). L p spces Let [,b] be finite intervl in R,where < b. We denote by L p [,b],1 p <, the spce of ll Lebesgue mesurble functions f for which f (t) p dt <, where ( 1 b p f p = f (t) dt) p, nd by L [,b] the set of ll functions mesurble nd essentilly bounded on [,b] with f = esssup{ f (x) : x [,b]}. Theorem 1.1 (INTEGRAL HÖLDER S INEQUALITY) Let p,q R such tht 1 p,q nd 1 p + 1 q = 1. Let f,g: [,b] R be integrble functions such tht f L p[,b] nd 1

2 2 1 PRELIMINARIES g L q [,b]. Then f (t)g(t) dt f p g q. (1.1) Equlity in (1.1) holds if nd only if A f (t) p = B g(t) q lmost everywhere, where A nd B re constnts. Spces of continuous nd bsolutely continuous functions We denote by C n [,b], n N, the spce of functions which re n times continuously differentible on [,b],thtis { } C n [,b]= f : [,b] R: f (k) C[,b],k =,1,...,n. In prticulr, C [,b]=c[,b] is the spce of continuous functions on [,b] with the norm nd for C[, b] f C n = n k= f (k) C = n k= f C = mx f (x). x [,b] mx f (k) (x), x [,b] Lemm 1.1 The spce C n [,b] consists of those nd only those functions f which re represented in the form 1 f (x)= (n 1)! (x t) n 1 (t)dt + n 1 k= c k (x ) k, (1.2) where C[,b] nd c k re rbitrry constnts (k =,1,...,n 1). Moreover, (t)= f (n) (t), c k = f (k) () (k =,1,...,n 1). (1.3) k! By C[,b] n we denote the subspce of the spce C n [,b] defined by { } C n [,b]= f C n [,b]: f (k) ()=,k =,1,...,n 1. For f C n [,b] nd < 1wedefine f (n) (x) f (n) (y) f n, = sup x y : x,y [,b],x y. Let >, N, n theintegrlprtof (nottion n =[])ndlet = n. ByD [,b] we denote the spce { } D [,b]= f C n [,b]: f n, <,

3 1.1 SPACES OF FUNCTIONS 3 nd by D [,b] the subspce of the spce D [,b] { } D [,b]= f D [,b]: f (k) ()=,k =,1,...,n. Specilly, for = n N we hve D n [,b]=c n [,b] nd D n[,b]=cn [,b]. The spce of bsolutely continuous functions on finite intervl [,b] is denoted by AC[,b]. It is known tht AC[,b] coincides with the spce of primitives of Lebesgue integrble functions L 1 [,b] (see Kolmogorov nd Fomin [53, Chpter 33.2]): f AC[,b] f (x)= f ()+ (t)dt, L 1 [,b], nd therefore n bsolutely continuous function f hs n integrble derivtive f (x)=(x) lmost everywhere n [,b]. We denote by AC n [,b], n N, the spce In prticulr, AC 1 [,b]=ac[,b]. { } AC n [,b]= f C n 1 [,b]: f (n 1) AC[,b]. Lemm 1.2 ThespceAC n [,b] consists of those nd only those functions which cn be represented in the form (1.2), where L 1 [,b] nd c k re rbitrry constnts (k =,1,...,n 1). Moreover, (1.3) holds. The next theorem hs numerous pplictions involving multiple integrls. Theorem 1.2 (FUBINI S THEOREM) Let (X,M, ) nd (Y,N,) be -finite mesure spces nd f -mesurble function on X Y. If f, then next integrls re equl ( ) ( f (x,y)d( )(x,y), f (x,y)d(y) d(x) nd X Y X Y Y X ) f (x,y)d(x) d(y). If f is complex function, then bove equlities hold with dditionl requirement X Y f (x,y) d( )(x,y) <. Next equlities re consequences of this theorem: dx d c f (x,y)dy = d c dy f (x,y)dx; dx f (x,y)dy = dy f (x,y)dx. (1.4) y

4 4 1 PRELIMINARIES The gmm nd bet functions The gmm function is the function of complex vrible defined by Euler s integrl of second kind (z)= t z 1 e t dt, R(z) >. (1.5) This integrl is convergent for ech z C such tht R(z) >. It hs next property from which follows For domin R(z) wehve (z + 1)=z(z), R(z) >, (n + 1)=n!, n N. (z + n) (z)=, R(z) > n; n N; z Z = {, 1, 2,...}, (1.6) (z) n where (z) n is the Pochhmmer s symbol defined for z C nd n N by (z) = 1; (z) n = z(z + 1) (z + n 1), n N. The gmm function is nlytic in complex plne except in, 1, 2,...which re simple poles. The bet function is the function of two complex vribles defined by Euler s integrl of the first kind B(z,w)= 1 It is relted to the gmm function with t z 1 (1 t) w 1 dt, R(z),R(w) >. (1.7) B(z,w)= (z)(w) (z + w), z,w Z, which gives B(z + 1,w)= z z + w B(z,w). Next we proceed with exmples of integrls often used in proofs nd clcultions in this book. Exmple 1.1 Let, > ndx [,b]. Then by substitution t = x s(x ) we hve (x t) 1 (t ) 1 dt = 1 (x ) + 1 s 1 (1 s) 1 ds = B(, )(x ) + 1. Anlogously, by substitution t = x + s(b x), it follows x (t x) 1 (b t) 1 dt = B(, )(b x) + 1.

5 1.2 SPACES OF FUNCTIONS 5 Exmple 1.2 Let, >, f L 1 [,b] nd x [,b]. Then interchnging the order of integrtion nd evluting the inner integrl we obtin Anlogously, t (x t) 1 (t s) 1 f (s)dsdt = x s= f (s) = B(, ) t=s (t x) 1 (s t) 1 f (s)dsdt = B(, ) The Lplce trnsform t (x t) 1 (t s) 1 dt ds (x s) + 1 f (s)ds. x (s x) + 1 f (s)ds. Let f : [,) R be function such tht mpping t e t f (t), >, is integrble on [,). Then for ech p the Lebesgue integrl F(p)= e pt f (t)dt (1.8) exists. The mpping f F is clled the Lplce trnsform nd noted with L,thtis L [ f ](p)=f(p). Sufficient conditions for the Lplce trnsform existence re tht function f is loclly integrble nd exponentilly bounded in, thtis f (t) Me t for t >, wherem, nd re constnt. The bsciss of convergence is the smllest vlue of for which f (t) Me t. Exmple 1.3 Let f : [,) R, f (t)=t,where > 1. Obviously f (t) = t < e t for t > nd. For 1 < <, the function f is loclly integrble nd t 1for t 1. Therefore, by substitution pt = x, the Lplce trnsform hs the form L [ f ](p)= e pt t dt = 1 p +1 e x x ( + 1) dx = p +1. We give some properties nd rules of the Lplce trnsform, nd importnt uniqueness theorem ([74, Teorem 6.3]): [ t ] convolution: L f (t )g()d (p)=l[ f ](p)l [g](p) differentition: [ L f (n)] (p)=p n L [ f ](p) n k=1 p n k f (k 1) () Theorem 1.3 (UNIQUENESS THEOREM) Let f,g: [,) R be two functions for which the Lplceov trnsform exists. If e pt f (t)dt = e pt g(t)dt for ech p on common re of convergence, then f (t)=g(t) for lmost every t [,).

6 6 1 PRELIMINARIES 1.2 Convex functions nd Jensen s inequlities Definitions nd properties of convex functions nd Jensen s inequlity, with more detils, could be found in monogrphs [61, 62, 67]. Let I be n intervl in R. Definition 1.1 A function f : I R is clled convex if f ((1 )x + y) (1 ) f (x)+ f (y) (1.9) for ll points x nd y in I nd ll [,1]. It is clled strictly convex if the inequlity (1.9) holds strictly whenever x nd y re distinct points nd (,1). If fisconvex (respectively, strictly convex) then we sy tht f is concve (respectively, strictly concve). If f is both convex nd concve, then f is sid to be ffine. Lemm 1.3 (THE DISCRETE CASE OF JENSEN S INEQUALITY) A rel-vlued function f defined on n intervl I is convex if nd only if for ll x 1,...,x n in I nd ll sclrs 1,..., n in [,1] with n k=1 k = 1 we hve f ( n k=1 k x k ) n k=1 k f (x k ). (1.1) The bove inequlity is strict if f is strictly convex, ll the points x k re distinct nd ll sclrs k re positive. Theorem 1.4 (JENSEN) Let f : I R be continuous function. Then f is convex if nd only if f is midpoint convex, tht is, ( ) x + y f (x)+ f (y) f (1.11) 2 2 for ll x,y I. Corollry 1.1 Let f : I R be continuous function. Then f is convex if nd only if f (x + h)+ f (x h) 2 f (x) (1.12) for ll x I nd ll h > such tht both x + h nd x hreini. Proposition 1.1 (THE OPERATIONS WITH CONVEX FUNCTIONS) (i) The ddition of two convex functions (defined on the sme intervl) is convex function; if one of them is strictly convex, then the sum is lso strictly convex. (ii) The multipliction of (strictly) convex function with positive sclr is lso (strictly) convex function.

7 1.2 CONVEX FUNCTIONS AND JENSEN S INEQUALITIES 7 (iii) The restriction of every (strictly) convex function to subintervl of its domin is lso (strictly) convex function. (iv) If f : I R is convex (respectively strictly convex) function nd g : R R is nondecresing (respectively n incresing) convex function, then g fisconvex (respectively strictly convex) (v) Suppose tht f is bijection between two intervls I nd J. If f is incresing, then f is (strictly) convex if nd only if f 1 is (strictly) concve. If f is decresing bijection, then f nd f 1 re of the sme type of convexity. Definition 1.2 If g is strictly monotonic, then f is sid to be (strictly) convex with respect to g if f g 1 is (strictly) convex. Proposition 1.2 If x 1,x 2,x 3 I re such tht x 1 < x 2 < x 3, then the function f : I R is convex if nd only if the inequlity holds. (x 3 x 2 ) f (x 1 )+(x 1 x 3 ) f (x 2 )+(x 2 x 1 ) f (x 3 ) Proposition 1.3 If f is convex function on n intervl I nd if x 1 y 1,x 2 y 2,x 1 x 2, y 1 y 2, then the following inequlity is vlid f (x 2 ) f (x 1 ) f (y 2) f (y 1 ). x 2 x 1 y 2 y 1 If the function f is concve, then the inequlity reverses. The following theorems concern derivtives of convex functions. Theorem 1.5 Let f : I R be convex. Then (i) f is Lipschitz on ny closed intervl in I; (ii) f + nd f exist nd re incresing in I, nd f f + (if f is strictly convex, then these derivtives re strictly incresing); (iii) f exists, except possibly on countble set, nd on the complement of which it is continuous. Proposition 1.4 Suppose tht f : I R is twice differentible function. Then (i) f is convex if nd only if f ; (ii) f is strictly convex if nd only if f nd the set of points where f vnishes does not include intervls of positive length. Next we need divided differences, commonly used when deling with functions tht hve different degree of smoothness.

8 8 1 PRELIMINARIES Definition 1.3 Let f : I R,n N nd let x,x 1,...,x n I be mutully different points. The n-th order divided difference of function t x,...,x n is defined recursively by [x i ; f ]= f (x i ), i =,1,...,n, [x,x 1 ; f ]= [x ; f ] [x 1 ; f ] = f (x ) f (x 1 ), x x 1 x x 1 [x,x 1,x 2 ; f ]= [x,x 1 ; f ] [x 1,x 2 ; f ], (1.13) x x 2. [x,...,x n ; f ]= [x,...,x n 1 ; f ] [x 1,...,x n ; f ]. x x n Remrk 1.1 The vlue [x,x 1,x 2 ; f ] is independent of the order of the points x, x 1 nd x 2. This definition my be extended to include the cse in which some or ll the points coincide. Nmely, tking the limit x 1 x in (1.13),weget lim [x,x 1,x 2 ; f ]=[x,x,x 2 ; f ]= f (x ) f (x 2 ) f (x )(x x 2 ) x 1 x (x x 2 ) 2, x 2 x provided tht f exists, nd furthermore, tking the limits x i x, i = 1,2 in(1.13), we get lim lim [x,x 1,x 2 ; f ]=[x,x,x ; f ]= f (x ) x 2 x x 1 x 2 provided tht f exists. Definition 1.4 A function f : I R is sid to be n-convex (n N ) if for ll choices of n + 1 distinct points x,...,x n I, the n-th order divided difference of f stisfies [x,...,x n ; f ]. (1.14) Thus the 1-convex functions re the nondecresing functions, while the 2-convex functions re precisely the clssicl convex functions. Definition 1.5 A function f : I (,) is clled log-convex if for ll points x nd y in I nd ll [,1]. f ((1 )x + y) f (x) 1 f (y) (1.15) If function f : I R is log-convex, then it is lso convex, which is consequence of the weighted AG-inequlity. We end this section with the integrl form of Jensen s inequlity. Theorem 1.6 (INTEGRAL JENSEN S INEQUALITY) Let (,A, ) be finite mesure spce, < () < nd let f : Ibe-integrble function. If : I R is convex function, then next inequlity holds ( ) 1 fd 1 ( f )d. (1.16) () ()

9 1.3 EXPONENTIAL CONVEXITY 9 If is strictly convex, then in (1.16) we hve equlity if nd only f is constnt -lmost everywhere on. 1.3 Exponentil convexity Following definitions nd properties of exponentilly convex functions comes from [28], lso [66]. Let I be n intervl in R. Definition 1.6 A function : I R is n-exponentilly convex in the Jensen sense on I if n ( ) xi + x j i j 2 i, j=1 holds for ll choices i R nd x i I, i = 1,...,n. A function : I R is n-exponentilly convex if it is n-exponentilly convex in the Jensen sense nd continuous on I. Remrk 1.2 It is cler from the definition tht 1 exponentilly convex functions in the Jensen sense re in fct nonnegtive functions. Also, n-exponentilly convex functions in the Jensen sense re k exponentilly convex in the Jensen sense for every k N, k n. By definition of positive semi-definite mtrices nd some bsic liner lgebr we hve the following proposition. Proposition 1.5 If is n n-exponentilly convex in the Jensen sense, then the mtrix [ ( )] xi + x k j is positive semi-definite mtrix for ll k N,k n. Prticulrly, 2 i, j=1 [ ( )] xi + x k j det for ll k N,k n. 2 i, j=1 Definition 1.7 A function : I R is exponentilly convex in the Jensen sense on I if it is n-exponentilly convex in the Jensen sense for ll n N. A function : I R is exponentilly convex if it is exponentilly convex in the Jensen sense nd continuous. Remrk 1.3 It is known (nd esy to show) tht : I (,) is log-convex in the Jensen sense if nd only if ( ) x + y 2 (x) (y) 2 holds for every, R nd x,y I. It follows tht function is log-convex in the Jensen sense if nd only if it is 2 exponentilly convex in the Jensen sense. Also, using bsic convexity theory it follows tht function is log-convex if nd only if it is 2 exponentilly convex.

10 1 1 PRELIMINARIES One of the min fetures of exponentilly convex functions is its integrl representtion given by Bernstein ([32]) in the following theorem. Theorem 1.7 The function : I R is exponentilly convex on I if nd only if (x)= for some non-decresing function : R R. 1.4 Opil-type inequlities e tx d(t), x I In 196. Opil published n inequlity involving integrls of function nd its derivtive, which now ber his nme ([64]). Over the lst five decdes, n enormous mount of work hs been done on Opil s inequlity: severl simplifictions of the originl proof, vrious extensions, generliztions nd discrete nlogues. More detils cn be found in the monogrph by Agrwl nd Png [5] which is dedicted to the theory of Opil-type inequlities nd its pplictions in theory of differentil nd difference equtions. We observe Beesck s, Wirtinger s, Willett s, Godunov-Levin s, Roznov s, Fink s, Agrwl-Png s nd Alzer s versions of Opil s inequlity. Theorem 1.8 (OPIAL S INEQUALITY) Let f C 1 [,h] be such tht f ()= f (h)=nd f (x) > for x (,h). then f (x) f (x) h [ dx f (x) ] 2 dx, (1.17) 4 where constnt h/4 is the best possible. The novelty of Opil s result is thus in estblishing the best possible constnt h/4. Exmple 1.4 It is esy to construct the function which stisfy equlity in (1.17). For instnce, let f be defined by cx, x h 2 f (x)= c(h x), h 2 x h where c > is rbitrry constnt. Although this function is not derivble in t = h/2, it could be pproximted by the function belonging to C 1 [,h] tht stisfy (1.17). Then constnt h/4 is the best possible. Opil s inequlity (1.17) holds even if function f hs discontinuity t t = h/2, provided tht f is bsolutely continuous on both of the subintervls [, h 2 ] nd [ h 2,h], with f () = f (h) =. Also, the positivity requirement of f on (,h) is unnecessry, tht is, next Beesck s inequlity holds ([31]).

11 1.4 OPIAL-TYPE INEQUALITIES 11 Theorem 1.9 (BEESACK S INEQUALITY) Let f AC[,h] be such tht f ()=. Then f (x) f (x) h dx 2 Equlity in (1.18) holds if nd only if f (x)=cx, where c is constnt. [ f (x) ] 2 dx. (1.18) Theorem 1.1 (WIRTINGER S INEQUALITY) Let f : [,h] R be such tht f L 2 [,h]. If f ()=f(h)=,then ( ) [ f (x)] 2 h 2 [ dx f (x) ] 2 dx. (1.19) Equlity in (1.19) holds if nd only if f (x)=csin x, where c is constnt. Remrk 1.4 A weker form of Opil s inequlity cn be obtined by combining Cuchy- Schwrz-Bunikowski s inequlity nd Wirtinger s inequlity: f (x) f (x) ( ) 1 dx f (x) 2 2 ( dx f (x) ) h dx h [ f (x) ] 2 dx. Next inequlity involving x (n), n 1, is given by Willett [75] (see lso [5, p. 128]). Theorem 1.11 (WILLETT S INEQUALITY) Let x C n [,h] be such tht x (i) ()=, i=,...,n 1,n 1. Then x(t)x (n) (t) dt hn x (n) (t) 2 dt. (1.2) 2 More generliztions nd extensions of Willett s inequlity re done by Boyd in [33]. Following generliztion of Opil s inequlity is due to Godunov nd Levin [46] (see lso [5, p. 74]). Theorem 1.12 (GODUNOVA-LEVIN S INEQUALITY) Let f be convex nd incresing function on [,) with f () =. Further, let x be bsolutely continuous on [,] nd x()=. Then, the following inequlity holds ( ) f ( x(t) ) x (t) dt f x (t) dt. (1.21) An extension of the inequlity (1.21) is embodied in the following inequlity by Roznov [69] (see lso [5, p. 82]). Theorem 1.13 (ROZANOVA S INEQUALITY) Let f, g be convex nd incresing functions on [,) with f ()=, nd let p(t), p (t) >,t [,] with p()=. Further, let x be bsolutely continuous on [,] nd x()=. Then, the following inequlity holds ( x p ) ( )) ( (t) x(t) ( x (t)g p f (p(t)g dt f p ) ) (t) (t)g (t) p(t) p dt. (1.22) (t) Moreover, equlity holds in (1.22) for the function x(t)=cp(t).

12 12 1 PRELIMINARIES Remrk 1.5 The condition in the two previous theorems tht function f is to be incresing is ctully unneeded, nd lso, the condition g is missing in Theorem 1.13 (it cn be esily seen from proofs of the theorems). Among inequlities of Opil-type, there is clss of inequlity involving higher order derivtives. First we hve Fink s inequlity ([45]). Theorem 1.14 (FINK S INEQUALITY) Let q 1, 1 p + 1 q = 1,n 2nd i j n 1. Let f AC n [,h] be such tht f ()= f ()= = f (n 1) ()= nd f (n) L q [,h]. Then ( f (i) (x) f ( j) (x) dx Ch 2n i j+1 q 2 h f (n) (x) dx) q 2 q, (1.23) where C = C(n,i, j,q) is given by [ ] C = 2 q 1 (n i 1)!(n j)![p(n j)+1] 1 p [p(2n i j 1)+2] 1 1 p. (1.24) Inequlity (1.23) is shrp for j = i + 1, where equlity in this cse is chieved for q > 1 nd function f such tht 1 f (x)= (n 1)! (x t) n 1 (h t) p q (n i 1) dt. Remrk 1.6 Agrwl nd Png proved in [65] tht Fink s inequlity does not hold for i = j, nd tht is not necessry to ssume tht f (k) ()=fork < i. Next inequlity is due to Agrwl nd Png ([65]). Theorem 1.15 (AGARWAL-PANG S INEQUALITY) Let n N nd f AC n [,h] be such tht f ()= f ()= = f (n 1) ()=. Letw 1 nd w 2 be positive, mesurble functions on [,h]. Letr i >, i=,...,n 1, nd let r = n 1 i= r i.lets k > 1 nd 1 s + 1 k s = 1 for k k = 1,2, nd q R such tht q > s 2. Further, let ( ) P = [w 2 (x)] s r 2q s 2 dx <, Then w 1 (x) n 1 i= ( Q = [w 1 (x)] s 1 dx ) 1 s 1 <. ( f (i) (x) r i + dx Ch s 1 h 1 w 2 (x) f (n) (x) dx) q r q, (1.25) where = n 1 Ir i + r, I = n i 1, = 1 s 2 1 q, nd C = C(n,{r i},w 1,w 2,s 1,s 2,q) is given by i= C QP n 1 i= (I!) r i [ ] [ I ri n Ir i s 1 + rs i= provided tht integrl on the right side in (1.25) exists. ] 1 s1,

13 1.4 OPIAL-TYPE INEQUALITIES 13 Alzer s inequlities re given in [1, 11], where second one includes higher order derivtives of two functions. Theorem 1.16 (ALZER S INEQUALITY 1) Let n N nd f C n [,b] be such tht f ()= f ()= = f (n 1) ()=. Let w be continuous, positive, decresing function on [,b]. Let r i,i=,...,n 1, nd n 1 i= r i = 1. Letp 1,q> nd = 1/(p + q). Then where w(x) ( n 1 i= A 1 = q q [n p f (i) i) (x) r f (n) (x) q dx A 1 w(x) f (n) (x) p+q dx, (1.26) n 1 i=1 ir i ] p (b ) (n n 1 ir i )p n 1 i=1 i= [ ( 1 ) 1 1 n i (n i 1)! Theorem 1.17 (ALZER S INEQUALITY 2) Let p,q>,r> 1 nd r > q. Let n N, k N, k n 1. Letw 1 nd w 2 > be mesurble functions on [,b]. Further, let f,g AC n [,b] be such tht f (i) ()=g (i) ()= for i =,...,n 1 nd let integrls w 2(x) f (n) (x) r dx nd w 2(x) g (n) (x) r dx exist. Then where g w 1 (x)[ (k) (x) p f (n) (x) q + f (k) (x) p g (n) (x) q ] dx ( A 2 w 2 (x)[ f (n) (x) r + g (n) (x) r ]dx [ 2M q A 2 = [(n k 1)!] p 2(p + q) s(x)= ] q [ r b [w 1 (x)] r q r [w 2 (x)] q (x u) r(n k 1) r 1 [w 2 (u)] 1 ( ) 1 2 M = p q r q, p q, 2 p r, p q. 1 r du, ) p+q r q r [s(x)] p(r 1) r q ] ri p, (1.27) dx ] r q r,.