Weighted inequalities for monotone and concave functions

Transcription

1 STUDIA MATHEMATICA 116 (2) (1995) Weighted inequlities for monotone nd concve functions by H A N S H E I N I G (Hmilton, Ont.) nd L E C H M A L I G R A N D A (Luleå) Abstrct. Chrcteriztions of weight functions re given for which integrl inequlities of monotone nd concve functions re stisfied. The constnts in these inequlities re shrp nd in the cse of concve functions, constitute weighted forms of Fvrd Berwld inequlities on finite nd infinite intervls. Relted inequlities, some of Hrdy type, re lso given. 1. Introduction. In 1939 L. Berwld [8] proved, vi generliztion of men vlue inequlity of J. Fvrd [18], tht if f is non-negtive concve continuous function on [, 1] ( 1 ) nd < p q <, then (1.1) f q (p + 1 (q + 1) 1/q f p, where the constnt (p + 1 (q + 1) 1/q is shrp. If p 1 this is clled Fvrd s inequlity. This inequlity my be interpreted s the converse of Hölder s inequlity nd in the limiting cse with q 1, p, it is reverse Jensen inequlity 1 f() d e ( 1 2 ep ) ln f() d, where the constnt e/2 is shrp. Closely relted to the Fvrd Berwld inequlity is n inequlity of Grüss [21]. It sserts tht the L 1 -norms of certin functions f, g re dominted by the L 1 -norm of the product fg. More generlly, one obtins (cf. Brnes [4] with r 1) the inequlity 1991 Mthemtics Subject Clssifiction: Primry 26D15, 26D7. Key words nd phrses: weighted integrl inequlities, weighted Hrdy inequlities, weighted Hrdy inequlities for monotone functions, weighted Fvrd Berwld inequlity, reverse Hölder inequlity, concve functions. ( 1 ) The intervl of integrtion in Berwld s result is [, b] insted of [, 1]. This generliztion my be chieved in this pper with minor modifictions. [133]

2 134 H. Heinig nd L. Mligrnd f p g q C fg r for certin indices p, q nd r. The object of this pper is to introduce weight functions u, v, w in these inequlities nd provide conditions on the weights which re equivlent to corresponding weighted forms of the Fvrd Berwld inequlity, Grüss inequlity nd the reverse Hölder inequlity. These nd corresponding vrints re given both on finite nd infinite intervls. To prove these results, number of uilliry inequlities re required. One, due to Clderón Scott [14], with specil cses of erlier origin, is proved here with shrp constnt. These inequlities re of the form (1.2) b f() dg() ( b f() γ d[g() γ ]) 1/γ, for < γ 1 nd f decresing, g incresing s well s some other vrints (cf. Th. 2.1). One of the consequences of this inequlity is tht it permits n esy proof of weight chrcteriztions of Hrdy type in the cse 1 p q from the specil cse of p q. Also using inequlity (1.2), we cn give nother proof of the imbeddings f q,u C f p,v for f nd f q,u C f p,v for f, which were proved erlier by Swyer [39], Stepnov [44] nd Heinig Stepnov [24]. All these results re given in Section 2. In Section 3, we study inequlities of the form (1.3) T 1 f q,u C T 2 f p,v, where T 1, T 2 re the positive integrl opertors T i f() k i (, t)f(t) dt, k i (, t), i 1, 2, f monotone nd C shrp constnt. Some of these results re essentilly known (cf. [2], [16], [24], [27], [33], [39], [42] [44]) nd in the min we prove them with shrp constnts. The results re then pplied in Sections 4 nd 5 to obtin shrp weighted inequlities for concve functions, weighted Fvrd Berwld inequlities s well s reverse Hölder inequlities. As specil cses we obtin results of Brnrd nd Wells [3], whose work motivted this study. All functions considered re ssumed mesurble nd non-negtive. We shll write f, respectively f to men tht the function f is incresing non-decresing, respectively decresing non-incresing. For f we define f 1 (t) inf{s : f(s) t}, inf, nd similrly for f, f 1 (t) inf{s : f(s) > t}, inf. Loclly integrble non-negtive weight functions on (, ) re denoted by u, v, w, nd the conjugte inde of p (, ) is denoted by p p/(p 1), even if < p 1, nd similrly for q. We shll sy f L p w, p >, if f p,w ( f() p w() d <. Finlly, χ E denotes the chrcteristic function of the set E. Inequlities (such s in Theorem 2.1) re interpreted to men tht if the right side is finite, so is the left, nd the inequlity holds.

3 Inequlities for monotone nd concve functions Integrl inequlities. In this section we prove inequlities frequently used in the sequel. Theorem 2.1. Let < b nd f on (, b) nd g continuous on (, b). () Suppose f on (, b) nd g on (, b) with lim + g(). Then for ny γ (, 1], (2.1) b f() dg() ( b f() γ d[g() γ ]) 1/γ. If 1 γ <, the inequlity in (2.1) is reversed. (b) Suppose f on (, b) nd g on (, b) with lim b g(). Then for ny γ (, 1], (2.2) b f() d[ g()] ( b If 1 γ <, the inequlity in (2.2) is reversed. We shll give two proofs of the theorem. f() γ d[ g() γ ]) 1/γ. P r o o f 1. It suffices to prove the theorem when the integrls on the right of (2.1) nd (2.2) re finite. Suppose < γ 1. () Since f() dg() 1 γ [f()γ g() γ ] 1/γ 1 f() γ d(g() γ ) 1 γ [ f(t) γ d(g(t) γ )] 1/γ 1f() γ d(g() γ ) integrting from to b yields d [ 1/γ, f(t) γ d(g(t) )] γ d b f() dg() ( ( b ) 1/γ f(t) γ d(g(t) γ b ) f() γ d[g() γ ]) 1/γ.

4 136 H. Heinig nd L. Mligrnd (b) Also, f() d[ g()] 1 γ [f()γ g() γ ] 1/γ 1 f() γ d[ g() γ ] 1 γ { b d { b d f(t) γ d[ g() γ ]} 1/γ 1f() γ d[ g() γ ] f(t) γ d[ g() γ ]} 1/γ, nd gin integrting from to b one obtins (2.2). It is esily seen tht inequlities in (2.1) nd (2.2) re reversed if 1 γ <. nd P r o o f 2. For t (, b), let h(t) k(t) ( t ( b t ) γ t f() dg() ) γ b f() d[ g()] Then h nd k re continuous nd for < γ 1, ( t h (t) γ f() γ d[g() γ ] t f() γ d[ g() γ ]. f() dg()) γ 1f(t)g (t) γf(t) γ g(t) γ 1 g (t) γ[f(t)g(t)] γ 1 f(t)g (t) γf(t) γ g(t) γ 1 g (t) Therefore h on (, b) nd so h(b) h(), which proves (2.1). Similrly with k(t). R e m r k 2.2. (i) If f nd g re s in Theorem 2.1, then F (γ) ( b ) 1/γ, ( b f() γ d[g() γ ] resp. G(γ).e. ) 1/γ f() γ d[ g() γ ] re decresing on (, ). (ii) Inequlity (2.2) with, b nd constnt 1/γ on the right side ws proved by Clderón Scott [14, Lemm 6.1]. (iii) If g() 1/p, < p 1, nd b, then (2.1) tkes the form ( f() 1/p 1 d p f() p d

5 Inequlities for monotone nd concve functions 137 for ll f, while g() yields ( b ) p b f() d p ( ) p 1 f() p d for ll f. The lst inequlity hs been obtined by vrious uthors (cf. Hrdy Littlewood Póly [22, p. 1], Lorentz [28, p. 39], Stein Weiss [41, Th. 3.11i], Mz y [31, Lemm 2.2], Bergh Burenkov Persson [7, Lemm 2.1]. (iv) The specil cse of (2.1) when g() p, p 1, nd f is positive incresing on (, b), b <, hs the form b f(b ) d p ( b ) p ( b f(b d f( d) p. This inequlity with some dditionl constnt C p (1, 2) ws proved by Grcí del Amo [2, Th. 5]. We cn lso write the bove inequlity in the form b ( b p, p f() p (b ) p 1 d f() d) where p 1 nd f is positive incresing function. This inequlity ws proved by Bushell Okrsiński [13] for nturl p, nd for ny p 1 quite recently by Wlter Weckesser [45]. (v) If γ, the reversed inequlities (2.1) nd (2.2) re meningless; however, one does hve the following: If f nd g re s in Theorem 2.1(), then b sup f()g() f() dg(). (,b) In fct, sup f()g() (,b) sup f() (,b) sup (,b) dg(t) f(t) dg(t) Similrly, if f nd g re s in Theorem 2.1(b) then sup f()g() (,b) b b f() d[ g()]. f() dg(). The inequlities given in Theorem 2.1 my be pplied in connection with Hrdy s inequlity. It is well known (cf. [36]) tht the weighted Hrdy

6 138 H. Heinig nd L. Mligrnd inequlity (2.3) ( ( ) q ) 1/q ( f(t) dt d C v()f() p d holds for ll f if nd only if for 1 p q <, A p,q (u, v) sup t> ( t nd for < q < p, p > 1, B p,q (u, v) { [( t ) 1/q ( t d ) 1/q ( t d v() 1 p d <, v() 1 p d) 1/q ] rv(t) 1 p dt} 1/r <, where 1/r 1/q 1/p. Moreover, if C C p,q (u, v) is the best constnt for which (2.3) holds, then A p,q (u, v) C p,q (u, v), respectively, B p,q (u, v) C p,q (u, v). Writing A p,p A p nd u s () ( u(t) dt)s 1, s >, we obtin from Theorem 2.1 the following: Proposition 2.3. () If 1 p q <, then A p,q (u, v) (p/q A p (u p/q, v) nd C p,q (u, v) (p/q C p (u p/q, v). (b) If < q < p < nd p > 1, then A p (u, v) (q/p) 1/q (r/p ) 1/r B p,q (u q/p, v) nd C p (u, v) (p/q) 1/q C p,q (u q/p, v). P r o o f. () First, for ll t >, we hve ( t ( t u p/q () d (q/p( (q/p( t t [ d v() 1 p d ( ) 1/q ( t d ) p/q ] ( t u(s) ds v() 1 p d, v() 1 p d which gives the equlity. Secondly, by Theorem 2.1(b) with γ p/q 1 nd f replced by

7 Inequlities for monotone nd concve functions 139 ( f(t) dt)q nd g replced by u(t) dt, we hve ( ) q ) q [ f(t) dt d f(t) dt d { ( ( (p/q) q/p{ ( C ) p [ ( f(t) dt d ( v()f() p d) q/p, ] u(s) ds ) p/q ]} q/p u(s) ds pup/q } q/p f(t) dt) () d which mens tht C p,q (u, v) (p/q C p (u p/q, v). (b) For ny α >, B p,q (u q/p, v) { [( ) 1/q ( 1/q ] rv() } 1/r u q/p (t) dt v(t) dt) 1 p 1 p d { [( p d ( ) ) q/p 1/q u(s) ds dt q dt t ( 1/q ] r } 1/r v(t) dt) 1 p v() 1 p d ( ) 1/q p { [( ( 1/q ] rv() } 1/r u(s) ds v(t) dt) 1 p 1 p d q ( ) 1/q p { α [( ( 1/q ] rv() } 1/r u(s) ds v(t) dt) 1 p 1 p d q ( ) 1/q p ( q α ( ) 1/q p ( ) { 1/p α [( ) 1+r/q ] } p 1/r u(s) ds d v(t) 1 p dt q r α ( ) 1/q ( ) p p 1/r ( ( α ) (1+r/q )(1/r) u(s) ds v(t) 1 p dt q r α ( ) 1/q ( ) p p 1/r ( ( α u(s) ds v(t) 1 p dt, q r α { α ( ) r/q ( )} 1/r u(s) ds v(t) 1 p dt d v(t) 1 p dt

8 14 H. Heinig nd L. Mligrnd which gives ( ( ) 1/q ( ) p p 1/r B p,q (u q/p, v) A p (u, v). q r Similrly, using Theorem 2.1(b) with γ q/p 1 nd f replced by ( f(t) dt)p, nd g replced by u(t) dt, we hve ) p ( ) p [ ] f(t) dt d f(t) dt d u(s) ds { ( ) q [ ( f(t) dt d ) q/p ]} p/q u(s) ds (p/q) p/q{ ( quq/p } p/q ( f(t) dt) () d C which mens tht C p (u, v) (p/q) 1/q C p,q (u q/p, v). ) v()f() p d, R e m r k 2.4. (i) Artol, Tlenti, Tomselli nd Muckenhoupt (cf. [32]) proved tht Hrdy s inequlity (2.3) with p q 1 holds if nd only if A p (u, v) <. Then Brdley [1], Kokilshvili nd Mz y [31] etended this result to the cse 1 p q. The cse 1 q < p ws proved by Mz y [31] nd Swyer [38], nd the cse < q < 1 < p by Sinnmon [4] (see lso [38]). For more informtion we refer to [36]. Our Proposition 2.3() shows tht if the Hrdy inequlity (2.3) is proved in the inde rnge p q 1, then it holds for 1 p < q <. In fct, if A p,q (u, v) <, then by Proposition 2.3(), C p,q (u, v) (p/q C p (u p/q, v) A p (u p/q, v) <. The best reltionship between the constnts C p,q (u, v) nd A p,q (u, v) when 1 < p < q < ws found by Mnkov [3]. (ii) Using Theorem 2.1() one obtins the sme implictions for the dul Hrdy opertor f(t) dt. If, for < p < nd for weight function v, we define the Lorentz spce Λ p (v) s the spce generted by the qusi-norm ( f Λp (v) f () p v() d, where f denotes the decresing rerrngement of f, then the imbedding Λ p (v) Λ q (u) is equivlent to corresponding weighted integrl inequlity for decresing functions. Such results re due to Swyer [39] nd Stepnov [44]. Heinig nd Stepnov [24, Th. 2.1(i)] proved similr result for incresing functions. Theorem 2.1 my lso be pplied in nturl wy in the proof of these results.

9 Inequlities for monotone nd concve functions 141 Proposition 2.5. Let < p q <. () The inequlity ( ) 1/q ( (2.4) f() q d C v()f() p d holds for ll f if nd only if (2.5) ( t ) 1/q ( t d C (b) The inequlity ( ) 1/q ( (2.6) f() q d D v() d t >. v()f() p d holds for ll f if nd only if ( ) 1/q ( (2.7) d D t t v() d t >. P r o o f. () Tht (2.4) implies (2.5) follows on tking f() χ [,t] (), t >, in (2.4). (2.5) (2.4). It suffices to prove this impliction for those functions f for which supp f (, N] (, ) nd v()f() p d <. Then stndrd limiting rguments give the result. If supp f (, N] (, ) nd v()f() p d <, then integrtion by prts, ssumption (2.4), Theorem 2.1(b) with γ p/q nd gin integrtion by prts yield { } p/q { f() q d [ f() q d ]} p/q u(t) dt {f() q u(t) dt + [ ] } p/q u(t) dt d( f() q ) { [ ] } p/q u(t) dt d( f() q ) { C p C q[ [ q/p } p/q v(t) dt] d( f() q ) ] v(t) dt d( f() p ) C p (b) The necessity follows t once with f() χ [t, ) (), t >. v()f() p d.

10 142 H. Heinig nd L. Mligrnd Conversely, similrly to () we cn ssume tht supp f [δ, ) (, ) nd v()f() p d <. Using integrtion by prts, ssumption (2.6), Theorem 2.1() with γ p/q nd gin integrtion by prts we obtin { } p/q { [ ]} p/q f() q d f() q d u(t) dt { f() q u(t) dt [ ] } p/q + u(t) dt d[f() q ] { ( ) } p/q u(t) dt d[f() q ] { D q( ( D p nd inequlity (2.6) is proved. q/p } p/q v(t) dt) d[f() q ] ) v(t) dt d[f() p ] D p v()f() p d, 3. Shrp weighted inequlities for monotone functions. We now prove weighted inequlities of the form T 1 f q,u C T 2 f p,v, where T 1 nd T 2 re positive integrl opertors on monotone functions. These estimtes re essentil in proving the weighted Berwld Fvrd inequlities given in the sequel. First we need the following known lemm (cf. Neugebuer [34, Lemms 2.1, 4.2] nd Crro Sori [15, Th. 2.1]): Lemm 3.1. Let k be loclly integrble on (, ) nd < r <. () If f on (, ), then f() r k() d r y r 1(f (b) If f on [, ), <, then f() r k() d r (c) If ϕ on (, ), then [ ] k() d ϕ(y) r dy ϕ(y) f() y r 1( [ 1 (y) f 1 (y) ) k() d dy. ) k() d dy. ] ϕ 1 (s) d(s r ) r ϕ 1 () k() d.

11 Inequlities for monotone nd concve functions 143 P r o o f. () By the Fubini theorem f() r k() d r r y r 1( y r 1(f {:f()>y} 1 (y) where the lst equlity follows since f is decresing. (b) In similr wy f() r k() d r ϕ(y) r f() y r 1( f() y r 1( ) k() d dy ) k() d dy, { [,):f()>y} f 1 (y) ) k() d dy. ) k() d dy (c) Interchnging the order of integrtion we hve [ ] [ ] k() d ϕ(y) r dy ϕ(y) r dy k() d. ϕ 1 () The following equlity is geometriclly obvious (cf. O Neil [35, p. 13] with r 1): r ϕ 1 (t 1/r ) dt r ϕ 1 () + ϕ 1 () ϕ(y) r dy. Applying these equlities together with r ϕ 1 (t 1/r ) dt ϕ 1 (s) d(s r ) we obtin the proof of (c). The proof of prt (b) of Theorem 3.2 is tken from [16]. We repet it here, with n emphsis on the constnt which in our cse plys n importnt role. (3.1) Theorem 3.2. Suppose k 1 (, t) nd k 2 (, t) re two non-negtive kernels. () Let < p 1 q. Then ( [ ] q ) 1/q k 1 (, t)f(t) dt d ( C [ v() ] p k 2 (, t)f(t) dt d

12 144 H. Heinig nd L. Mligrnd holds for ll f on [, ) if nd only if ( [ α ] q ) 1/q (3.2) k 1 (, t) dt d ( C [ α ] p v() k 2 (, t) dt d holds for ll α >. (b) Let < p 1 nd p q. Then (3.3) ( [ ] q ) 1/q ( k 1 (, t)f(t) dt d C v()f() p d holds for ll f on [, ) if nd only if ( [ α (3.4) ] q ) 1/q ( α k 1 (, t) dt d C v() d holds for ll α >. (c) Let 1 p q. Then ( ) 1/q ( (3.5) f(t) q d C [ v() ] p k 2 (, t)f(t) dt d holds for ll f on [, ) if nd only if (3.6) ( α holds for ll α >. ) 1/q ( d C [ α v() ] p k 2 (, t) dt d P r o o f. The necessity prts of (), (b) nd (c) follow on tking f(t) χ [,α] (t), α > fied. () To prove sufficiency we pply Lemm 3.1() (with r 1), Minkowski s inequlity twice nd (3.2) to obtin ( [ ] q ) 1/q k 1 (, t)f(t) dt d { [ ( f 1 (y) ( f 1 (y) ) ] q } 1/q k 1 (, t) dt dy d ) k 1 (, t) dt dy q,u f 1 (y) k 1 (, t) dt dy q,u

13 { Inequlities for monotone nd concve functions 145 { C [ { C { C { C ( C [ v() ( f 1 (y) ( f 1 (y) [ v() [ v() ( f 1 (y) ( f 1 (y) ) q ] 1/q } k 1 (, t) dt d dy ) p ] 1/p } k 2 (, t) dt d dy ) p 1/p,v} 1/p k 2 (, t) dt d ) p } 1/p k 2 (, t) dt 1/p,v d ( f 1 (y) ) ] p } 1/p k 2 (, t) dt dy d k 2 (, t)f(t) dt] p d. The lst equlity follows from Lemm 3.1() with r 1. (b) By Lemm 3.1() (with r 1), ( ( ) q ) 1/q k 1 (, t)f(t) dt d { [ ( f 1 (y) ) q } 1/q. k 1 (, t) dt dy] d Assume first tht q 1. Since the innermost integrl is decresing function of y, Theorem 2.1() pplies with γ q (cf. Remrk 2.2) so tht the lst integrl is less thn or equl to { q [ y q 1(f { q 1 (y) { C q ) q ] } 1/q k 1 (, t) dt dy d y q 1[ y q 1(f 1 (y) ( f 1 (y) ) q ] } 1/q k 1 (, t) dt d dy v() d) q/p dy } 1/q, where the lst inequlity follows from (3.4). Since p/q 1 we pply gin

14 146 H. Heinig nd L. Mligrnd Theorem 2.1() with γ p/q to see tht the lst epression is not lrger thn { C p y p 1(f 1 (y) ) } 1/p ( v() d dy C Here the equlity follows gin from Lemm 3.1(). If q > 1, let g f q. Then by Lemm 3.1(), k 1 (, t)f(t) dt 1 q k 1 (, t)g(t) 1/q dt y 1/q 1(g 1 (y) Minkowski s inequlity nd (3.4) now shows tht ( ( ) q ) 1/q k 1 (, t)f(t) dt d 1 q { 1 q 1 q 1 q C q [ y 1/q 1(g y 1/q 1(g 1 (y) 1 (y) ) k 1 (, t) dt dy g 1 (y) y 1/q 1 k 1 (, t) dt dy q,u y 1/q 1[ y 1/q 1(g 1 (y) ( g 1 (y) v() d dy. f() p v() d. ) k 1 (, t) dt dy. ) ] q } 1/q k 1 (, t) dt dy d q,u k 1 (, t) dt) q d ] 1/q dy But p 1, so gin Theorem 2.1() with γ p pplies nd hence the lst epression is not lrger thn { p C q y p/q 1(g 1 (y) { C ) } 1/p v() d dy } 1/p { g() p/q v() d C f() p v() d} 1/p.

15 Inequlities for monotone nd concve functions 147 (c) Let g f q. Then by Lemm 3.1(), [ ] p/q [ f() q d ( g 1 (y) ) d dy] p/q. Theorem 2.1() with γ p/q 1 nd (3.6) show tht this is not lrger thn p q C p p q C p y p/q 1(g [ p q 1 (y) y p/q 1[ d) p/q dy y p/q 1(g ( g 1 (y) 1 (y) ) pv() ] k 2 (, t) dt d dy ) ] p k 2 (, t) dt dy v() d C p nd gin by Theorem 2.1() with γ 1/p 1 we hve A() p q [ [ y p/q 1(g ( g 1 (y) ( g 1 (y) 1 (y) Thus from the bove nd Lemm 3.1(), [ ] p/q f() q d C p k 2 (, t) dt) p dy k 2 (, t) dt) p d(y p/q ) ) ] p k 2 (, t) dt d(y 1/q ) 1 1 (y) ) p q y1/q 1(g k 2 (, t) dt dy]. C p [ [ A()v() d g 1 1 (y) p q y1/q 1 k 2 (, t) dt dy] v() d g(t) 1/q k 2 (, t) dt] pv() d C p [ f(t)k 2 (, t) dt] pv() d. This proves the theorem.

16 148 H. Heinig nd L. Mligrnd The net ssertions for incresing functions on [, ) will be proved by similr rguments. Theorem 3.3. Suppose k 1 (, t) nd k 2 (, t) re two non-negtive kernels. () Let < p 1 q. Then ] q ) 1/q (3.7) k 1 (, t)f(t) dt d ( [ ( [ ] p D v() k 2 (, t)f(t) dt d holds for ll f on [, ), <, if nd only if ( [ ] q ) 1/q (3.8) k 1 (, t) dt d α ( [ ] p D v() k 2 (, t) dt d α holds for ll α (, ). (b) Let < p 1 nd p q. Then ( [ ] q ) 1/q ( (3.9) k 1 (, t)f(t) dt d D v()f() p d holds for ll f on [, ), <, if nd only if ( [ ] q ) 1/q ( (3.1) k 1 (, t) dt d D α α v() d holds for ll α (, ). (c) Let 1 p q. Then ( ) 1/q ( (3.11) f() q d D [ ] p v() k 2 (, t)f(t) dt d holds for ll f on [, ), <, if nd only if ( ) 1/q ( [ ] p (3.12) d D v() k 2 (, t) dt d holds for ll α (, ). α P r o o f. The necessity prts of (), (b) nd (c) follow on tking f(t) χ [α,) (t), α > fied. Sufficiency. () The proof is similr to tht of Theorem 3.2(), we need only use Lemm 3.1(b) twice (with r 1): α

17 Inequlities for monotone nd concve functions 149 { [ ] q } 1/q k i (, t)f(t) dt d { [ f() ( f 1 (y) ) q } 1/q, k i (, t) dt dy] d i 1, 2. (b) Similrly, by Lemm 3.1(b) (with r 1), { [ ] q } 1/q k 1 (, t)f(t) dt d { [ f() ( f 1 (y) ) q } 1/q. k 1 (, t) dt dy] d Assume gin, first, tht q 1. Since the inner integrl is decresing function of y we my pply Lemm 2.1() with γ q so the lst term is not lrger thn { q [ f() y q 1( f 1 (y) { f() q { f() D q y q 1[ ) q ] } 1/q k 1 (, t) dt dy d y q 1[ ( f 1 (y) f 1 (y) ) q ] } 1/q k 1 (, t) dt d dy v() d] q/p dy } 1/q, where the lst inequlity follows from (3.1). But since p/q 1 we pply Theorem 2.1() with γ p/q to see tht the lst epression is not lrger thn { f() D p y p 1[ f 1 (y) If q > 1, let g f q. Then by Lemm 3.1(b), k 1 (, t)f(t) dt ] } 1/p ( 1/p. v() d dy D f() p v() d) 1 q k 1 (, t)g(t) 1/q dt g() y 1/q 1( g 1 (y) ) k 1 (, t) dt dy.

18 15 H. Heinig nd L. Mligrnd By Minkowski s inequlity nd (3.1), { [ ] q } 1/q k 1 (, t)f(t) dt d 1 q 1 q D q { [ g() y 1/q 1( { g() y 1/q 1[ g() y 1/q 1( g 1 (y) g 1 (y) ( g 1 (y) v() d dy. ) ] q } 1/q k 1 (, t) dt dy d ) q ] 1/q } k 1 (, t) dt d dy But since p 1, Theorem 2.1 pplies gin nd hence by Lemm 3.1(b) the lst term is not lrger thn { g() p D y p/q 1( ) } 1/p v() d dy q D g 1 (y) ( ( 1/p. g() p/q v() d D f() p v() d) (c) The proof is similr to tht of Theorem 3.2(c) nd therefore omitted. R e m r k 3.4. Theorems 3.2(b) nd 3.3(b) were proved in different wy by Stepnov [43], Mysnikov Persson Stepnov [33] nd, in the cse of < p q 1, by Li [27]. Our method of proof is tken from the pper by Crro Sori [15] nd Li [27]. Theorem 3.2(c) ws lso proved by Mysnikov Persson Stepnov [33] nd Li [27]. The choice of k 1 (, t) χ [,] (t) nd k 1 (, t) χ [,] (t), in Theorem 3.2(b), respectively Theorem 3.3(b), gives Corollry 3.5. Let < p 1 nd p q. () The inequlity ( ( ) q ) 1/q ( f(t) dt d C f() p v() d holds for ll f if nd only if for every α >, ( ) 1/q ( α 1/p. (min{, α}) q d C v() d)

19 Inequlities for monotone nd concve functions 151 (b) For <, the inequlity ( ( ) q ) 1/q ( f(t) dt d D f() p v() d holds for ll f if nd only if for every α (, ), ( ) 1/q ( (min{, α}) q d D α v() d. R e m r k 3.6. If p q 1 nd u v, then the result corresponding to Corollry 3.5() ws proved by Neugebuer [34, Th. 2.2]. In the cse when both the integrl opertors re equl nd of the form f(t) dt, Theorem 3.2() etends to the rnge p q 1. Theorem 3.7. If either < p 1 q < or 1 p q <, then ( ( ) q ) 1/q ( ( ) p (3.13) f(t) dt d C v() f(t) dt d holds for ll f if nd only if ( ) 1/q ( (3.14) (min{, α}) q d C v()(min{, α}) p d holds for every α >. P r o o f. If < p 1 q, then the proof follows immeditely from Theorem 3.2() by tking k 1 (, t) k 2 (, t) χ [,] (t). If 1 p q, then (3.14) with α ϕ(y) mens (min{, ϕ(y)}) p d C p v()(min{, ϕ(y)}) p d. Integrting from to with respect to y nd using Lemm 3.1() nd (c), we obtin ( ) (min{, ϕ(y)}) p d dy [ ϕ(y) ] p d + ϕ(y) p d dy ϕ 1 () p d + [ ϕ(y) [ ] ϕ 1 (s) d(s p ) d. ] ϕ 1 (s) d(s p ) p ϕ 1 () d

20 152 H. Heinig nd L. Mligrnd Similrly for the epression with weight v we obtin from (3.14), [ ] [ ] ϕ 1 (s) d(s p ) d C p ϕ 1 (s) d(s p ) v() d. Tking ϕ 1 (s) f(s)(s 1 s f(t) dt)p 1 we hve ϕ 1 (s) d(s p ) ( f(t) dt)p nd so [ ] p f(t) dt d C p [ f(t) dt] pv() d. R e m r k 3.8. The sme result with different constnt in the rnge < p q < nd q 1 ws proved by Stepnov [43, Th. 3.3]. 4. Shrp weighted inequlities for concve functions. Consider the Green kernel { (1 t) if t 1, (4.1) K(, t) t(1 ) if t 1. Then the function (4.2) f() 1 K(, t)g(t) dt, where g rnges over ll non-negtive functions in L 1 [, 1], constitute dense subset of non-negtive concve functions on [, 1] (cf. [3], [5]). This fct is used in the proof of the following theorem: (4.3) Theorem 4.1. Suppose either < p 1 q or 1 p q. Then ( 1 ) 1/q ( 1 f() q d C v()f() p d holds for ny concve function f on [, 1] if nd only if (4.4) ( 1 ) 1/q ( 1 K(, α) q d C v()k(, α) p d holds for ll α (, 1), where K(, α) min{, α} min{1, 1 α}. P r o o f. The necessity follows t once on tking f() K(, α), α (, 1) fied, in (4.3). To prove sufficiency in the cse of < p 1 q, it suffices to prove (4.3) for those functions f hving representtion (4.2) nd then stndrd limiting rguments give the generl result. We my lso ssume tht the left side of (4.3) is finite on tking first suitble dense subset. This restriction gin cn be removed by limiting procedures.

21 Inequlities for monotone nd concve functions 153 Assuming this, we use Hölder s inequlity, (4.4) nd Minkowski s inequlity to obtin f q q,u C [ 1 f() q 1 [ 1 g(t) [ 1 g(t) 1 g(t) ( 1 C ( 1 C ( 1 C ( 1 C [ 1 ] K(, t)g(t) dt d ] K(, t)f() q 1 d dt ] 1/q [ 1 K(, t) q d ] 1/p K(, t) p v() d dt f q/q q,u ) K(, t) p 1/p f q 1 v() d 1/p,g ] 1/q f() q d dt q,u f K(, t) p q 1 v() 1/p,g d q,u ( 1 ) p f K(, t)v( g(t) dt d q 1 q,u f f() p q 1 v() d q,u nd the result follows on division by f q 1 q,u. C f p,v f q 1 q,u Sufficiency in the cse 1 p q. For p let the differentil opertor on C 2 (, 1) be given by L p [y] (1 ) p d { p+1 (1 ) 1 p d } d d ( p y) (1 )y + (p 1)(2 1)y + p(1 p)y nd consider the Rdon-concve functions of order p, R p Q p (cf. [17], [37]), i.e. the closure in the topology of loclly uniform convergence on (, 1) of Q p {f C 2 [, 1] : L p [f], f() f(1) }. For every continuous function g : [, 1] R the boundry vlue problem L p [f] g, f() f(1),

22 154 H. Heinig nd L. Mligrnd hs solution where G p (, t) 1 p f() 1 G p (, t)g(t) dt, { p t p if < < t < 1, (1 ) p (1 t) p if < t < 1. Our result now follows from the following two fcts (cf. lso [17] nd [37]): 1. If 1 p q, (4.4) holds nd f R p, then ( 1 ) p/q 1 f() q/p d C In fct (nd similrly to the bove), f q/p 1 q/p,u p 1 p [ 1 f() q/p 1 [ 1 g(t) [ 1 g(t) 1 1 g(t) [ 1 ] G p (, t)g(t) dt d ] G p (, t)f() q/p 1 d dt ] p/q [ 1 G p (, t) q/p d f()v() d. ] p/q q/p 1 qg q (, t) d dt f q/p,u g(t)t p (1 t) p[ p Cp g(t)t p (1 t) p[ 1 C p 1 C p 1 C p 1 [ 1 g(t) [ 1 f() q/p d] 1 p/q dt ] p/q K(, t) q q/p 1 d dt f q/p,u ] G p (, t)v() d G p (, t)g(t) dt f()v() d f q/p 1 q/p,u. ] K(, t) p v() d dt f q/p 1 q/p,u dt f q/p 1 q/p,u ] v() d f q/p 1 q/p,u

23 Inequlities for monotone nd concve functions If p 1 nd f R 1, i.e. f is positive concve function on [, 1], then f p R p. It is enough to prove tht if y Q 1 (i.e. y nd y() y(1) ), then y p Q p. Since L p [y p ] (1 )p[(p 1)y p 2 (y ) 2 + y p 1 y ] + (p 1)(2 1)py p 1 y + p(1 p)y p (p 1)y p 1 L p [y] + p(p 1)y p 2 (y y)[(1 )y + y] nd y () y() ty (t) dt, (1 )y () + y() 1 (t 1)y (t) dt it follows tht L p [y p ] nd so y p Q p. R e m r k 4.2. (i) Theorem 4.1 in the cse p 1 ws lso proved by Mligrnd Pečrić Persson [29, Th. 3]. In the cse < p 1 q, Theorem 4.1 is still true for ll functions which hve representtion (4.2) with the non-negtive kernel K(, t). In prticulr, if { 1 if t 1, K(, t) if < t 1, then f() g(t) dt is incresing on [, 1]. Similrly, if { if t 1, K(, t) 1 if < t 1, then f() 1 g(t) dt is decresing on [, 1]. Therefore Theorem 4.1 lso gives the proof of Proposition 2.5 but only in the inde rnge < p 1 q. (ii) If either < p 1 q or 1 p q nd v() 1, then h(t) [ 1 [ t [ t ] 1/q [ 1 K(, t) q d q (1 t) q d + 1 p (1 t) p d + (p + 1 (q + 1) 1/q, t ] 1/p K(, t) p d ] 1/q t q (1 ) q d 1 t ] 1/p t p (1 ) p d which is the quotient, i.e. the constnt C of (4.4), nd hence Theorem 4.1 gives the Fvrd Berwld inequlity f q (p+1 (q+1) 1/q f p for ny positive concve function f on [, 1]. Berwld [8] proved this inequlity in the full rnge < p q. In prticulr, if q 1 then f (p+1)1/p f p, < p 1. As p this inequlity becomes the reverse Jensen inequlity given in the introduction.

24 156 H. Heinig nd L. Mligrnd (iii) If v() α, α nd p 1 in Theorem 4.1, then the function h(t) [ 1 ] 1/q /[ 1 K(, t) q α d [(1 t) q t α+1 1 q + α (1 t)t α+1 α + 2 t + 1 tα+1 α + 1 ] K(, t) α d (1 ) q α d 1 tα+2 α + 2 ] 1/q cn be shown to be decresing on (, 1). Hence h(t) h() nd therefore the constnt of (4.3) is ( 1 (α + 1)(α + 2) (1 ) q α d) 1/q (α + 1)(α + 2)B(q + 1, α + 1) 1/q, B being the Bet function. This is result of Brnrd Wells [3, Th. 1]. We cn pply Corollry 3.5 nd Theorem 3.7 to prove Fvrd Berwld inequlities for incresing nd decresing concve functions. Note tht if f is concve on [, ) then f is necessrily incresing. R e m r k 4.3. (i) From Theorem 3.7 it follows t once tht if either < p q or 1 p q <, then f q,u C f p,v holds for ny concve incresing function f on [, ) if nd only if (3.14) holds for ll α >. (ii) If 1 p q <, then it follows from Corollry 3.5 nd Theorem 2.1() tht ( 1 ) 1/q 1 f() q d D v()f() d holds for ny concve decresing function f on [, 1] if nd only if ( 1 ) 1/q 1 [min(1, 1 α)] q d D is stisfied for ll α (, 1). We conclude this section with three emples. v() min(1, 1 α) d

25 Inequlities for monotone nd concve functions 157 Emple 4.4. If f is concve incresing on [, 1], then by Remrk 4.3(i) with p 1, q 2 nd v() α, α >, we obtin (4.5) ( 1 ) 1/2 1 f() 2 α α + 2 d α + 3 f() α d. The constnt (α + 2)/ α + 3 is best possible, s my be seen on tking f() in (4.5). Emple 4.5. If f is concve on [, ), then for t >, ( ) 1/q (4.6) e t f() q d t 1 1/q Γ (q + 1) 1/q e t f() d, 1 < q <. The constnt t 1 1/q Γ (q + 1) 1/q is best possible, s my be seen on tking f() in (4.6). In order to obtin the constnt in (4.6) it clerly suffices to tke t 1. By Remrk 4.3(i) with p 1, we must show tht sup α> {( ) 1/q / } e [min(, α)] q d e min(, α) d {( α sup e q d + α q e α) 1/q/ } (1 e α ) Γ (q + 1) 1/q. α> But if h(α) ( α e q d + α q e α )/(1 e α ) q, then h is incresing on (, ) since h (α) (1 e α ) q{ qα q 1 e α + qe α (1 e α ) q 1[ α nd the bove supremum equls lim α h(α) 1/q Γ (q + 1) 1/q. Emple 4.6. If f is concve on [, ), then ( ) 1/q (4.7) e 2 f() q d e q d + α q e α]} >, ( e q d) 1/q ( q /q Γ 2 ) 1/q e 2 f() d, 1 < q <,

26 158 H. Heinig nd L. Mligrnd nd the constnt 2 1 1/q Γ ((q + 1)/2) 1/q is shrp, s my be seen on tking f() in (4.7). Agin by Remrk 4.3(i) we only need to show tht {( ) 1/q / } sup e 2 [min(, α)] q d e 2 min(, α) d α> {( α sup e 2 q d + α q α> ( α e 2 d + α For tht it suffices to show tht k(α) ( α e 2 q d + α q α α α ) 1/q e 2 d ) 1 } e 2 d 2 1 1/q Γ )/( α e 2 d e 2 d + α ( ) 1/q q + 1. α 2 ) q e 2 d is n incresing function on (, ), for then the bove supremum is ( ) 1/q / ( ) 1/q q + 1 lim α k(α)1/q e 2 q d e 2 d 2 1 1/q Γ. 2 But ( α k (α) q e 2 d + α α [α q 1 α e 2 d nd hence the function k is incresing s required. ) q 1 ( ) e 2 d e 2 d α α ] e 2 q d > 5. Weighted reverse Hölder inequlities. Closely relted to the Fvrd Berwld inequlity is Grüss inequlity [21] (see lso [23]). It sserts tht the L 1 -norms of f nd g re dominted by the L 1 -norm of the product fg, where f, g re from some clss of functions. This inequlity together with Fvrd s gives reverse Hölder inequlity (cf. [3], [4], [6], [9], [26]). We now discuss weighted versions of Grüss inequlity nd lso give weight chrcteriztions for which such inequlities hold. In our first result we ssume w L 1 (I), where I R is n intervl, nd we let P be the clss of positive mesurble functions on I. Theorem 5.1. () If f 2,w C 2 f 1,w for ny f P, then (5.1) C 2 f 1,w g 1,w fg 1,w for ny f, g P, where C2 2/w(I) C2 2 nd w(i) w() d. I

27 Inequlities for monotone nd concve functions 159 (b) If f r,w C r f 1,w for every r > 1 nd f P, then (5.2) C p,q f p,w g q,w fg 1,w for 1 p, q <, f, g P nd C p,q C 2 /(C p C q ). P r o o f. () Let f 1,w g 1,w 1. Then f + g 2 2,w f 2 2,w + 2 fg 1,w + g 2 2,w C 2 2 f 2 1,w + 2 fg 1,w + C 2 2 g 2 1,w 2C fg 1,w. On the other hnd, Hölder s inequlity shows tht so tht nd hence 2 f + g 1,w f + g 2,w w(i) 1/2, 2 fg 1,w f + g 2 2,w 2C w(i) 2C2 2, fg 1,w ( ) 2 w(i) C2 2 f 1,w g 1,w. (b) The proof follows immeditely from () nd the ssumption. For concve functions the constnts given in Theorem 5.1 re not lwys shrp. We illustrte this fct by emples given net. Emple 5.2. () Let P {f : f is concve on [, 1]}. If w() α, α, then Remrk 4.2(iii) shows tht the constnt C r equls (α+1)(α+2) B(r + 1, α + 1) 1/r, where B is the Bet function. Since we hve C 2 [2(α + 1)(α + 2)/(α + 3)] 1/2, it follows tht C2 2(α + 1)/(α + 3) nd C p,q 2[(α + 1)(α + 2) 2 (α + 3)] 1 B(p + 1, α + 1) 1/p B(q + 1, α + 1) 1/q. In prticulr, C 2,2 (α + 2) 1 but it is known tht the best constnt is {(α + 1)/[2(α + 3)]} 1/2 (cf. Brnrd Wells [3]). If α the constnt C p,q 1 6 (p + 1)1/p (q + 1) 1/q is shrp (cf. Brnes [4]). (b) If P {f : f is concve on [, )} nd w() e 2, then Emple 4.6 shows tht C r 2 1 1/r Γ ((r + 1)/2) 1/r. Hence we hve C2 (4 π)π 1/2 > nd ( ) 1/p ( ) 1/q p + 1 q + 1 Cp,q (4 π)π 1/2 2 1/p+1/q 2 Γ Γ. 2 2 To obtin shrp constnts in inequlity (5.2) for specific clss of functions, sy concve functions, we require properties of these functions which re not s generl s those of P.

28 16 H. Heinig nd L. Mligrnd Theorem 5.3. Suppose f, g re non-negtive concve functions on [, 1] nd < r 1 p, q <. If u, v nd w re weight functions on [, 1], then (5.3) ( 1 holds if nd only if ( 1 f() p d ) 1/q v()g() q d ( 1 C ) 1/r w()[f()g()] r d [ 1 (5.4) C sup K(, t)p d] 1/p [ 1 K(, s)q v() d] 1/q s,t (,1) [ 1 [K(, t)k(, <, s)]r w() d] 1/r where { (1 y) if y 1, K(, y) y(1 ) if y 1. P r o o f. Necessity. Substitute f() K(, t) nd g() K(, s), where s, t (, 1) re rbitrry, into (5.3). Then the necessity follows. Sufficiency. As noted erlier, it suffices to prove (5.3) if f nd g hve the representtion f() 1 K(, t)f 1 (t) dt, g() 1 K(, s)g 1 (s) ds, where f 1, g 1 re some non-negtive functions in L 1 [, 1]. Applying Minkowski s inequlity twice nd (5.3) one obtins f p,u g q,v [ 1 [ C ( 1 [ 1 f 1 (t) 1 1 ( 1 v() ) p ] 1/p K(, t)f 1 (t) dt d ) q ] 1/q K(, s)g 1 (s) ds d ] 1/p 1 K(, t) p d dt [ 1 f 1 (t)g 1 (s) [ 1 g 1 (s) ] 1/p [ 1 K(, t) p d dt { 1 f 1 (t)g 1 (s) [K(, t)k(, s)] r w() d} 1/r ds dt K(, s) q v() d] 1/q ds K(, s) q v() d] 1/qds dt

29 { 1 C { 1 C C fg r,w. Inequlities for monotone nd concve functions 161 [ 1 w() [ 1 w() 1 ] r } 1/r f 1 (t)g 1 (s)k(, t)k(, s) ds dt d ] r [ 1 f 1 (t)k(, t) dt ] r } 1/r g 1 (s)k(, s) ds R e m r k 5.4. (i) The method of proof of Theorem 5.3 with the Green function K(, y) is usully clled the Bellmn or Bellmn Weinberger method. Bellmn proved such result in the cse of u v w 1, r 1 nd 1/p + 1/q 1 ( cf. [5], [6], [3], [26], [37], [46], [47]). This method ws lso used erlier by Bückner [11] who proved the cse u v w, r 1 nd p q 2. (ii) Computtions of the supremum (5.4) or (5.6) re often quite tedious. If v() w() 1 in Theorem 5.3, then the supremum over < s t < 1 of H(s, t) ( 1 ( 1 K(, t) p d ) 1/q /( 1 K(, s) q d ) 1/r [K(, t)k(, s)] r d [ (1 s) (p + 1) 1/p (q + 1) 1/q r s r+1 t(1 s) 2r + 1 t + r (1 ) r d + tr (1 t) r+1 ] 1/r 2r + 1 s is not esy to compute (cf. [3], [6], [25], [37], [46] nd [47]). (iii) If f nd g re non-negtive concve incresing functions on [, ), <, then the result corresponding to Theorem 5.3 is the following: If < r 1 p, q <, then ( ( ) 1/q (5.5) f() p d v()g() q d holds if nd only if ( ) 1/r D w()[f()g()] r d [ (5.6) D sup K(, t)p d] 1/p [ K(, s)q v() d] 1/q s,t (,) [ [K(, t)k(, <, s)]r w() d] 1/r

30 162 H. Heinig nd L. Mligrnd where K(, y) { if y <, y if y <. Finlly, if for p, q 1 nd α, H p,q,α denotes the best constnt H in the reverse Hölder inequlity ( 1 H ( 1 f() p α d ) 1/q 1 g() q α d f()g() α d, f, g concve positive functions on [, 1], then the known results re the following: H 2,2, 1/2 (Frnk Pick [19]), H 1,1, 2/3 (Grüss [21]), H p,p, 1 6 (p + 1)1/p (p + 1 (Bellmn [6]), H p,q, 1 6 (p + 1)1/p (q + 1) 1/q (Brnes [4]), α + 1 H 2,2,α 2(α + 2) (Brnrd Wells [3]), H p,p,α min{v α (p), V α (p ), W α (p)}/[(α + 2)(α + 3)] (Wng Chen [47]), where V α (p) (p α B(p + 1, α + 1) 1/p, W α (p) 2/[(α + 1)B(p + 1, α + 1 B(p + 1, α + 1 ]. Also if H + p,q,α is the corresponding constnt for incresing concve functions f, g on [, 1], then nd H + 2,2, 3/2 (Krft, Bückner [11]) H + p,q, min { 1 2 (p+1)1/p, 1 2 (q+1)1/q, 1 3 (p+1)1/p (q+1) 1/q} (Petschke [37]). Other results my lso be obtined from Remrk 5.4(iii); in prticulr, we get (α + 1)(α + 3) H 2,2,α +. α + 2 Acknowledgements. This reserch ws supported in prt by NSERC grnt A-4837 nd Luleå University. The first nmed uthor pprecites the kind hospitlity of the Deprtment of Mthemtics, Luleå University, Luleå, where the reserch ws lmost finished in Jnury Februry 1993.

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33 Inequlities for monotone nd concve functions 165 [47] H.-T. Wng nd S.-Y. Chen, Inverse Hölder inequlities with weight t α, J. Mth. Anl. Appl. 176 (1993), DEPARTMENT OF MATHEMATICS MCMASTER UNIVERSITY HAMILTON, ONTARIO L8S 4K1 CANADA E-mil: HEINIG@MCMAIL.CIS.MCMASTER.CA DEPARTMENT OF MATHEMATICS LULEÅ UNIVERSITY S LULEÅ, SWEDEN E-mil: LECH@SM.LUTH.SE Received September 27, 1994 (3342) Revised version Mrch 2, 1995