Balanced Canavati type Fractional Opial Inequalities

Transcription

1 Blnced Cnvti type Frctionl Opil Ineulities George A. Anstssiou eprtment of Mthemticl Sciences University of Memphis Memphis, T 385, U.S.A. gnstss@memphis.edu edicted to the 65th irthdy of Professor Heiner Gonsk Astrct Here we present L p, p >, frctionl Opil type ineulities suject to high order oundry conditions. They involve the right nd left Cnvti type generlised frctionl derivtives. These derivtives re mied together into the lnced Cnvti type generlised frctionl derivtive. This lnced frctionl derivtive is introduced nd ctivted here for the rst time. AMS Suject Clssi ction : 6A33, 6, 65. Key Words nd Phrses: Opil ineulity, frctionl ineulity, Cnvti frctionl derivtive, oundry conditions. Introduction This rticle is inspired y the fmous theorem of. Opil [], 96, which follows Theorem Let (t) C ([; h]) e such tht () (h), nd (t) > in (; h) : Then h In (), the constnt h 4 the optiml function j (t) (t)j dt h 4 h ( (t)) dt: () is the est possile. Ineulity () holds s eulity for ct; t h (t) ; h c (h t) ; t h;

2 where c > is n ritrry constnt. To prove esier Theorem, Beesck [4] proved the following well-known Opil type ineulity which is used very commonly. This is nother insirtion to our work. Theorem Let (t) e solutely continuous in [; ], nd (). Then j (t) (t)j dt ( (t)) dt: () Ineulity () is shrp, it is ttined y (t) ct, c > is n ritrry constnt. Opil type ineulities re used lot in proving uniueness of solutions to di erentil eutions, lso to give upper ounds to their solutions. By themselves hve mde gret suject of intensive reserch nd there eists gret literture out them. Typicl nd gret sources on them re the monogrphs [], []. We de ne here the lnced Cnvti type frctionl derivtive nd we prove relted Opil type ineulities suject to oundry conditions. These hve smller constnts thn in other Opil ineulities when using trditionl frctionl derivtives. Bckground Let >, n : [] (integrl prt of ), nd : n ( < < ). The gmm function is given y () R e t t dt. Here [; ] R, ; [; ] such tht, where is ed. Let f C ([; ]) nd de ne the left Riemnn- Liouville integrl (J f) () : () ( t) f (t) dt; (3). We de ne the suspce C ([:]) of C n ([; ]): n o C ([; ]) : f C n ([; ]) : J f (n) C ([ ; ]) : (4) For f C ([; ]), we de ne the left generlized -frctionl derivtive of f over [ ; ] s f : J f (n) ; (5) see [], p. 4, nd Cnvti derivtive in [5]. otice tht f C ([ ; ]) : We need the following generliztion of Tylor s formul t the frctionl level, see [], pp. 8-, nd [5].

3 Theorem 3 Let f C ([; ]), [; ] ed. (i) If then f () f ( )+f ( ) ( )+f ( ) ( ) +:::+f (n ) ( ) ( ) n (n ) (6) + J f () ; ll [; ] : : (ii) If < < we get f () J f () ; ll [; ] : (7) We will use (6) nd (7). Furthermore we need: Let >, m [], m, < <, f C ([; ]), cll the right Riemnn-Liouville frctionl integrl opertor y J f () : () (J ) f (J) dj; (8) [; ], see lso [3], [6], [7], [8], []. e ne the suspce of functions C ([; ]) : n f C m ([; ]) : J o f (m) C ([; ]) : (9) e ne the right generlized -frctionl derivtive of f over [; ] s f : ( ) m J f (m), () see [3]. We set f f. otice tht f C ([; ]) : From [3], we need the following Tylor frctionl formul. Theorem 4 Let f C ) If, we get f () mx k f (k) ( ) k ) If < <, we get We will use () nd (). We introduce new concept ([; ]), >, m : []. Then ( ) k + J f () ; 8 [; ] : () f () J f () ; 8 [; ] : () e nition 5 Let f C ([; ]), [; ], >, m : []. Assume tht f C + ; nd f C ; +. We de ne the lnced Cnvti type frctionl derivtive y f () : f (), for + ; f (), for < + : (3) 3 3

4 3 Min Result We give our min result Theorem 6 Let f C ([; ]), >, m : []. Assume tht f C nd f C ; +. Assume further tht + ; f (k) () f (k) (), k ; ; :::; m ; (4) p; > : p +, nd > : (i) Cse of <. Then jf ()j j f ()j d (5) (+ p) p( )+ ( ) ( p ) () [(p ( ) + ) (p ( ) + )] p (ii) Cse of >. Then j f ()j d : jf ()j j f ()j d (6) (+ p( )+ ) ( ) ( p ) () [(p ( ) + ) (p ( ) + )] p (iii) When p, >, then j f ()j d : (+ ) ( ) h p i () ( ) jf ()j j f ()j d (7) j f ()j d Remrk 7 Let us sy tht, then y (7) we otin jf ()j jf ()j d ( ) 4 : (f ()) d ; (8) tht is reproving nd recovering Opil s ineulity (), see [], see lso Olech s result [9]. 4 4

5 Proof. of Theorem 6. Let ; +, we hve y ssumption f (k) (), k ; ; :::; m nd Theorem 3 tht f () () ( ) f () d: (9) Let + ; ; we hve y ssumption f (k) (), k ; ; :::; m nd Theorem 4 tht f () () ( ) f () d: () Using Hölder s ineulity on (9) we get jf ()j () ( ) j f ()j d Set Then nd () ( ) p d p j f ()j d p( )+ p ( ) j () (p ( ) + ) f ()j d : () p z () : Therefore y () we hve j f ()j d; (z () ). z () j f ()j ; j f ()j (z ()), ll + : jf ()j j f ()j p( )+ p ( ) () (p ( ) + ) p (z () z ()) ; () ll + : et working similrly with () we otin jf ()j () ( ) f () d () ( ) p p d f () d 5 5

6 Set Then () : p( )+ p ( ) () (p ( ) + ) p f () d () f () f () d : (3) f () d; ( () ). nd f () (), ll + : Therefore y (3) we hve jf ()j f () p( )+ p ( ) () (p ( ) + ) p () () ; (4) ll + : et we integrte () over [; ] to otin jf ()j j f ()j d () (p ( ) + ) p ( ) p( )+ d () (p ( ) + ) p So we hve proved for ll + : By (6) we get () (p ( p( )+ ( ) p p( )+ p ( ) (z () z ()) d p p( )+ p ( ) ) + ) p (p ( ) + ) p () [(p ( ) + ) (p ( ) + )] p p( )+ ( ) p jf ()j j f ()j d () [(p ( ) + ) (p ( ) + )] p + jf ()j j f ()j d z () z () z () d j f ()j d : (5) j f ()j d ; (6) 6 6

7 (p( )+) p ( ) [ p( )+ p + ] () [(p ( ) + ) (p ( ) + )] p Similrly we integrte (4) over [; ] to otin + j f ()j d : (7) jf ()j f () d () (p ( () (p ( ) + ) p We hve proved tht ) + ) p () (p ( p( )+ ( ) p p( )+ p ( ) ( ) p( )+ d p p( )+ p ( ) ) + ) p (p ( ) + ) p jf ()j () [(p ( ) + ) (p ( ) + )] p for ll + : By (9) we get (p( )+) p ( ) + jf ()j [ p( )+ p + ] () [(p ( ) + ) (p ( ) + )] p Adding (7) nd (3) we get 4 jf ()j j f ()j d + j f ()j d Assume <, then. Therefore we get + + f () d f () d + () () d () () d ( ()) : (8) f () d ; (9) f () d : (3) (+ p) p( )+ ( ) ( p ) () [(p ( ) + ) (p ( ) + )] p f () 3 d 5 : (:) (3) () (+ p) p( )+ ( ) ( p ) () [(p ( ) + ) (p ( ) + )] p 7 7

8 " + j f ()j d + f () # d (3) + (+ p) p( )+ ( ) ( p ) () [(p ( ) + ) (p ( ) + )] p So for < we hve proved (5). Assume now >, then < <. Therefore we get j f ()j d : (33) () (+ p) p( )+ ( ) ( p ) () [(p ( ) + ) (p ( ) + )] p " + j f ()j d + f () # d + (+ p( )+ ) ( ) ( p ) () [(p ( ) + ) (p ( ) + )] p j f ()j d : (34) So when > we hve estlished (6). (iii) The cse of p, see (7), is ovious, it derives from (5) immeditely. References [] R.P. Agrwl nd P.Y.H. Png, Opil Ineulities with Applictions in i erentil nd i erence Eutions, Kluwer, ordrecht, London, 995. [] G.A. Anstssiou, Frctionl i erentition Ineulities, Reserch Monogrph, Springer, ew York, 9. [3] G.A. Anstssiou, On Right Frctionl Clculus, Chos, Solitons nd Frctls, 4 (9), [4] P.R. Beesck, On n integrl ineulity of. Opil, Trns. Amer. Mth. Soc. 4 (96), [5] J.A. Cnvti, The Riemnn-Liouville Integrl, ieuw Archief Voor Wiskunde, 5 () (987), [6] A.M.A. El-Syed, M. Ger, On the nite Cputo nd nite Riesz derivtives, Electronic Journl of Theoreticl Physics, Vol. 3, o. (6),

9 [7] G.S. Frederico,.F.M. Torres, Frctionl Optiml Control in the sense of Cputo nd the frctionl oether s theorem, Interntionl Mthemticl Forum, Vol. 3, o. (8), [8] R. Goren o, F. Minrdi, Essentils of Frctionl Clculus,, Mphysto Center, Minrdiotes/fmk.ps. [9] C. Olech, A simple proof of certin result of. Opil, Ann. Polon. Mth. 8 (96), []. Opil, Sur une ineglite, Ann. Polon. Mth. 8 (96), 9-3. [] S.G. Smko, A.A. Kils, O.I. Mrichev, Frctionl Integrls nd erivtives, Theory nd Applictions, (Gordon nd Brech, Amsterdm, 993) [English trnsltion from the Russin, Integrls nd erivtives of Frctionl Order nd Some of Their Applictions (uk i Tekhnik, Minsk, 987)]. 9 9

10 Cnvti frctionl Ostrowski type ineulities George A. Anstssiou eprtment of Mthemticl Sciences University of Memphis Memphis, T 385, U.S.A. gnstss@memphis.edu Astrct Here we present Ostrowski type ineulities involving left nd right Cnvti type generlised frctionl derivtives. Comining these we otin frctionl Ostrowski type ineulities of mied form. Then we estlish Ostrowski type ineulities for ordinry nd frctionl derivtives involving comple vlued functions de ned on the unit circle. AMS Suject Clssi ction : 6A33, 67, 6, 65. Keywords nd Phrses: devition of function, frctionl derivtive, Ostrowski ineulity, function verge. Introduction In 938, A. Ostrowski [] proved the following importnt ineulity: Theorem Let f : [; ] R e continuous on [; ] nd di erentile on (; ) whose derivtive f : (; ) R is ounded on (; ), i.e., kf k : jf (t)j < +. Then sup t(;) f (t) dt f () " ( ) # ( ) kf k ; () for ny [; ]. The constnt 4 is the est possile. Since then there hs een lot of ctivity round these ineulities with importnt pplictions to numericl nlysis nd proility. In this rticle we present vrious generl Ostrowski type ineulities involving frctionl derivtives of Cnvti type. At the end we give pplictions to comple vlued functions de ned on the unit circle.

11 Bckground Let >, n : [] (integrl prt of ), nd : n ( < < ). The gmm function is given y () R e t t dt. Here [; ] R, ; [; ] such tht, where is ed. Let f C ([; ]) nd de ne the left Riemnn- Liouville integrl (J f) () : () ( t) f (t) dt; (). We de ne the suspce C ([:]) of C n ([; ]): n o C ([; ]) : f C n ([; ]) : J f (n) C ([ ; ]) : (3) For f C ([; ]), we de ne the left generlized -frctionl derivtive of f over [ ; ] s f : J f (n) ; (4) see [4], p. 4, nd Cnvti derivtive in [6]. otice tht f C ([ ; ]) : We need the following generliztion of Tylor s formul t the frctionl level, see [4], pp. 8-, nd [6]. Theorem Let f C ([; ]), [; ] ed. (i) If then f () f ( )+f ( ) ( )+f ( ) ( ) +:::+f (n ) ( ) ( ) n (n ) (5) + J f () ; ll [; ] : : (ii) If < < we get f () J f () ; ll [; ] : (6) We will use (5). Furthermore we need: Let >, m [], m, < <, f C ([; ]), cll the right Riemnn-Liouville frctionl integrl opertor y J f () : () (J ) f (J) dj; (7) [; ], see lso [5], [9], [], [], [3]. e ne the suspce of functions C ([; ]) : n f C m ([; ]) : J o f (m) C ([; ]) : (8)

12 e ne the right generlized -frctionl derivtive of f over [; ] s f : ( ) m J f (m), (9) see [5]. We set f f. otice tht f C ([; ]) : From [5], we need the following Tylor frctionl formul. Theorem 3 Let f C ) If, we get ([; ]), >, m : []. Then f () mx k f (k) ( ) k ) If < <, we get ( ) k + J f () ; 8 [; ] : () f () J f () ; 8 [; ] : () We will use (). In [4], pp , nd [3], we proved the rst frctionl Ostrowski ineulity. Theorem 4 Let <, is ed. Let f C ([; ]),, n : []. Assume f (i) ( ), i ; :::; n. Then f (y) dy f ( ) f ;[;] ( ) : () ( + ) Ineulity () is shrp, nmely it is ttined y f () : ( ),, [; ] : (3) When < the ssumption f (i) ( ), i ; :::; n is void. 3 Min Results We give Theorem 5 Sme ssumptions s in Theorem 4. Then f (y) dy f ( ) f ( L([ ;]) ) : (4) ( + ) 3

13 Proof. By (5) we get Hence I.e. f (y) f ( ) () (y ) () y jf (y) f ( )j () y (y w) f (w) dw; 8 y : (5) y f (w) (y ) dw () (y ) () jf (y) f ( )j (y ) () (y w) f (w) dw f L([ ;]) : Therefore we get f (y) dy f ( ) (f (y) (y ) dy proving the clim. We continue with f (w) dw f L([ ;]) ; 8 y [ ; ] : (6) jf (y) f ( )j dy (6) f L([ ;]) ( ) ( + ) () ( ) f L([ ;]) ; f ( )) dy f L([ ;]) Theorem 6 Sme ssumptions s in Theorem 4. Let p; > : p +. Then f (y) dy f ( ) f L([ ;]) ( () (p ( ) + ) p + p ) + p : (7) Proof. We notice tht jf (y) f ( )j () y (y w) f (w) dw () 4 3

14 Tht is jf (y) () () y f ( )j (y w) p( ) dw p( )+ (y ) p ( ) + p (y ) ( )+ p () (p ( ) + ) p f L([ ;]) () p y f (w) dw f (w) dw f L([ ;]) : (y ) ( )+ p ; 8 y [ (p ( ) + ) ; ] : (8) p Conseuently we otin f (y) dy f ( ) proving the clim. Comining Theorems 4-6 we derive jf (y) f ( )j dy (8) f L([ ;]) (y () (p ( ) + ) ) ( )+ p dy p f L([ ;]) ( ) + p ; (9) () (p ( ) + ) p + p Proposition 7 Let ll s in Theorem 6. Then ( f (y) dy f ( ) min f ;[;] ( ( + ) ) ; () f L([ ;]) ( ) ( + ) f L([ ; ( ;]) ) + p () (p ( ) + ) p + p We continue with right Cnvti frctionl Ostrowski ineulities. Theorem 8 Let, m [], f C ([; ]). Assume f (k) ( ), k ; :::; m ; which is void when <. Then f () d f () f ;[;] ( ) : () ( + ) 9 ; : 5 4

15 Proof. Let [; ]. By () we get Then, s efore, we get Hence it holds f () f () () jf () f ()j f () d f ;[;] ( + ) ( ) proving the clim. We continue with (J ) f (J) dj: () ( ) ( + ) f ;[;] ; 8 [; ] : (3) f () jf () f ()j d (3) ( ) d f ;[;] ( ) ( ( + )) ( + ) f ;[;] ( ) ; (4) ( + ) Theorem 9 Sme ssumptions s in Theorem 8. Then f () d f () f L([;]) ( ) : (5) ( + ) Hence Proof. We hve gin jf () f ()j () (J ) f (J) dj (6) ( ) () f L([;]), 8 [; ] : f L([;]) ( ()) ( ) proving the clim. We lso hve f () d f () ( ) d jf () f ()j d (6) f L([;]) ( + ) ( ) ; (7) 6 5

16 Theorem Sme ssumptions s in Theorem 8. Let p; > : p + Then f () d f () f L([;]) ( () (p ( ) + ) p + p ) + p : (8) Proof. As efore we otin f L([;]) jf () f ()j () (p ( ) + ) p ( ) + p ; 8 [; ] : (9) Hence f () d f () jf () f ()j d (9) f L([;]) ( ) + p d () (p ( ) + ) p ( ) f L([;]) ( ) + p ; (3) (p ( ) + ) p () + p proving the clim. Comining Theorems 8- we derive Proposition Here ll s in Theorem. Then ( f () d f () min f ;[;] ( ) ; (3) ( + ) f L([;]) ( ) f 9 ( ) + p L([;]) ; ( + ) () (p ( ) + ) p + ; : p We lso give optimlity of (). Proposition Ineulity () is shrp, nmely it is ttined y m : [] : f (J) ( J),, J [; ], (3) 7 6

17 Proof. We hve tht f (m) (J) ( ) m ( ) ( ) ::: ( m + ) ( m + ) ( J) m : (33) We lso notice J ( m) ( + m) f (m) () (7) ( + m) (J ) +m f (m) (J) dj (J ) m ( ) m ( ) ( ) ::: ( m + ) ( m + ) ( J) m dj ( )m ( ) ( ) ::: ( m + ) ( m + ) ( + m) ( J) ( m+) (J ) (+m ) dj ( )m ( ) ( ) ::: ( m + ) ( m + ) ( + m) ( m + ) ( + m ) () ( ) ( ) m ( ) ( ) ::: ( m + ) ( m + ) ( m + ) ( ) Tht is ( ) m ( ) ( ) ::: ( m + ) ( m + ) ( ) Therefore it holds ::: ( ) m ( + ) ( ) : (34) J +m f (m) () ( ) m ( + ) ( ) : (35) f () (9) ( ) m ( ) m ( + ) ( ) ( + ) : (36) So tht f ;[;] ( + ) : (37) We lso notice tht f (k) ( ), k ; :::; m, nd f (). We oserve further tht nd R:H:S: () L:H:S: () ( ) + ; (38) ( + ) ( + ) ( ( ) ) + ; (39) proving the clim. et we present mied Cnvti frctionl Ostrowski type ineulities. 8 7

18 Theorem 3 Let, m [], [; ] ed, f C ([; ]) with f C ([; ]) nd f C ([; ]). Assume tht f (k) (), k ; :::; m, which is void when <. Then f (y) dy f () (4) n ( ) ( + ) f ( )+ ;[;] + k fk ( )+o ;[;] Hence ( ) + + ( ) + ( ) ( + ) Proof. Let [; ]. By () we get f (y) f () () jf (y) f ()j () 8 y [; ] : Similrly, y (5) we get Hence f (y) f () () jf (y) f ()j () 8 y [; ] : We oserve tht y y y m n f ;[;] ; k fk ;[;] o : (4) (J y) f (J) dj; 8 y [; ] : (4) (J y) f (J) dj ( y) y f (y) dy ( + ) f ;[;] ; (43) (y w) ( f) (w) dw; 8 y [; ] : (44) (y w) j f (w)j dw k fk ;[;] (y ) ( + ) ; f () (45) jf (y) f ()j dy (46) ( ) y ((43), (45)) jf (y) f ()j dy + jf (y) f ()j dy ( f ;[;] ( y) dy + k fk ) ;[;] (y ) dy ( + ) ( + ) (47) 9 8

19 n ( ) ( + ) f ( ;[;] )+ + k fk ( ;[;] )+o ; (48) proving the clim. We continue with the optimlity of Theorem 3. Proposition 4 Ineulities (4), (4) re shrp, nmely re ttined y f (J) ( J), J [; ] ; (J ), J [; ] ; (49) where, [; ] is ed. See tht f (k) ( ) f (k) ( + ), k ; ; :::; m : We hve tht f ;[;] f ;[;] ( + ) : (5) Furthermore we notice L:H:S: (4) n ( ) + + ( ) +o ; (5) ( + ) ( ) nd ( ) + + ( ) + R:H:S: (4) ( ) ( + ) ( ) + + ( ) + ( + ) proving the clim. We continue with ( + ) ( ) ; (5) Theorem 5 All s in Theorem 3. Then ( [; ]) f (y) dy f () (53) n f ( ) ( + ) ( ) L([;]) + k fk ( )o L([;]) ( ) + ( ) n m ( ) ( + ) f o ; k L([;]) fk L([;]) : (54) Proof. Let [; ]. From (4) we get (y [; ]) jf (y) f ()j () y (J y) f (J) dj (55) 9

20 Tht is ( y) () y f (J) ( y) dj () f (J) dj: (56) jf (y) f ()j ( y) f L([;]) ; 8 y [; ] : (57) () Similrly from (44) we get jf (y) f ()j () (y ) () y (y w) j( f) (w)j dw (58) k fk L([;]) ; 8 y [; ] : (59) From (46) we otin f (y) dy f () (6) ( f L([;]) ( y) dy + k fk ) L([;]) (y ) dy () () (6) n ( ) ( + ) f ( ) L([;]) + k fk ( )o L([;]) ( ) + ( ) n m ( ) ( + ) f o ; k L([;]) fk L([;]) ; (6) proving the clim. We lso give Theorem 6 All s in Theorem 3. Let p; > : p +. Then ( [; ]) f (y) dy f () (63) ( ) () (p ( ) + ) p + p n ( ) + p f o + ( )+ p L([;]) k fk ( ) + p + ( ) + p A ( ) () (p ( ) + ) p + p n m f o ; k L([;]) fk L([;]) : (64)

21 Proof. By (55) we get jf (y) f ()j () y (J y) p( ) p dj f L([;]) ( y) + p () (p ( ) + ) p Similrly from (58) we derive jf (y) f ()j () By (46) we derive y (y ) + p () (p ( ) + ) p f (y) dy f L([;]) ; 8 y [; ] : (65) (y w) p( ) dw p k fk L([;]) k fk L([;]) ; 8 y [; ] : (66) f () () ( ) (p ( ) + ) p ( y) + p dy f + L([;]) (y ) + p dy k fk L([;]) ( ) () (p ( ) + ) p + p n ( ) + p f o + ( )+ p L([;]) k fk ( ) + p + ( ) + p ( ) () (p ( ) + ) p proving the clim. + p A m Corollry 7 All s in Theorem 3. Then f (y) dy ( ) + + ( ) + ( ) () p + ) (67) (68) n f L([;]) ; k fk L([;])o ; (69) f () (7) m n f L([;]) ; k fk L([;]) o :

22 Proof. By Theorem 6. Comining Theorems 3, 5, 6 we derive Theorem 8 Here ll s in Theorem 3. Let ny p; > : p +. Then f (y) dy f () min ( ) ( + ) ( ) ( + ) n f ;[;] ( )+ + k fk ;[;] ( )+o ; n f L([;]) ( ) + k fk L([;]) ( )o ; (7) ( ) () (p ( ) + ) p + p n ( ) + p f oo + ( )+ p L([;]) k fk L([;]) ( ( ) + + ( ) + n min m f o ( ) ( + ) ; k ;[;] fk ;[;] ; ( ) + ( ) n m ( ) ( + ) f o ; k L([;]) fk L([;]) ; ( ) + p + ( ) + p A ( ) () (p ( ) + ) p + p n m f oo ; k L([;]) fk L([;]) : 4 Applictions Ineulities for comple vlued functions de ned on the unit circle were studied etensively y S. rgomir, see [7], [8]. We give here our version for these functions involved in Ostrowski type ineulities, y pplying results of this rticle. Let t [; ] [; ), the unit circle rc A z C : z e it ; t [; ], nd f : A C e continuous function. Clerly here there eist functions u; v : A R continuous, the rel nd the comple prt of f, respectively, such tht f e it u e it + iv e it : (73) So tht f is continuous, i u; v re continuous. Cll g (t) f e it, l (t) u e it, l (t) v e it, t [; ]; so tht g : [; ] C nd l ; l : [; ] R re continuous functions in t. 3

23 If g hs derivtive with respect to t, then l, l hve lso derivtives with respect to t. In tht cse (i.e. g (t) l (t) + il (t)), which mens f t e it u t e it + iv t e it ; (74) f t (cos t + i sin t) u t (cos t + i sin t) + iv t (cos t + i sin t) : (75) Let us cll cos t, y sin t. Then Similrly we nd tht u t e it u t (cos t + i sin t) u t ( + iy) u t eit ( sin t) + cos So tht ut v t e ( sin t) cos t: (77) e e e it ;[;] + ;[;] ;[;] nd vt e e e it ;[;] + ;[;] ;[;] Conseuently it holds f t e it e e e e it ;[;] ;[;] ;[;] ;[;] Since g is continuous on [; ], then R f eit dt eists. Furthermore it holds f e it dt u e it dt + i Let now t [; ]. We oserve tht f e it dt f e it u e it dt + i v e it dt: (8) v e it dt u e it iv e it (8) 4 3

24 u e it dt u e it + v e it dt v e it (y ()) " 4 + t + # h ( ) ( ) u t e it + ;[;] v t e it i ;[;] : (83) We hve proved the following version of Ostrowski ineulity for comple functions. Theorem 9 Let f C (A; C) with its rel nd comple prt u e it ; v e it C ([; ]) s functions of t, where t [; ] [; ): Then f e it dt f e it " 4 + t + # h ut ( ) ( ) e it + vt ;[;] e it i ;[;] (84) " 4 + t + # ( ) ( ) (85) e e e e it ;[;] ;[;] ;[;] Ineulity (84) is shrp. ;[;] An eplntion follows net. For z C fg we cll principl vlue of log (z) the comple vlued function where Arg (z) < : For t [; ) we hve tht Log (z) : ln jzj + iarg (z) ; (86) Log e it it: (87) Let here < <, nd t. Here l (t) nd l (t) t, with l (t) : otice in generl tht 4 + t + ( ) ( ) ( t ) + (t ) ; (88) ( ) for ny t [; ], see [9], p. 498 nd []. 5 4

25 Hence we hve L:H:S: (84) Log e it dt itdt tdt : (89) Furthermore it holds R:H:S: (84) : (9) By (89) nd (9) we conclude tht ineulity (84) is ttined y Log t t on [; ], tht is shrp ineulity. We now move t the frctionl level. Let t [; ], we rewrite (8) s follows f e it dt f e it l (t) dt l (t ) + l (t) dt l (t ) : (9) By pplying Theorem 8 to ech of the lst two summnds we derive the following comple frctionl Ostrowski ineulity. Theorem Let f C (A; C), t; t [; ] [; ); ny p; > : p +. Let, m [], with l ; l Ct ([; t ]) nd l ; l Ct ([t ; ]). Assume tht l (k) (t ) l (k) (t ), k ; :::; m, which is void when <. Then f e it dt f e it min ( ) ( + ) ( ) ( + ) min n t l ;[;t] (t ) + + t l ;[t;] ( t ) +o ; n t l L([;t]) (t ) + t l L([t ;]) ( t ) o ; + p ( ) () (p ( ) + ) p n (t ) + p t l L([;t + ( t ]) ) + p t l L([t ;]) ( ) ( + ) ( ) ( + ) oo + n t l ;[;t] (t ) + + t l ;[t;] ( t ) +o ; n t l L([;t]) (t ) + t l L([t ;]) ( t ) o ; ( ) () (p ( ) + ) p + p 6 5

26 n (t ) + p ( ( t ) + + (t ) + min ( ) ( + ) ( t ) + (t ) t l L([;t + ( t ]) ) + oo p t l L([t ;]) ( ) ( + ( t ) + p + (t ) + p ( ) () (p ( ) + ) p (9) n m t l ;[;t] ; l o t ;[t;] ; n m t l L([;t ; ]) t l o L([t ; ;]) + p n m t l L([;t ; ]) t l L([t ;]) ( ( t ) + + (t ) + min ( ) ( + ) ( t ) + (t ) ( ) ( + ( t ) + p + (t ) + p ( ) () (p ( ) + ) p A oo + n m t l ;[;t] ; l o t ;[t;] ; n m t l L([;t ; ]) t l o L([t ; ;]) + p n m t l L([;t ; ]) t l L([t ;]) A oo : (93) References [] G.A. Anstssiou, Ostrowski type ineulities, Proc. AMS 3 (995), [] G. Anstssiou, Quntittive Approimtion, Chpmn & Hll / CRC, Boc Rton, ew York,. [3] G. Anstssiou, Frctionl Ostrowski type ineulities, Communictions in Applied Anlysis, 7 (3), o., 3-8. [4] G.A. Anstssiou, Frctionl i erentition Ineulities, Reserch Monogrph, Springer, ew York, 9. [5] G.A. Anstssiou, On Right Frctionl Clculus, Chos, Solitons nd Frctls, 4 (9), [6] J.A. Cnvti, The Riemnn-Liouville Integrl, ieuw Archief Voor Wiskunde, 5 () (987),

27 [7] S.S. rgomir, Ostrowski s Type Ineulities for Comple Functions e ned on unit Circle with Applictions for Unitry Opertors in Hilert Spces, rticle no. 6, 6 th vol. 3, RGMIA, Res. Rep. Coll., [8] S.S. rgomir, Generlized Trpezoidl Type Ineulities for Comple Functions e ned on Unit Circle with Applictions for Unitry Opertors in Hilert Spces, rticle no. 9, 6th vol. 3, RGMIA, Res. Rep. Coll., [9] A.M.A. El-Syed, M. Ger, On the nite Cputo nd nite Riesz derivtives, Electronic Journl of Theoreticl Physics, Vol. 3, o. (6), [] G.S. Frederico,.F.M. Torres, Frctionl Optiml Control in the sense of Cputo nd the frctionl oether s theorem, Interntionl Mthemticl Forum, Vol. 3, o. (8), [] R. Goren o, F. Minrdi, Essentils of Frctionl Clculus,, Mphysto Center, Minrdiotes/fmk.ps. [] A. Ostrowski, Üer die Asolutweichung einer di erentiren Funcktion von ihrem Integrlmittelwert, Comment. Mth. Helv., (938), 6-7. [3] S.G. Smko, A.A. Kils, O.I. Mrichev, Frctionl Integrls nd erivtives, Theory nd Applictions, (Gordon nd Brech, Amsterdm, 993) [English trnsltion from the Russin, Integrls nd erivtives of Frctionl Order nd Some of Their Applictions (uk i Tekhnik, Minsk, 987)]. 8 7

28 Frctionl Poly type integrl ineulity George A. Anstssiou eprtment of Mthemticl Sciences University of Memphis Memphis, T 385, U.S.A. gnstss@memphis.edu Astrct Here we estlish frctionl Poly type integrl ineulity with the help of generlised right nd left frctionl derivtives. The mzing fct here is tht we do not need ny oundry conditions s the clssicl Poly integrl ineulity reuires. AMS Suject Clssi ction : 6A33, 6, 65. Keywords nd Phrses: Poly integrl ineulity, frctionl derivtive. Introduction We mention the following fmous Poly s integrl ineulity, see [7], [8, p, 6], [9] nd [, p. 83]. Theorem Let f () e di erentile nd not identiclly constnt on [; ] with f () f (). Then the eists t lest one point [; ] such tht jf ()j > 4 ( ) f () d: () In [], Feng Qi presents the following very interesting Poly type integrl ineulity (), which generlizes (). Theorem Let f () e di erentile nd not identiclly constnt on [; ] with f () f () nd M jf ()j. Then sup [;] f () d ( ) M; () 4 where ( ) 4 in () is the est constnt. 8

29 In this short note we present frctionl Poly type integrl ineulity, similr to (), without the oundry conditions f () f (). For the lst we need the following frctionl clculus ckground. Let >, m [] ; m; < < ; f C ([; ]), [; ] R, [; ]. The gmm function is given y () R e t t dt. We de ne the left Riemnn-Liouville integrl J + f () () ( t) f (t) dt; (3). We de ne the suspce C+ ([:]) of C m ([; ]): n o C+ ([; ]) f C m ([; ]) : J + f (m) C ([; ]) : (4) For f C+ ([; ]), we de ne the left generlized -frctionl derivtive of f over [; ] s +f : J + f (m) ; (5) see [], p. 4. Cnvti rst in [3] introduced the ove over [; ]. otice tht +f C ([; ]) : We need the following left frctionl Tylor s formul, see [], pp. 8-, nd in [3] the sme over [; ] tht ppered rst. Theorem 3 Let f C + ([; ]). (i) If ; then f () f ()+f () ( )+f () + () (ii) If < < ; we hve f () We will use (7). otice tht () ( ) +:::+f (m ) () ( t) +f (t) dt; ll [; ] : ( t) +f (t) dt ( )m (m ) (6) ( t) +f (t) dt; ll [; ] : (7) +f ( (t) d t) +f ( ) ( ), where [; ] ; (8) y rst integrl men vlue theorem. Hence, when < <, we get f () +f ( ) ( ) ; ll [; ] : (9) ( + ) 9

30 Furthermore we need: Let gin >, m [], m, f C ([; ]), cll the right Riemnn- Liouville frctionl integrl opertor y J f () : () (t ) f (t) dt; () [; ], see lso [], [4], [5], [6], []. e ne the suspce of functions C ([; ]) : n f C m ([; ]) : J o f (m) C ([; ]) : () e ne the right generlized -frctionl derivtive of f over [; ] s f ( ) m J f (m), () see []. We set f f. otice tht f C ([; ]) : From [], we need the following right Tylor frctionl formul. Theorem 4 Let f C (i) If, we get ([; ]), >, m : []. Then f () mx k f (k) () k (ii) If < <, we get f () J f () () ( ) k + J f () ; ll [; ] : (3) (t ) f (t) dt; ll [; ] : (4) We will use (4). otice tht (t ) f (t) dt f (t (t) d ) f ( ) ( ), where [; ] ; (5) y rst integrl men vlue theorem. Hence, when < <, we otin f () f ( ) ( ) ; ll [; ] : (6) ( + ) 3 3

31 Min Result We present the following frctionl Poly type integrl ineulity without ny oundry conditions. Theorem 5 Let < <, f C ([; ]). Assume f C + f C + ;. Set Then ; + nd n M (f) m +f ;[; + ] ; f o ;[ + ;] : (7) f () d jf ()j d M (f) ( )+ ( + ) : (8) Ineulity (8) is shrp, nmely it is ttined y ( ) ; ; + f () ( ), + ; ; < < : (9) Clerly here non zero constnt functions f re ecluded. Proof. By (9) we get jf ()j + f ;[; + ] ( ) ; for ny ; + : () ( + ) By (6) we derive jf ()j f ( ) + ;[ + ;] ; for ny ( + ) ; : () Hence we get jf ()j d + (y (), ()) +f + ;[; + ] ( ) d + ( + ) + f ;[; + + ] + ( ( + )) ( + ) jf ()j d + jf ()j d + f ;[ + ;] ( ) d ( + ) + f ;[ + ;] ( ( + )) ( + ) + f ;[; + ] + f ;[ + ;] ( + ) + () + : (3) 4 3

32 So we hve proved tht jf ()j d m n +f ;[; + ] ; f ;[ + ;] o ( ) + ( + ) ; (4) proving (8). otice tht f + f + + ; so tht f C ([; ]) : Here m. We see tht J + ( ) () J + ( ) () (y [3], p. 56) ( ) ( ) ( t) (t ) dt ( t) ( ) (t ) (+) dt ( ) ( ) ( + ) () ( ) ( + ) ( ) : Hence Therefore Furthermore we hve + ( ) ( + ) ; for ll + ( ; + : (5) ) ;[; + ] ( + ) : (6) J ( ) () ( ) (t ) ( t) dt (y [3], p. 56) ( ) ( t) (+) (t ) ( ) dt ( ) ( + ) ( ) () ( ) ( + ) ( ) : Therefore ( ) ( + ) ; for ll + ; ; (7) 5 3

33 nd Conseuently we nd tht Applying f into (8) we otin: ( ) ;[ + ;] ( + ) : (8) R.H.S.(8) for f ( + ) while we get the sme result from L.H.S. (8) for f + proving shrpness of (8). We mke M (f ) ( + ) : (9) ( )+ ( )+ ( + ) ( + ) ; (3) f () d ( ) d + ( + ) ( )+ d ( + ) ; (3) Remrk 6 When, thus m [], nd y ssuming tht f (k) () f (k) (), k ; ; :::; m, we cn prove the sme sttements s in Theorem 5. If we set there we derive ectly Theorem. So we generlize Theorem. Agin here f (m) cnnot e constnt di erent thn zero, euivlently, f cnnot e non-trivil polynomil of degree m. References [] G.A. Anstssiou, Frctionl i erentition Ineulities, Reserch Monogrph, Springer, ew York, 9. [] G.A. Anstssiou, On Right Frctionl Clculus, Chos, Solitons nd Frctls, 4 (9), [3] J.A. Cnvti, The Riemnn-Liouville Integrl, ieuw Archief Voor Wiskunde, 5 () (987), [4] A.M.A. El-Syed, M. Ger, On the nite Cputo nd nite Riesz derivtives, Electronic Journl of Theoreticl Physics, Vol. 3, o. (6), [5] G.S. Frederico,.F.M. Torres, Frctionl Optiml Control in the sense of Cputo nd the frctionl oether s theorem, Interntionl Mthemticl Forum, Vol. 3, o. (8),

34 [6] R. Goren o, F. Minrdi, Essentils of Frctionl Clculus,, Mphysto Center, Minrdiotes/fmk.ps. [7] G. Poly, Ein mittelwertstz für Funktionen mehrerer Veränderlichen, Tohoku Mth. J. 9 (9), -3. [8] G. Poly nd G. Szegö, Aufgen und Lehrsätze us der Anlysis, Volume I, Springer-Verlg, Berlin, 95. (Germn) [9] G. Poly nd G. Szegö, Prolems nd Theorems in Anlysis, Volume I, Clssics in Mthemtics, Springer-Verlg, Berlin, 97. [] G. Poly nd G Szegö, Prolems nd Theorems in Anlysis, Volume I, Chinese Edition, 984. [] Feng Qi, Poly type integrl ineulities: origin, vrints, proofs, re nements, generliztions, euivlences, nd pplictions, rticle no., 6th vol. 3, RGMIA, Res. Rep. Coll., [] S.G. Smko, A.A. Kils, O.I. Mrichev, Frctionl Integrls nd erivtives, Theory nd Applictions, (Gordon nd Brech, Amsterdm, 993) [English trnsltion from the Russin, Integrls nd erivtives of Frctionl Order nd Some of Their Applictions (uk i Tekhnik, Minsk, 987)]. [3] E.T. Whittker nd G.. Wtson, A Course in Modern Anlysis, Cmridge University Press,

35 Univrite Frctionl Poly type integrl ineulities George A. Anstssiou eprtment of Mthemticl Sciences University of Memphis Memphis, T 385, U.S.A. gnstss@memphis.edu Astrct Here we estlish series of vrious frctionl Poly type integrl ineulities with the help of generlised right nd left frctionl derivtives. We give ppliction to comple vlued functions de ned on the unit circle. AMS Suject Clssi ction : 6A33, 6, 65. Keywords nd Phrses: Poly integrl ineulity, frctionl derivtive. Introduction We mention the following fmous Poly s integrl ineulity, see [], [3, p, 6], [4] nd [5, p. 83]. Theorem Let f () e di erentile nd not identiclly constnt on [; ] with f () f (). Then the eists t lest one point [; ] such tht jf ()j > 4 ( ) f () d: () In [6], Feng Qi presents the following very interesting Poly type integrl ineulity (), which generlizes (). Theorem Let f () e di erentile nd not identiclly constnt on [; ] with f () f () nd M jf ()j. Then sup [;] f () d ( ) M; () 4 where ( ) 4 in () is the est constnt. 35

36 The ove motivte the current pper. In this rticle we present univrite frctionl Poly type integrl ineulities in vrious cses, similr to (). For the lst we need the following frctionl clculus ckground. Let >, m [] ; m; < < ; f C ([; ]), [; ] R, [; ]. The gmm function is given y () R e t t dt. We de ne the left Riemnn-Liouville integrl J + f () () ( t) f (t) dt; (3). We de ne the suspce C+ ([:]) of C m ([; ]): n o C+ ([; ]) f C m ([; ]) : J + f (m) C ([; ]) : (4) For f C+ ([; ]), we de ne the left generlized -frctionl derivtive of f over [; ] s +f : J + f (m) ; (5) see [], p. 4. Cnvti rst in [5], introduced the ove over [; ]. otice tht +f C ([; ]) : We need the following left frctionl Tylor s formul, see [], pp. 8-, nd in [5] the sme over [; ] tht ppered rst. Theorem 3 Let f C + ([; ]). (i) If ; then f () f ()+f () ( )+f () + () (ii) If < < ; we hve f () () ( ) +:::+f (m ) () ( t) +f (t) dt; ll [; ] : ( )m (m ) (6) ( t) +f (t) dt; ll [; ] : (7) Furthermore we need: Let gin >, m [], m, f C ([; ]), cll the right Riemnn- Liouville frctionl integrl opertor y J f () : () (t ) f (t) dt; (8) [; ], see lso [], [8], [9], [], [7]. e ne the suspce of functions C ([; ]) : n f C m ([; ]) : J o f (m) C ([; ]) : (9) 36

37 e ne the right generlized -frctionl derivtive of f over [; ] s f ( ) m J f (m), () see []. We set f f. otice tht f C ([; ]) : From [], we need the following right Tylor frctionl formul. Theorem 4 Let f C (i) If, we get f () mx k f (k) () k (ii) If < <, we get f () J f () () We need from [3] ([; ]), >, m : []. Then ( ) k + J f () ; ll [; ] : () (t ) f (t) dt; ll [; ] : () e nition 5 Let f C ([; ]), [; ], >, m : []. Assume tht f C + ; nd f C+ ; +. We de ne the lnced Cnvti type frctionl derivtive y f () : f (), for + ; +f () ; for < + : (3) In [4] we proved the following frctionl Poly type integrl ineulity without ny oundry conditions. ; + nd Theorem 6 Let < <, f C ([; ]). Assume f C + f C + ;. Set n M (f) m +f ;[; + ] ; f o ;[ + ;] : (4) Then f () d jf ()j d (5) + f ;[; + ] + f ;[ + ;] + M (f) ( )+ ( + ) ( + ) : (6) Ineulities (5), (6) re shrp, nmely they re ttined y ( ) ; ; + f () ( ), + ; ; < < : (7) Clerly here non zero constnt functions f re ecluded. 3 37

38 The lst result lso motivtes this work. Remrk 7 (see [4]) When, thus m [], nd y ssuming tht f (k) () f (k) (), k ; ; :::; m, we cn prove the sme sttements (5), (6), (7) s in Theorem 6. If we set there we derive ectly Theorem. So we hve generlized Theorem. Agin here f (m) cnnot e constnt di erent thn zero, euivlently, f cnnot e non-trivil polynomil of degree m. We continue here with other interesting univrite frctionl Poly type ineulities. Min Results We present our rst min result. Theorem 8 Let, m [], f C ([; ]). Assume f C+ ; + nd f C + ;, such tht f (k) () f (k) (), k ; ; :::; m. Set n M (f) m + f L([; + ]) ; f o L([ + ;]) : (8) Then f () d k fk ( + ) L([;]) Here f cnnot e non-trivil polynomil of degree m. Proof. By ssumption nd Theorem 3 we hve f () () ( t) +f (t) dt; ll lso it holds, y ssumtion nd Theorem 4, tht f () By () we get () jf ()j ( ) () (t ) f (t) dt; ll () + ( t) +f (t) dt + f (t) dt; 4 M (f) ( + ) ( ) : () ; + ; () + ; : () ll ; + : (3) 38

39 By () we derive jf ()j ( ) () Conseuently we hve + () (t ) f (t) dt f (t) dt; ll + ; : (4) + jf ()j d () + ( ) d +f L([; + ]) (5) nd ( + ) +f L([; + ]) ; + jf ()j d () + ( ) d f L([ + ;]) (6) ( + ) f L([ + ;]) : Therefore we otin y dding (5) nd (6) tht h jf ()j d ( + ) +f L([; + ]) + f i L([ + ;]) k fk ( + ) L([;]) n m + f L([; + ]) ; f o ( ) L([ + ;]) ( + ) ; (8) proving the clim. We continue with Theorem 9 Let p; > : p +, >, m [], f C ([; ]). Assume f C+ ; + nd f C + ;, such tht f (k) () f (k) (), k ; ; :::; m. When < <, the lst oundry conditions re void. Set n M 3 (f) m + f L([; + ]) ; f o L([ + ;]) : (9) Then jf ()j d () (p ( ) + ) p + p + p (3) 5 39

40 h + f L([; + ]) + f i L([ + ;]) () (p ( M 3 (f) ) + ) p + p Agin here f cnnot e non-trivil polynomil of degree m. Proof. By Theorem 3 we hve ( ) + p : (3) jf ()j () ( t) +f (t) dt Tht is jf ()j () (p( )+) p ( ) () (p ( ) + ) p ( ) + p () (p ( ) + ) p Similrly from Theorem 4 we get ( t) p( ) p dt +f (t) dt + f L([; + ]) ; for ll ; + : (3) + f L([; + ]) ; for ll ; + (33) jf ()j () (t ) f (t) dt Tht is jf ()j () ( ) ( )+ p () (p ( ) + ) p (t ) p( ) dt ( ) + p () (p ( ) + ) p p f (t) dt (34) f + L([ + ;]) ; for ll ; : (35) Conseuently we otin y (33) tht + f + L([ + ;]) ; for ll ; : jf ()j d (36) () (p ( ) + ) p + ( ) + p d +f L([; + ]) 6 4

41 () (p ( ) + ) p + p + p +f L([; + ]) : (37) Similrly it holds y (36) tht + jf ()j d () (p ( () (p ( ) + ) p ) + ) p + + p ( ) + p d + p f L([ + ;]) f L([ + ;]) : (38) Adding (37) nd (38) we hve jf ()j d () (p ( ) + ) p + p h + f L([; + ]) + f i L([ + ;]) ]) ; f o L([ + ;]) m n + f L([; + () (p ( ) + ) p + p + p (39) ( ) + p ; (4) proving the clim. Comining Theorem 6, Remrk 7, Theorem 8 nd Theorem 9, we otin Theorem Let ny p; > : p +,, m [], f C ([; ]). Assume f C+ ; + nd f C + ;, such tht f (k) () f (k) (), k ; ; :::; m. Then jf ()j d 8 >< + f ;[; + ] + f ;[ + min ;] + ; >: ( + ) k fk ( + ) ; (4) L([;]) h +f L([; + ]) + i 9 f L([ + ;]) + p > () (p ( ) + ) p + >; p 7 4

42 ( min M (f) ( )+ ( + ) ; M (f) ( + ) ( ) ; () (p ( M 3 (f) ) + ) p + p 9 ( ) + p ; ; (4) where M (f) s in (4), M (f) s in (8) nd M 3 (f) s in (9). Here f cnnot e non-trivil polynomil of degree m. Corollry Here ll s in Theorem. Then f () d jf ()j d 8 >< + f ;[; + ] + f ;[ + min ;] >: ( + ) + ( ) ; (43) ( + ) ( ) k fk ; L([;]) h + f L([; + ]) + f i 9 L([ + ;]) > ( ) + p () (p ( ) + ) p + p + p >; ( ) min M (f) ( + ) ; M (f) ( + ) ( ) ; 9 M 3 (f) ( ) + p () (p ( ) + ) p + p ; : (44) In 938, A. Ostrowski [] proved the following importnt ineulity: Theorem Let f : [; ] R e continuous on [; ] nd di erentile on (; ) whose derivtive f : (; ) R is ounded on (; ), i.e., kf k : jf (t)j < +. Then sup t(;) f (t) dt f () " ( ) # ( ) kf k ; (45) for ny [; ]. The constnt 4 is the est possile. 8 4

43 In (45) for + we get f (t) dt + f 4 kf k : (46) We hve proved the following Theorem 3 Let f C ([; ]), with f f (t) dt ( ) 4 +. Then kf k ; (47) where the constnt 4 is the est possile. So we reproved () with only one initil condition. 3 Appliction Ineulities for comple vlued functions de ned on the unit circle were studied etensively y S. rgomir, see [6], [7]. We give here our version for these functions involved in Poly type ineulity, y pplying result of this rticle. Let t [; ] [; ), the unit circle rc A z C : z e it ; t [; ], nd f : A C e continuous function. Clerly here there eist functions u; v : A R continuous, the rel nd the comple prt of f, respectively, such tht f e it u e it + iv e it : (48) So tht f is continuous, i u; v re continuous. Cll g (t) f e it, l (t) u e it, l (t) v e it, t [; ]; so tht g : [; ] C nd l ; l : [; ] R re continuous functions in t. If g hs derivtive with respect to t, then l, l hve lso derivtives with respect to t. In tht cse (i.e. g (t) l (t) + il (t)), which mens f t e it u t e it + iv t e it ; (49) f t (cos t + i sin t) u t (cos t + i sin t) + iv t (cos t + i sin t) : (5) Let us cll cos t, y sin t. Then u t e it u t (cos t + i sin t) u t ( + iy) u t eit ( sin t) + cos 9 43

44 Similrly we nd tht v t e ( sin t) cos t: (5) Since g is continuous on [; ], then R f eit dt eists. Furthermore it holds We hve here tht f e it dt f e it dt u e it dt + i u e it dt + v e it dt jl (t)j dt + We give the following ppliction of Theorem. v e it dt: (53) f e it dt (54) jl (t)j dt: (55) Theorem 4 Let f C (A; C), [; ] [; ); ny p; > : p +,, m []. Assume l ; l C+ ; + nd l ; l C + ;, such tht l (k) () l (k) () l (k) () l (k) (), k ; ; :::; m. Then f e it dt f e it dt 8 >< + l ;[; + ] + l ;[ + min ;] + ; (56) >: ( + ) k l k ( + ) ; L([;]) h +l L([; + ]) + l i 9 L([ + ;]) + p > () (p ( ) + ) p + >; + p 8 >< + l ;[; + ] + l ;[ + min ;] + ; >: ( + ) k l k ( + ) ; L([;]) h +l L([; + ]) + l i L([ + ;]) () (p ( ) + ) p + p + p 9 > >; 44

45 min ( M (l ) ( )+ ( + ) ; M (l ) ( + ) ( ) ; 9 M 3 (l ) ( ) + p () (p ( ) + ) p + p ; + (57) ( min M (l ) ( )+ ( + ) ; M (l ) ( + ) ( ) ; () (p ( M 3 (l ) ) + ) p + p 9 ( ) + p ; ; where M (l i ) s in (4), M (l i ) s in (8) nd M 3 (l i ) s in (9), i ; : Here l ; l cnnot e non-trivil polynomils of degree m. References [] G.A. Anstssiou, Frctionl i erentition Ineulities, Reserch Monogrph, Springer, ew York, 9. [] G.A. Anstssiou, On Right Frctionl Clculus, Chos, Solitons nd Frctls, 4 (9), [3] G.A. Anstssiou, Blnced Cnvti type frctionl Opil ineulities, to pper, J. of Applied Functionl Anlysis, 4. [4] G.A. Anstssiou, Frctionl Poly type integrl ineulity, sumitted, 3. [5] J.A. Cnvti, The Riemnn-Liouville Integrl, ieuw Archief Voor Wiskunde, 5 () (987), [6] S.S. rgomir, Ostrowski s Type Ineulities for Comple Functions e ned on unit Circle with Applictions for Unitry Opertors in Hilert Spces, rticle no. 6, 6 th vol. 3, RGMIA, Res. Rep. Coll., [7] S.S. rgomir, Generlized Trpezoidl Type Ineulities for Comple Functions e ned on Unit Circle with Applictions for Unitry Opertors in Hilert Spces, rticle no. 9, 6th vol. 3, RGMIA, Res. Rep. Coll., [8] A.M.A. El-Syed, M. Ger, On the nite Cputo nd nite Riesz derivtives, Electronic Journl of Theoreticl Physics, Vol. 3, o. (6),

46 [9] G.S. Frederico,.F.M. Torres, Frctionl Optiml Control in the sense of Cputo nd the frctionl oether s theorem, Interntionl Mthemticl Forum, Vol. 3, o. (8), [] R. Goren o, F. Minrdi, Essentils of Frctionl Clculus,, Mphysto Center, Minrdiotes/fmk.ps. [] A. Ostrowski, Üer die Asolutweichung einer di erentiren Funcktion von ihrem Integrlmittelwert, Comment. Mth. Helv., (938), 6-7. [] G. Poly, Ein mittelwertstz für Funktionen mehrerer Veränderlichen, Tohoku Mth. J. 9 (9), -3. [3] G. Poly nd G. Szegö, Aufgen und Lehrsätze us der Anlysis, Volume I, Springer-Verlg, Berlin, 95. (Germn) [4] G. Poly nd G. Szegö, Prolems nd Theorems in Anlysis, Volume I, Clssics in Mthemtics, Springer-Verlg, Berlin, 97. [5] G. Poly nd G Szegö, Prolems nd Theorems in Anlysis, Volume I, Chinese Edition, 984. [6] Feng Qi, Poly type integrl ineulities: origin, vrints, proofs, re nements, generliztions, euivlences, nd pplictions, rticle no., 6th vol. 3, RGMIA, Res. Rep. Coll., [7] S.G. Smko, A.A. Kils, O.I. Mrichev, Frctionl Integrls nd erivtives, Theory nd Applictions, (Gordon nd Brech, Amsterdm, 993) [English trnsltion from the Russin, Integrls nd erivtives of Frctionl Order nd Some of Their Applictions (uk i Tekhnik, Minsk, 987)]. 46

47 Multivrite Generlised Frctionl Poly type integrl ineulities George A. Anstssiou eprtment of Mthemticl Sciences University of Memphis Memphis, T 385, U.S.A. gnstss@memphis.edu Astrct Here we present set of multivrite generlised frctionl Poly type integrl ineulities on the ll nd shell. We tret oth the rdil nd non-rdil cses in ll possiilities. We give lso estimtes for the relted verges. AMS Suject Clssi ction : 6A33, 6, 65. Keywords nd Phrses: multivrite Poly integrl ineulity, rdil generlised frctionl derivtive, ll, shell. Introduction We mention the following fmous Poly s integrl ineulity, see [9], [, p, 6], [] nd [, p. 83]. Theorem Let f () e di erentile nd not identiclly constnt on [; ] with f () f (). Then the eists t lest one point [; ] such tht jf ()j > 4 ( ) f () d: () In [3], Feng Qi presents the following very interesting Poly type integrl ineulity (), which generlizes (). Theorem Let f () e di erentile nd not identiclly constnt on [; ] with f () f () nd M jf ()j. Then sup [;] f () d ( ) M; () 4 47

48 where ( ) 4 in () is the est constnt. The ove motivte the current pper. In this rticle we present multivrite frctionl Poly type integrl ineulities in vrious cses, similr to (). For the lst we need the following frctionl clculus ckground. Let >, m [] ([] is the integrl prt), m; < < ; f C ([; ]), [; ] R, [; ]. The gmm function is given y () R e t t dt. We de ne the left Riemnn-Liouville integrl J + f () (). We de ne the suspce C + ([:]) of C m ([; ]): C + ([; ]) ( t) f (t) dt; (3) n o f C m ([; ]) : J + f (m) C ([; ]) : (4) For f C+ ([; ]), we de ne the left generlized -frctionl derivtive of f over [; ] s +f : J + f (m) ; (5) see [], p. 4. Cnvti rst in [5], introduced the ove over [; ]. otice tht +f C ([; ]) : We need the following left frctionl Tylor s formul, see [], pp. 8-, nd in [5] the sme over [; ] tht ppered rst. Theorem 3 Let f C + ([; ]). (i) If ; then f () f ()+f () ( )+f () ( ) +:::+f (m ) () ( )m (m ) (6) + () (ii) If < < ; we hve f () () ( t) +f (t) dt; ll [; ] : ( t) +f (t) dt; ll [; ] : (7) Furthermore we need: Let gin >, m [], m, f C ([; ]), cll the right Riemnn- Liouville frctionl integrl opertor y J f () : () (t ) f (t) dt; (8) 48

49 [; ], see lso [], [6], [7], [8], [5]. e ne the suspce of functions C ([; ]) : n f C m ([; ]) : J o f (m) C ([; ]) : (9) e ne the right generlized -frctionl derivtive of f over [; ] s f ( ) m J f (m), () see []. We set f f. otice tht f C ([; ]) : From [], we need the following right Tylor frctionl formul. Theorem 4 Let f C (i) If, we get ([; ]), >, m : []. Then f () mx k f (k) () k (ii) If < <, we get f () J f () () We need from [3] ( ) k + J f () ; ll [; ] : () (t ) f (t) dt; ll [; ] : () e nition 5 Let f C ([; ]), [; ], >, m : []. Assume tht f C + ; nd f C+ ; +. We de ne the lnced Cnvti type frctionl derivtive y f () : f (), for + ; +f () ; for < + : (3) In [4] we proved the following frctionl Poly type integrl ineulity without ny oundry conditions. ; + nd Theorem 6 Let < <, f C ([; ]). Assume f C + f C + ;. Set n M (f) m + f ;[; + ] ; f o ;[ + ;] : (4) Then f () d jf ()j d (5) + f ;[; + ] + f ;[ + ;] + M (f) ( )+ ( + ) ( + ) : (6) 3 49

50 Ineulities (5), (6) re shrp, nmely they re ttined y ( ) ; ; + f () ( ), + ; ; < < : (7) Clerly here non zero constnt functions f re ecluded. The lst result lso motivtes this work. Remrk 7 (see [4]) When, thus m [], nd y ssuming tht f (k) () f (k) (), k ; ; :::; m, we cn prove the sme sttements (5), (6), (7) s in Theorem 6. If we set there we derive ectly Theorem. So we hve generlized Theorem. Agin here f (m) cnnot e constnt di erent thn zero, euivlently, f cnnot e non-trivil polynomil of degree m. We present Poly type integrl ineulities on the ll nd shell. Min Results We need Remrk 8 We de ne the ll B (; R) f R : jj < Rg R,, R >, nd the sphere S : f R : jj g; where jj is the Eucliden norm. Let d e the element of surfce mesure on S nd let d : S For R fg we cn write uniuely r, where r jj > nd r S, jj. ote tht R B(;R) dy R is the Leesgue mesure of the ll. Following [4, pp. 49-5, eercise 6] nd [6, pp , Theorem 5..] we cn write F : B (; R) R Leesgue integrle function tht R F () d F (r) r dr d; (8) B(;R) S we use this formul lot. Initilly the function f : B (; R) R is rdil; tht is, there eists function g such tht f () g (r), where r jj, r [; R], 8 B (; R), >, m []. Here we ssume tht g C ([; R]) with g C+ ; R nd 4 5

51 g CR R ; R, such tht g (k) () g (k) (R), k ; ; :::; m of < < then the lst oundry conditions re void. By ssumption here nd Theorem 3 we hve. In cse g (s) () s (s t) +g (t) dt; (9) ll s ; R ; lso it holds, y ssumption nd Theorem 4, tht g (s) () R s (t s) R g (t) dt; () ll s R ; R : By (9) we get jg (s)j + g ;[; R for ny s ; R : Tht is for ny s ; R : Similrly we otin R ] jg (s)j g ;[ R ;R] () for ny s R ; R : I.e. it holds for ny s R ; R : et we oserve tht R s () () jg (s)j () jg (s)j B(;R) s s R s (s t) + g (t) dt (s t) dt + g ;[; R (t s) dt ( + ) ] + g ;[; R ( + ) (t s) R g (t) dt ] s ; () s ; () R g ;[ R ;R] (R s) ; (3) ( + ) R g ;[ R ;R] (R s) ; (4) ( + ) f (y) dy B(;R) jf (y)j dy (8) 5 5

52 R S R jg (s)j s R ds d jg (s)j s ( R jg (s)j s ds + R " X k ( + ) g ;[ R ;R] ( + ) : k g ;[ R ;R] R R R R R R ds jg (s)j s jg (s)j s ( +g ;[; R ] (R s) s 8 < + g ;[; R ( + ) k R R " X k We hve proved tht R ( + ) : + ] R R ds S d (5) ds ) s + (y () nd (4)) ds+ + R ds + R + R (R s) (+) s 8 < + g ;[; R ( + ) k R ( + ) ( k) k ( + + k) 8 R + < + g ;[; R ] :( + ) ( + ) + g ;[ R ;R] ( ) B(;R) R + + " X k f (y) dy ( ) R g ;[ R ;R] ] ) g ;[ R ;R] (6) k R ds#) + R + (7) #) + k R #) : (8) k ( + + k) B(;R) jf (y)j dy 8 < + g ;[; R ] :( + ) ( + ) + (9) " X k #) : k ( + + k) 6 5

53 Consider now We hve s in [4] tht nd Similrly s in [] we get nd Tht is Conseuently we nd tht s, s ; R g (s) ; (R s) ; s R ; R ; > : +s ( + ), ll s + s ;[; R ] ( + ) : R (R s) ( + ) ; ll s (3) ; R ; (3) R ; R ; (3) R (R s) ;[ R ;R] ( + ) : (33) +g ;[; R ] R g ;[ R ;R] ( + ) : (34) R.H.S. (9) ( ) ( + ) R + + " X k ( + ) + k ( + + k) #) : (35) Let f : B (; R) R e rdil such tht f () g (s), s jj, s [; R], 8 B (; R): Then we hve L.H.S. (9) f (y) dy (8) ( R R g (s) s ( R s + ds + + ( + ) + R R B(;R) ds R R (R s) s ds (R s) s R + + ( + ) + 7 ) (36) R + R ) ds 53

54 X k k X k k R R k ( ) ( + ) R " X k R (R s) (+) s R + + ( + ) + k ( + ) ( k) ( + + k) R + + ( + ) + k ( + + k) proving (9) shrp, infct it is ttined. We hve proved the following min result. #) k R ds) (37) ) + k R (35) R.H.S. (9), (38) Theorem 9 Let f : B (; R) R e rdil; tht is, there eists function g such tht f () g (s), s jj, s [; R], 8 B (; R), >. Assume tht g C ([; R]), with g C+ ; R nd g C R R ; R, such tht g (k) () g (k) (R), k ; ; :::; m, m []. When < < the lst oundry conditions re void. Then f (y) dy jf (y)j dy B(;R) R + + ( ) R g ;[ R ;R] B(;R) 8 < + g ;[; R ] :( + ) ( + ) + (39) " X k #) : k ( + + k) Ineulities (39) re shrp, nmely they re ttined y rdil function f such tht f () g (s), ll s [; R] ; where We continue with s, s ; R g (s) ; (R s) ; s R ; R : (4) 8 54

55 Remrk (Continution of Remrk 8) Here we ssume tht. By (9) we get jg (s)j s () + g L([; R ]) ; (4) ll s ; R : Also, y (), we otin ll s R ; R : Hence s in (5) we get B(;R) (R s) jg (s)j () R g L([ R ;R]) ; (4) jf (y)j dy ( R jg (s)j s ds + R R R R R jg (s)j s jg (s)j s ds ) ds (43) (y (4), (4)) ( R s + ds +g () L([; R ]) + (44) (R s) s ds R g ) L([ R ;R]) (cting the sme s efore, see (6)-(8)) 8 R + < + g L([; R ]) :( + ) () + We hve proved + ( ) R g " X L([ R ;R]) R + + k " X ( ) k gk L([ R ;R]) #) (3) (45) k ( + k) ( k gk L([; R ]) ( + ) () + k #) : (46) k ( + k) 9 55

56 Theorem Here ll terms nd ssumptions s in Theorem 9, however with. Then 8 jf (y)j dy R + < + g L([; R ]) :( + ) () + B(;R) We continue with + ( ) R g " X L([ R ;R]) k #) : (47) k ( + k) Remrk (Also continution of Remrk 8) Let here p; > : p +, with > : By (9) we hve jg (s)j () s (s t) + g (t) dt s (s t) p( ) p s dt () () (p ( ) + ) p s ll s ; R : Similrly y () we otin + p +g (t) dt +g L([; R ]) ; (48) jg (s)j () R s (t s) R g (t) dt () R s (t s) p( ) dt p R s R g (t) dt ll s R ; R : Hence it holds () (R s) + p () (p ( ) + ) p B(;R) ( R jg (s)j s ds + (p ( ) + ) p ( R R g L([ R ;R]) ; (49) jf (y)j dy (5) R R jg (s)j s ds ) (y (48), (49)) s + + p ds + g L([; R ]) + (5) 56

57 () () " X k X 4 () k ( ) R R k (R s) + p s ds R g ) L([ R ;R]) (p ( ) + ) p k R (p ( ) + ) p ( ) k ( k ) () R R 8 < R : + (+ ) (R s) (+ p ) s R g L([ R ;R]) o 8 < : k R (p ( ) + ) p X 4 k k + R (+ ) + g L([; R ]) + k R ds# (+ ) +g L([; R ]) + + p ( k) + p + k R g L([ R ;R]) o 8 < : + R (+ ) 3 + R p + k 5 (5) (+ ) + g L([; R ]) + ( ) + R + p p + k R g L([ R ;R]) ; R + (p ( ) + ) p + + p X 4 k k We hve proved the following (5) 8 < +g L([; R ]) + : p + k R g L([ R ;R]) ; : (53) Theorem 3 Let p; > : p +, >. All other terms nd ssumptions s in Theorem 9. Then jf (y)j dy B(;R) 57

58 ( ) () R + (p ( ) + ) p + + p X 4 k k Comining Theorems 9,, 3 we derive 8 < +g L([; R ]) + : p + k R g L([ R ;R]) ; : (54) Theorem 4 Let ny p; > : p + nd. And let f : B (; R) R e rdil; tht is, there eists function g such tht f () g (s), s jj, s [; R], 8 B (; R). Assume tht g C ([; R]), with g C+ ; R nd g CR R ; R, such tht g (k) () g (k) (R), k ; ; :::; m, m []. When < < the lst oundry conditions re void. Then f (y) dy jf (y)j dy ( ) 8 < min : B(;R) R + + ( ) R g ;[ R ;R] () + B(;R) 8 < + g ;[; R ] :( + ) ( + ) + " X k #) ; k ( + + k) 8 R + < + g L([; R ]) :( + ) () + ( ) R g " X L([ R ;R]) R + (p ( ) + ) p + + p X 4 k k #) ; k ( + k) k 8 < +g L([; R ]) + : p + k R g L([ R ;R]) ;; : (55) ote 5 It holds V ol (B (; R)) R : (56) The corresponding estimte on the verge follows 58