ON WEIGHTED INTEGRAL AND DISCRETE OPIAL TYPE INEQUALITIES

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1 M atheatcal I equaltes & A pplcatos Volue 19, Nuber 4 16, do:1.7153/a ON WEIGHTED INTEGRAL AND DISCRETE OPIAL TYPE INEQUALITIES MAJA ANDRIĆ, JOSIP PEČARIĆ AND IVAN PERIĆ Coucated by C. P. Nculescu Abstract. I ths paper soe ultdesoal tegral ad dscrete Opal-type equaltes due to Agarwal, Pag ad Sheg are cosdered. Thers geeralzatos ad extesos usg subultplcatve covex fuctos, approprate tegral represetatos of fuctos, approprate suato represetatos of dscrete fuctos ad equaltes volvg eas are preseted. 1. Itroducto I 196, Z. Opal [1] proved ext tegral equalty: Let xt C 1 [,h] be such that x=xh=adxt > fort,h. The xtx t h dt x t dt, 1 4 where costat h 4 s the best possble. Over the last fve decades, a eorous aout of work has bee doe o ths tegral equalty, dealg wth ew proofs, varous geeralzatos, extesos ad dscrete aalogues. Opal s equalty s recogzed as fudaetal result the aalyss of qualtatve propertes of soluto of dfferetal equatos see [3, 9] ad the refereces cted there. The a of ths paper s to geeralze ad exted soe tegral ad dscrete Opaltype equaltes due to Agarwal, Pag ad Sheg [1,, 6]. To establsh these equaltes, we wll use soe eleetary techques such as approprate tegral represetatos of fuctos, approprate suato represetatos of the dscrete fuctos ad equaltes volvg eas. We start each secto wth equalty volvg a subultplcatve covex fucto. Recall that fucto f : [, [, s called subultplcatve fucto f t satsfes the equalty f xy f x f y, for all x,y [, Matheatcs subject classfcato 1: 6B5, 6D15. Keywords ad phrases: Opal s equalty, Jese s equalty, tegral equalty, dscrete equalty, ultdesoal equalty. c D l,zagreb Paper MIA

2 196 MAJA ANDRIĆ, JOSIP PEČARIĆ AND IVAN PERIĆ see for exaple [8]. Oe of such subultplcatve fuctos, whch s also covex ad creasg, s f x=x p loge + x,where p The obtaed results wll gve a specal case proveets of correspodg equaltes [1,, 6], ad, at the sae te, they wll splfy proofs of the correspodg theores [4, 5, 7]. For the followg equaltes we preset obtaed geeralzatos, extesos ad proveets: frst s a result by Agarwal ad Pag fro [1], observed Secto. Recall, AC[,h] s the space of all absolutely cotuous fuctos o [,h]. Also, let B deotes the beta fucto. THEOREM 1. [1] Let λ 1 be a gve real uber ad let p be a oegatve ad cotuous fucto o [,h]. Further, let x AC[,h] be such that x=xh=. The the followg equalty holds pt xt λ dt 1 λ 1 th t ptdt x t λ dt. For a costat fucto p, the equalty reduces to xt λ dt hλ λ + 1 B, λ + 1 x t λ dt. 3 Next s a ultdesoal Pocaré-type equalty by Agarwal ad Sheg fro [6], observed Secto 3. Ths equalty volves a specal class of cotuous fuctos, a class G, whose defto ad propertes are gve at the begg of Secto 3. THEOREM. [6] Let λ, μ 1 ad let u G. The the followg equalty holds ux λ dx Kλ, μ gradux λ μ dx, where Kλ, μ= λ B, 1 + λ λ C G b a λ, 4 μ { 1, α 1, Cα = 1 α 5, α 1. Fally, a dscrete equalty, observed Secto 4, s a result by Agarwal ad Pag fro []. A defto of a class G for the dscrete case ad a defto of forward dfferece operator Δ are gve at the begg of Secto 4. THEOREM 3. [] Let λ 1 ad let u G. The the followg equalty holds ux λ λ Kλ Δ ux, x=

3 ON WEIGHTED OPIAL-TYPE INEQUALITIES 197 where ad C s defed by 5. Kλ = 1 C λ 1 X 1 1 x =1 x X x λ 1 6. Itegral equaltes oe varable Frst we gve a geeralzato of Theore 1 volvg subultplcatve covex fuctos. I a specal case Corollary 4 ths theore wll prove result fro Theore 1. THEOREM 4. Let N ad let f be creasg, subultplcatve covex fuctos o [,, = 1,...,. Let p be a oegatve ad tegrable fucto o [,h]. Further, let x AC[,h] be such that x =x h = for = 1,...,. The the followg equalty holds pt pt f x t dt t f t + h t 1 dt f h t f x t dt. 7 Proof. As [1], for each fxed, = 1,...,, fro the hypotheses of the theore we have t x t= x sds, x t= x sds. t Sce f s a creasg ad covex fucto, we use Jese s equalty to obta 1 t f x t f t x t s ds 1 t f t x t s ds ad by subultplcatvty of f follows f x t 1 t t f t f x s ds = f t t t f x s ds. 8

4 198 MAJA ANDRIĆ, JOSIP PEČARIĆ AND IVAN PERIĆ Aalogously we obta Multplyg 8 by.e. 1 f x t f h t t f t t f t + 1 h t t 1 h t t = f h t h t t h t x s ds f h t x s ds f h t f x s ds t f x s ds. 9 h t ad 9by f h t ad addg these equaltes, we fd h t f x t f x f h t s ds, t f x t f t + h t 1 f x f h t s ds. 1 Ths gves us [ t f x t f t + h t 1 f x f h t s ] ds. 11 Now ultplyg 11by p ad tegratg o [, h] we obta pt pt f x t dt whch s the equalty 7. [ t f t + h t 1 f x f h t s ] ds dt, REMARK 1. For a specal class of a subultplcatve covex fuctos f o [, wth f = = 1,...,, Theore 4 also holds. Naely, subultplcatvty of a fucto ples ts postvty, ad f f s a covex, oegatve fucto o [, wth f =, the f s obvously a creasg fucto. COROLLARY 1. Let N ad let f be creasg, subultplcatve covex fuctos o [,, = 1,...,. Let p be a oegatve ad tegrable fucto o [,h]. Further, let x AC[,h] be such that x =x h= for = 1,...,. The the followg equalty holds pt 1 pt f x t dt f t f h t t h t 1 dt f x t dt. 1

5 ON WEIGHTED OPIAL-TYPE INEQUALITIES 199 Proof. The equalty 1 follows by the haroc-geoetrc equalty t f t + h t 1 f h t For = 1 we have two followg results. 1 f t f h t. t h t COROLLARY. Let f be a creasg, subultplcatve covex fucto o [, ad let p be a oegatve ad tegrable fucto o [,h]. Further, let x AC[, h] be such that x =xh =. The the followg equalty holds pt f xt dt t pt f t + h t 1 dt f h t f x t dt. 13 COROLLARY 3. Let f be a creasg, subultplcatve covex fucto o [, ad let p be a oegatve ad tegrable fucto o [,h]. Further, let x AC[, h] be such that x =xh =. The the followg equalty holds pt f xt dt 1 1 f t f h t pt dt f x t dt. 14 t h t Next result was prove by Bretć adpečarć [7]. Here t s erely a cosequece, a specal case of Corollary as we ca see fro ts proof. By the harocgeoetrc equalty, t s clear that 15 proves. COROLLARY 4. Let λ 1 be a gve real uber ad let p be a oegatve ad cotuous fucto o [,h]. Further, let x AC[,h] be such that x=xh=. The the followg equalty holds pt xt λ dt t 1 λ +h t 1 λ 1 ptdt x t λ dt. 15 Proof. The equalty 15 wll follow f we use the fucto f t=t λ ad apply Corollary. 3. Multdesoal tegral equaltes Let be a bouded doa R defed by = [a j,b j ]. Let x =x 1,...,x be a geeral pot ad dx = dx 1...dx. For ay cotuous real-valued fucto u defed o we deote uxdx the -fold tegral b 1 b a 1 a ux 1,...,x dx 1...dx. Let D k ux 1,...,x = x ux k 1,...,x ad D k ux 1,...,x =D 1 D k ux 1,...,x,1 k. We deote by G the class of cotuous fuctos u : R for whch D ux exsts wth ux x j =a j = ux x j =b j =, 1 j.

6 13 MAJA ANDRIĆ, JOSIP PEČARIĆ AND IVAN PERIĆ Further, let ux;s j =ux 1,...,x j 1,s j,x j+1,...,x,ad gradux μ = ux x j μ 1 μ Also let α =α 1,...,α ad α λ =α1 λ,...,αλ, λ R. I partcular, b a = b 1 a 1,...,b a ad b a λ =b 1 a 1 λ,...,b a λ. For the geoetrc ad the haroc eas of α 1,...,α we wll use G α ad H α, respectvely. Let M [k] α deote the ea of order k of α 1,...,α. We start wth a weghted exteso of Theore volvg subultplcatve covex fucto. Aga, a specal case Corollary 6 ths theore wll prove result fro Theore. THEOREM 5. Let f be a creasg, subultplcatve covex fucto o [,. Let p bea oegatvead tegrablefuctoo ad u G. The the followg equalty holds where α =α 1,...,α ad α = px f ux dx 1 H α f b a x a f x a +. ux dx, 16 x b x 1 pxdx, = 1,...,. f b x Proof. For each fxed, = 1,...,, wehave x ux= ux;s ds a s ad b ux= ux;s ds. x s Frst we use Jese s equalty sce f s a creasg covex fucto ad the subultplcatvty of f, to obta 1 x f ux f x a x a ux;s a s ds 1 x f x a x a ux;s a s ds 1 x f x a f ux;s x a a s ds = f x a x f ux;s x a s ds 17 a

7 ON WEIGHTED OPIAL-TYPE INEQUALITIES 131 ad aalogously f ux f b x b f ux;s b x x s ds, 18 for = 1,...,. Multplyg 17 by equaltes, we fd x a f x a + b x f ux f b x x a f x a ad 18 by b x f b x b a f ux;s s ds, ad addg these.e. x a f ux f x a + b x 1 b f ux;s f b x a s ds, 19 for = 1,...,. Now ultplyg 19by p adtegratgo we obta b x a px f ux dx a f x a + b x 1 pxdx f b x f ux x dx,.e. b x a a f x a + b x 1 1 pxdx px f ux dx f b x f ux x dx, 1 for = 1,...,. Notce that α 1 = b a x a f x a + b x 1 1 pxdx, = 1,...,. f b x Now, by sug these equaltes 1, we fd α 1 px f ux dx f ux x dx, whch s the sae as the equalty 16. COROLLARY 5. Let f be a creasg, subultplcatve covex fucto o [,. Let p be a oegatve ad tegrable fucto o ad let u G. The the followg equalty holds px f ux dx 1 H β f ux dx, x

8 13 MAJA ANDRIĆ, JOSIP PEČARIĆ AND IVAN PERIĆ where β =β 1,...,β ad β = b a 1 f x a f b x pxdx, = 1,...,. x a b x Proof. By haroc-geoetrc equalty we have x a f x a + b x 1 f b x 1 f x a f b x. x a b x Applyg ths ad usg H 1 γ= 1 H γ, the equalty follows. Next result was prove by Agarwal, BretćadPečarć[4]. Here we use Theore 5 appled o a costat fucto p ad the fucto f t=t λ to prove the equalty 3. By the haroc-geoetrc equalty, t follows that Corollary 6 proves Theore. COROLLARY 6. Let λ, μ 1 ad let u G. The the followg equalty holds ux λ dx K 1 λ, μ gradux λ μ dx, 3 where K 1 λ, μ= 1 λ H Iλ C b a λ, 4 μ ad C s defed by 5. Iλ = 1 t 1 λ +1 t 1 λ 1 dt 5 Proof. We follow steps fro the proof of Theore 5, usg the fucto f t=t λ, up to the equalty, whch s ow equal to b ux λ dx x a 1 λ +b x 1 λ 1 dx a for = 1,...,. However, sce b a x a 1 λ +b x 1 λ 1 dx =b a λ 1 the equalty 6 ca be wrtte as =b a λ Iλ, ux λ dx b a λ Iλ ux x λ dx 6 t 1 λ +1 t 1 λ 1 dt λ ux x dx. 7

9 ON WEIGHTED OPIAL-TYPE INEQUALITIES 133 Multplyg both sdes of the equalty 7 byb a λ, = 1,...,, adthe sug these equaltes, we obta b a λ ux λ dx Iλ ux x λ dx,.e. ux λ dx 1 Iλ H b a λ ux x λ dx. 8 Our result ow follows fro 8 ad the eleetary equalty a α Cα α a, a Multdesoal dscrete equaltes Let x,x N be such that x X,.e., x X, = 1,...,. Let =[,X], where [,X] N. We deote by G the class of fuctos u : R, whch satsfes codtos ux x = = ux x =X =, = 1,...,. Foru we defe forward dfferece operators Δ, = 1,...,, as Δ ux =ux 1,...,x 1,x + 1,x +1,...,x ux. As a prevous secto, let ux;s stad for ux 1,...,x 1,s,x +1,...,x, α = α 1,...,α ad let H α deote the haroc ea of α 1,...,α. Also, let deote X j 1 x j =1. Frst we preset a weghted exteso of Theore 3 volvg subultplcatve covex fuctos. THEOREM 6. Let N ad let f j be subultplcatve covex fuctos o [, wth f j =, j= 1,...,. Let p be a oegatve fucto o ad let u j G for j = 1,...,. The the followg equalty holds px where α =α 1,...,α ad X 1 α = x =1 px f j u j x 1 H α x f j x + x= f j Δ u j x, 3 X x 1, = 1,...,. 31 f j X x

10 134 MAJA ANDRIĆ, JOSIP PEČARIĆ AND IVAN PERIĆ Proof. For each fxed = 1,..., ad each fxed j j = 1,..., we have x 1 u j x= Δ u j x;s, s = X 1 u j x= Δ u j x;s. s =x Fro the dscrete case of Jese s equalty sce f j s a creasg covex fucto ad the subultplcatvty of f j,wehave x 1 1 f j u j x f j x Δ u j x;s x ad aalogously for = 1,...,. We ultply 3 by resultg equaltes, to obta.e. x f j x + s = 1 x 1 x f j x Δ u j x;s s = 1 x 1 x f j x f j Δ u j x;s s = = f x jx 1 x f Δ j u j x;s 3 s = f j u j x f X jx x 1 X x f Δ j u j x;s 33 s =x x f j x ad 33 by X x f j X x. The we add these X X x 1 f j u j x f j X x f Δ j u j x;s, s = x f j u j x f j x + for = 1,...,. Ths gves us f j u j x [ x f j x + for = 1,...,. Now ultplyg 35by p we get px px[ f j u j x x f j x + X x 1 X 1 f j X x f Δ j u j x;s 34 s = X x 1 X 1 f j X x f Δ j u j x;s ] 35 s = X x f j X x 1 ][ X 1 s = f j Δ u j x;s ]

11 ON WEIGHTED OPIAL-TYPE INEQUALITIES 135 for = 1,...,. Sug fro x = 1tox = X 1, we get px X 1 x =1 px f j u j x x f j x + X x f j X x 1 x= f j Δ u j x 36 for = 1,...,. Multplyg both sdes of the equalty 36 byα 1 ad the addg these equaltes, we obta α 1 px f j u j x x= f j Δ u j x, whch s the sae as the equalty 3. COROLLARY 7. Let N ad let f j be subultplcatve covex fuctos o [, wth f j =, j = 1,...,. Let p be a oegatve fucto o ad let u j G for j = 1,...,. The the followg equalty holds px where β =β 1,...,β ad X 1 β = x =1 f j u j x 1 H β px x= f j Δ u j x, 37 1 f j x f j X x, = 1,...,. 38 x X x Proof. By haroc-geoetrc equalty we have x f j x + X x 1 f j X x 1 f j x f j X x. x X x Applyg ths ad usg H 1 β = 1 H β, the equalty 37 follows. Next are specal results for = 1. COROLLARY 8. Let f be a subultplcatve covex fucto o [, wth f =. Let p be a oegatve fucto o ad u G. The the followg equalty holds px f ux 1 H α where α =α 1,...,α s defed by 31. x= f Δ ux, 39

12 136 MAJA ANDRIĆ, JOSIP PEČARIĆ AND IVAN PERIĆ COROLLARY 9. Let f be a subultplcatve covex fucto o [, wth f =. Let p be a oegatve fucto o ad u G. The the followg equalty holds px f ux 1 where β =β 1,...,β s defed by 38. H β x= f Δ ux, 4 A proveetof Theore3 s the followg result, gve also Agarwal, Bretć ad Pečarć [5]. Here we use Corollary 8 appled o a costat fucto p ad the fucto f t=t λ to prove the equalty 41. Thus aga, by the haroc-geoetrc equalty, t follows that Corollary 1 for μ = provestheore3. COROLLARY 1. Let λ, μ 1 ad let u G. The the followg equalty holds where h x,x,λ = ad C s defed by 5. ux λ λ μ K 1 λ, μ Δ ux μ, 41 x= K 1 λ, μ= 1 λ C H hx,x,λ, μ 4 hx,x,λ =h 1 x,x,λ,...,h x,x,λ, 43 X 1 x 1 λ +X x 1 λ 1, = 1,..., x =1 Proof. Fro Corollary 8 usg the fucto f t=t λ ad a costat fucto p we have ux λ 1 H hx,x,λ Δ ux λ. 44 x= The equalty 41 ow follows fro 44 ad the eleetary equalty 9. Ackowledgeet. Ths work has bee fully supported by Croata Scece Foudato uder the project REFERENCES [1] R. P. AGARWAL AND P. Y. H. PANG, Rearks o the geeralzato of Opal s equalty, J. Math. Aal. Appl., , [] R. P. AGARWAL AND P. Y. H. PANG, Sharp dscrete equaltes depedet varables, Appl. Math. Cop., , [3] R. P. AGARWAL AND P. Y. H. PANG, Opal Iequaltes wth Applcatos Dfferetal ad Dfferece Equatos, Kluwer Acadec Publshers, Dordrecht, Bosto, Lodo, [4] R. P. AGARWAL, J. PEČARIĆ AND I. BRNETIĆ, Iproved tegral equaltes depedet varables, Coputers Math. Applc., 33, , 7 38.

13 ON WEIGHTED OPIAL-TYPE INEQUALITIES 137 [5] R. P. AGARWAL, J. PEČARIĆ AND I. BRNETIĆ, Iproved dscrete equaltes depedet varables, Appl. Math. Lett., 11, 1998, [6] R. P. AGARWAL AND Q. SHENG, Sharp tegral equaltes depedet varables, Nolear Aal., , [7] I. BRNETIĆ AND J. PEČARIĆ, Soe ew Opal-type equaltes, Math. Iequal. Appl., 1, , [8] J. GUSTAVSSON, L. MALIGRANDA AND J. PEETRE, A subultplcatve fucto, Nederl. Akad. Wetesch. Idag. Math., 51, , [9] D. S. MITRINOVIĆ, J. PEČARIĆ AND A. M. FINK, Iequaltes Ivolvg Fuctos ad Ther Itegrals ad Dervatves, Kluwer Acadec Publshers, Dordrecht, [1] Z. OPIAL, Sur ue égalté, A. Polo. Math., 8 196, 9 3. Receved March 17, 16 Maja Adrć Faculty of Cvl Egeerg, Archtecture ad Geodesy Uversty of Splt Matce hrvatske 15, 1 Splt, Croata e-al: aja.adrc@gradst.hr Josp Pečarć Faculty of Textle Techology, Uversty of Zagreb Prlaz barua Flpovća 8a, 1 Zagreb, Croata e-al: pecarc@eleet.hr Iva Perć Faculty of Food Techology ad Botechology Uversty of Zagreb Perottjeva 6, 1 Zagreb, Croata e-al: perc@pbf.hr Matheatcal Iequaltes & Applcatos a@ele-ath.co