SOME HARDY TYPE INEQUALITIES WITH WEIGHTED FUNCTIONS VIA OPIAL TYPE INEQUALITIES R. P. AGARWAL, D. O REGAN 2 AND S. H. SAKER 3 Abstrct. In this pper, we will prove severl new ineulities of Hrdy type with eplicit constnts. The min results will be proved using generliztions of Opil s ineulity.. Introduction The clssicl Hrdy ineulity (see []) sttes tht for f integrble over ny nite intervl ( ) nd f p integrble nd convergent over ( ) nd p > then (.) p f(t)dt d p p p f p ()d: The constnt (p= (p )) p is the best possible. Some etensions of Hrdy s ineulity were considered in Beesc [5]. Our im in this pper is to prove some ineulities with weighted functions of Hrdy type using Opil type ineulities. 2. Min Results Throughout the pper, ll functions re ssumed to be positive nd mesurble nd ll the integrls which pper in the ineulities re ssumed to eist nd be nite. To obtin ineulities of Hrdy type we loo t ineulities for R( b)f ()F ()d R( b) = R b r(t) dt nd F () = R f(t)dt. Ech Opil type ineulity will give Hrdy type ineulity. We will use number of Opil type ineulities to illustrte this point. Boyd nd Wong [8] proved if p > nd if y is n bsolutely continuous function on [ b] with y() = (or y(b) = ), then (2.) (t) jy(t)j p y (t) dt (p ) w(t) y (t) p dt 2 Mthemtics Subject Clssi ction. 26A5, 26D, 26D5, 39A3, 34A4. Key words nd phrses. Hrdy s ineulity, Opil s ineulity.
2 R. P. AGARWAL, D. O REGAN 2 AND S. H. SAKER 3 nd w re nonnegtive functions in C [ b], nd such tht the boundry vlue problem hs solution ((t) u (t) p ) = w (t)u p (t) with u() = nd (b) [u (b)] p = w(b) u p (b), for which u > in [ b] (let be the smllest eigenvlue of the boundry vlue problem). Applying the ineulity (2.) on the term (p ) R b R( b)f p ()F ()d we hve (2.2) (p ) R( b)f p ()F ()d s(t)(f ()) p d r nd s re nonnegtive functions, r 2 C[ b], s 2 C [ b], nd such tht the boundry vlue problem hs solution (2.3) (R( b) u () p ) = s ()u p () (R( b) = R b r(t)dt nd note R 2 C [ b] since r 2 C[ b]) with u() = nd u(b) =, for which u > in [ b] (let be the smllest eigenvlue of the boundry vlue problem). Theorem 2.. Assume tht r s re nonnegtive functions with r 2 C[ b] s 2 C [ b] nd p >. Then p r() f(t)dt d s() (f()) p d for ll integrble functions f is the smllest eigenvlue of the boundry vlue problem (2.3): Proof. Let F () = R f(t)dt: Since f is integrble on [ b] then F is bsolutely continuous on [ b]. Note F () =, F () = f() > nd p r() f(t)dt d = r()f p ()d: Integrtion by prts gives p r() f(t)dt d = R( b)f p () b (p ) R( b)f p ()F ()d R( b) = R b r(t)dt. Using R(b b) = nd F () = we hve p (2.4) r() f(t)dt d = (p ) R( b)f p ()F ()d: Now (2.2) estblishes the result.
HARDY TYPE INEQUALITIES 3 Boyd in [7] etended the results of [8]. In Theorem 2. of [7] the uthor estblished ineulities (best possible constnts) of the form s(t) jy(t)j p y (t) dt b r(t) y (t) p dt (p ) p >, >, with r s 2 C ( b) nd r >, s >.e. on ( b) here is the smllest eigenvlue of n pproprite boundry vlue problem (ssuming certin conditions re stis ed see [7]). With these conditions (with = nd > ) we obtin using the procedure before nd in Theorem 2. p r() f(t)dt d p r(t) (f(t)) dt is the smllest eigenvlue of n pproprite boundry vlue problem. Insted of this ineulity (nd presenting the conditions to gurntee the eistence of ) we will consider two specil cses of this result, one found in [7] nd the other in [6]. In the following, we pply n ineulity due to Boyd [7] nd the Hölder ineulity. The Boyd ineulity sttes tht: If y is bsolutely continuous on [ b] with y() = (or y(b) = ), then (2.5) jy(t)j y (t) dt N( s)(b >, s >, < s, (2.6) N( s) := nd I( s) := (s ) s (s )( ) (I( s)) := ) y (t) s dt s (s ) (s ) s (s )( ) s( ) (s)=s s t [ ( )t]t = dt: Apply the Hölder ineulity nd ineulity (2.5) to obtin (2.7) R( b)f p ()F ()d R p p b ( b)d F p () F () d N (p s)(b ) p R p p ( b)d F () s d p s p >, =p = =, s > nd < < s here N(p s) is determined from (2.6) by putting = p nd =.
4 R. P. AGARWAL, D. O REGAN 2 AND S. H. SAKER 3 Theorem 2.2. Assume tht r is nonnegtive mesurble function on ( b) p > s >, < < s nd =p = =. Then p r() f(t)dt p d C (f()) s s d for ll integrble functions f here C = (p) N (p s)(b Proof. The result follows from (2.4) nd (2.7). R ) p b Rp p ( b)d : As in the proof of Theorem 2., by putting F () = R b f(t)dt we hve the following result. Theorem 2.3. Assume tht r is nonnegtive mesurble function on ( b) p > < < s nd =p = =. Then p r() f(t)dt p d C (f()) s s d for ll integrble functions f here C = (p) N (p s)(b nd R( ) = R r(t)dt. R ) p b Rp p ( )d When = s eution (2.5) becomes (2.8) jy(t)j y (t) dt L( )(b ) y (t) dt (2.9) L( ) := @ A nd (2.) is the Gmm function. Apply ineulity (2.8) to obtin F p () F () d L(p )(b ) p p F () d (2.) L(p ) = (p) p p @ p p p A : p
HARDY TYPE INEQUALITIES 5 Using (2.), we see tht R( b)f p ()F ()d R p p b ( b)d F p () F () d L (p )(b ) p R p ( b)d F () d p p p > nd =p = = : This gives us the following results. Theorem 2.4. Assume tht r is nonnegtive mesurble function on ( b) p >, > nd =p = =. Then p r() f(t)dt p d C (f()) d for ll integrble functions f here C = (p) L (p )(b nd L(p ) is de ned s in (2.). R ) p b Rp p ( b)d Theorem 2.5. Assume tht r is nonnegtive mesurble function on ( b) p >, > nd =p = =. Then p r() f(t)dt p d C (f()) d for ll integrble functions f here C = (p) L (p )(b nd L(p ) is de ned s in (2.). R ) p b Rp p ( )d Finlly we pply n Opil type ineulity due to Beesc [6] to prove ineulities of Hrdy type. The ineulity due to Beesc is given in the following theorem. Theorem 2.6. Let r s be nonnegtive, mesurble functions on ( ). Further ssume tht >, p >, < <, nd let y be bsolutely continuous in [ ] such tht y() = : Then (2.2) r(t) jy(t)j p y (t) dt K (p ) s(t) y (t) dt (p)=
6 R. P. AGARWAL, D. O REGAN 2 AND S. H. SAKER 3 K (p ) = (2.3) p (r(y)) (s(y)) y s (t)dt p( )=( ) dy! : If insted [ ] is replced by ( ) nd y() = is replced by y() =, then (2.4) r(t) jy(t)j p y (t) (p)= dt K 2 (p ) s(t) y (t) dt K 2 (p ) = (2.5) p (r(y)) (s(y)) y s (t)dt p( )=( ) dy! Now, we pply ineulity (2.2) nd (2.4). For completeness we pply (2.2) with > to obtin (p)= (2.6) R( b) F p () F ()d K (p ) s()(f ()) d (2.7) K (p ) = p (R( b)) (s()) p s (t)dt d : : Theorem 2.7. Let p >, > nd let r s be nonnegtive mesurble functions on ( b): Then p (p)= r() f(t)dt d (p ) K (p ) s()(f()) d for ll integrble functions f here K (p ) is de ned s in (2.7). Proof. The result follows from (2.4) nd (2.6). Theorem 2.8. Let p >, > nd let r s be nonnegtive mesurble functions on ( b): Then p (p)= r() f(t)dt d (p ) K 2 (p ) s()(f()) d
for ll integrble functions f here K 2 (p ) = p HARDY TYPE INEQUALITIES 7 (R( )) (s()) References p s (t)dt d! [] R. P. Agrwl nd P. Y. H. Png, Opil ineulities with Applictions in Di erentil nd Di erence Eutions, Kluwer, Dordrechet (995). [2] R. P. Agrwl, M. Bohner, D. O Regn nd S. H. Ser, Some Wirtinger-type ineulities on time scles nd their pplictions, Pci c J. Mth. 252 (2), -8. [3] K. F. Andersen nd H. P. Heining, Weighted norm ineulities for certin integrl opertor, Sim. J. Mth. Anl. 4 (983), 834-844. [4] J. A. Oguntuse nd C. O. Imoru, New generliztions of Hrdy s integrl ineulties, J. Mth. Anl. Appl. 24 (2), 73-82. [5] P. R. Bessc, Hrdy s ineulity nd its etensions, Pci c J. Mth. (96), 39-6. [6] P. R. Bessc, Elementry proofs of some Opil-type integrl ineulities, J. d Anlyse Mth. 36 (979), -4. [7] D. Boyd, Best constnts in clss of integrl ineulities, Pc. J. Mth. 3 (969), 367-383. [8] D. Boyd nd J. S. W. Wong, An etension of Opil s ineulity, J. Mth. Anl. Appl. 9 (967), -2. [9] J. Clvert, Some generliztions of Opil s ineulity, Proc. Amer. Mth. So. 8 (967), 72-75. [] G. H. Hrdy, J. E. Littlewood nd G. Poly, Ineulities, 2nd Ed. Cmbridge Univ. Press 952. [] G. H. Hrdy, Notes on theorem of Hilbert, Mth.. 6 (92), 34-37. [2] A. Kufner nd Lrs-Eri Persson, Weighted Ineulities of Hrdy Type, World Scienti c Publishing (23). [3] A. Kufner, L. Mligrnd nd L. Persson, The Hrdy ineulities: About its History nd Some Relted Results, Pilsen (27). [4] N. Levinson, On n ineulity of Opil nd Beesc, Proc. Amer. Mth. Soc. 5 (964), 565-566. [5] P. Mroni, Sur l ineglité d Opil-Beesc, C. R. Acd. Sci. Pris Ser A-B. 264 (967), A62-A64. [6] C. Olech, A simple proof of certin result of.opil, Ann. Polon. Mth. 8 (96), 6-63. [7]. Opil, Sur uné ineglité, Ann. Polon. Mth. 8 (96), 92-32. [8] B. Opic nd A. Kufner, Hrdy-type ineulities, Longmn Scienti c& Technicl, Hrlow, ESSe, UK, (989). [9] G. S. Yng, A note on some integrodi erentil ineulities, Soochow J. Mth. 9 (983), 23-236. Deprtment of Mthemtics, Tes A nd M University-Kingsville, Tes, 78363, USA., 2 School of Mthemtics, Sttistics nd Applied Mthemtics, Ntionl University of Irelnd, Glwy, Irelnd., 3 Deprtment of Mthemtics, Fculty of Science, Mnsour University, Mnsour 3556, Egypt, shser@mns.edu.eg. :