Opial inequality in q-calculus

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1 Mirkovićetl.Jourl of Iequlitie d Applictio 8) 8:347 R E S E A R C H Ope Acce Opil iequlity i q-clculu Ttj Z. Mirković *, Slobod B. Tričković d Miomir S. Stković 3 * Correpodece: tmirkovic75@gmil.com College of Applied Profeiol Studie, Vrje, Serbi Full lit of uthor iformtio i vilble t the ed of the rticle Abtrct I thirticle wegive q-log of the Opil iequlity for q-decreig fuctio. Uig cloed form of the retricted q-itegrl ee Guchm i Comput. Mth. Appl. 47:8 3, 4), we etblih ew itegrl iequlity of the q-opil type. MSC: 8P68; 6D Keyword: q-derivtive; q-itegrl; Opil iequlity Itroductio I 96, Opil ] etblihed the followig importt itegrl iequlity. Theorem. Let f C, h], where f ) = f h)=d f t)>for t, h). The h f x)f x) dx h 4 h The cott h i the bet poible. 4 f x) ) dx. ) Itegrl iequlitie of the form ) hve iteret i itelf, d lo hve importt pplictio i the theory of ordiry differetil equtio d boudry vlue problem ee,, 4]). I the yer therefter, umerou geerliztio, exteio d vritio of the Opil iequlity hve ppered ee, 4]). The oe cotiig frctiol derivtive i ivetigted well ee 3, 5]). I the cotiuou ce, the Opil iequity, i it modified form, tte tht if f x) i bolutely cotiuou fuctio with f ) =,df L, b) where d b re fiite, the f x)f x) dx b ) f x) ) dx, with equlity ttied oly if f x)=cx ). I recet pper 4], Yg proved the followig geerliztio of the Opil iequlity. Theorem. If f x) i bolutely cotiuou o, b] with f ) =, d if p, q, the f x) p f x) q dx q b )p p + q f x) p+q dx. The Author) 8. Thi rticle i ditributed uder the term of the Cretive Commo Attributio 4. Itertiol Licee which permit uretricted ue, ditributio, d reproductio i y medium, provided you give pproprite credit to the origil uthor) d the ource, provide lik to the Cretive Commo licee, d idicte if chge were mde.

2 Mirkovićetl.Jourl of Iequlitie d Applictio 8) 8:347 Pge of 8 Prelimirie Here we preet ecery defiitio d fct from the q-clculu. We followthe termiology d ottio ued i the book 8, 9,, 3]. I wht follow, q i rel umber tifyig < q <,dq-turl umber i defied by ] q = q q = q + + q +,..., N. Defiitio. Let f befuctiodefiedoitervl, b) R,othtqx, b)for ll x, b). For <q <, wedefietheq-derivtive D q f )x)= f x) fqx), x ; D q f ) = lim D q f x). ) x qx x I the pper 7],Jcko defied q-itegrl, which i the q-clculuber hi me. Defiitio. The q-itegrl o, ] i f x) d q x = q) j= q j f q j). O thi bi, i the me pper, Jcko defied itegrl o, b]: f x) d q x = f x) d q x f x) d q x. 3) For poitive iteger d =, uig the left-hd ide itegrl of 3), i the pper 6], Guchmitroducedthe q-retricted itegrl f x) d q x = f x) d q x = b q) q j f q j b ). 4) j= Defiitio.3 The rel fuctio f defied o, b] i clled q-icreig q-decreig) o, b]iff qx) f x)fqx) f x)) for x, qx, b]. It i ey to ee tht if the fuctio f i icreig decreig), the it i q-icreig q-decreig) too. 3 Reult d dicuio Our mi reult re cotied i three theorem. Theorem 3. Let f C, ] be q-decreig fuctio with f bq )=.The, for y p, D q f x) f x) p d q x b ) p D q f x) p+ d q x. 5)

3 Mirkovićetl.Jourl of Iequlitie d Applictio 8) 8:347 Pge 3 of 8 Proof Uig Defiitio. d 4), we hve Dq f x) b f x) p dq x = f x) fqx) x qx f x) p dq x = b q) q j f bq j ) fbq j+ bq j bq j+ f bq j) p j= = f bq j) f bq j+) f bq j) p j= f bq ) p f bq j ) f bq j+). j= I view of f )= j= f bqj+ ) f bq j ) d Hölder iequlity, we obti f bq j+) f bq j) p f bq j) f bq j+) j= = j= f bq j ) f bq j+) j= ) p f bq j ) f bq j+) j= f bq j) f ) p+ bq j+) p f bq j) f bq j+) p+. j= j= By elemetry clcultio, we eily trform the right-hd ide of the lt iequlity ito p b p q) p q j ) p f bq j ) fbq j+ ) p+. bq j bq j+ p j= However, becue of < q <,wehve q j ) p f bq j ) fbq j+ ) p+ bq j bq j+ p j= j= f bq j ) f bq j+ ) p+ bq j bq j+ p, meig tht p b p q) p q j ) p f bq j ) fbq j+ ) p+ p b p q) p bq j bq j+ p j= q b Dq f x) p+ d q x. Sice ] q = q q,wehve p q) p q ) p, d we rrive t the iequlity Dq f x) f x) p dq x b p q ) p q b Dq f x) p+ d q x.

4 Mirkovićetl.Jourl of Iequlitie d Applictio 8) 8:347 Pge 4 of 8 After iterchgig the boudrie i the right-hd ide itegrl, d replcig with,wefid b p q ) p Dq f x) p+ dq x =b ) p Dq f x) p+ d q x, which prove the theorem. Remrk 3. I prticulr, by tkig p =,theiequlity9) i Theorem reduce to the followig Opil iequlity i q-clculu: D q f x) f x) d q x b ) D q f x) d q x. The followigtheoremrecoceredwith q-mootoic fuctio. Theorem 3.3 If f x) d gx) re bolutely cotiuou q-decreig fuctio o, b) d f bq )=d gbq )=,the f x)dq gx)+gqx)d q f x) ] d q x b Proof Replcig )ithe itegrl Dq f x) ) + Dq gx) ) ] dq x. 6) f x)dq gx)+gqx)d q f x) ] d q x, we obti f x) gx) gqx) ] f x) fqx) + gqx) d q x, x qx x qx whece, uig the Guchm q-retricted itegrl, we hve b q) q j f bq j) gbq j ) gbq j+ ) + q j g ) bq j+) f bq j ) fbq j+ ) bq j bq j+ bq j bq j+ j= = f bq j ) g bq j) g bq j+)) + g bq j+) f bq j) f bq j+))]. j= Deotig f bq j )=f bq j+ ) f bq j )d gbq j )=gbq j+ ) gbq j ), we c rewrite the lt um i the form of j= f bqj ) gbq j )+gbq j+ ) f bq j )], d we fid j= f x)dq gx)+gqx)d q f x) ] d q x = f ) g ). Uig the elemetry iequlity b + b ), d coiderig tht f ) = f bq j), g ) = g bq j), j= j=

5 Mirkovićetl.Jourl of Iequlitie d Applictio 8) 8:347 Pge 5 of 8 by virtue of the Schwrz iequlity, we fid f ) g ) = f bq j)) + j= j= g ] bq j)) j= f bq j)) + g ] bq j)) j= f bq j )) + g bq j )) ], j= whece,becue f d g re q-decreig fuctio, we obti the iequlity f x)dq gx)+gqx)d q f x) ] d q x f bq j )) + g bq j )) ] j= = b q) Dq f x) ) + Dq gx) ) ] dq x. However, ice ] q = q q, there follow q) q,owehve f x)dq gx)+gqx)d q f x) ] d q x b q ) Dq f x) +D q gx) ] d q x = b Dq f x) ) + Dq gx) ) ] dq x. Thereby 6)iproved. Theorem 3.4 If f x) d gx) re bolutely cotiuou q-decreig fuctio o, b) d tify f bq )=f )=,gbq )=g )=,the we hve the iequlity f x) gx) t dq x b )+t Dq f x) ) +t dq x + t Dq gx) ) ) +t dq x. 7) Proof For k N, we hve the followig idetitie: f bq k) k = f bq i), f bq k) = f bq i), 8) i= i=k g bq k) = g bq i), g bq k) = g bq i). 9) i= From 8)d9)we oberve tht f bq k ) i= i=k f bq i, g bq k ) g bq i ). ) i=

6 Mirkovićetl.Jourl of Iequlitie d Applictio 8) 8:347 Pge 6 of 8 From ) d uig the elemetry iequlity z +t + tw +t )z w t, where z, w d, t >rerelumber,wefid f bq k ) g bq k ) t )+t f bq i ) +t ) + t i= g bq i ) ) +t ]. ) Uig Hölder iequlity o the right ide of ) with idice, i= +t +t,wehve f bq k ) g bq k ) t )+t +t f bq i ) +t + t g bq i ) ). +t ) Summig the iequlity )from to,we obti i= f bq k ) g bq k ) t )+t f bq k ) +t + t g bq k ) ). +t 3) k= k= k= i= After multiplyig the left-hd ide of 3)byb q)q k, we trform it ito the form of b q)q k f bqk ) gbq k ) t, b q q k k= d fter multiplyig the right-hd ide of 3) byb q)q k ) +t, we trform it ito the form of ) +t b q) q k ) +t f bq k ) fbq k+ ) +t b +t q) +t q k ) +t )+t k= + t q k ) ) +t gbq k ) gbq k+ ) +t. b +t q) +t q k ) +t k= Thu we obti ew form of the iequlity 3). Multiplyig both ide by b q), we hve b q)q k f bq k ) g bq k ) t k= )+t ) +t b q) k= k= ) gbq k ) gbq k+ ) +t + t. b +t q) +t q k ) +t f bq k ) f bq k+ ) +t b +t q) +t q k ) +t

7 Mirkovićetl.Jourl of Iequlitie d Applictio 8) 8:347 Pge 7 of 8 Subtitutig the left-hd ide for the correpodig q-retricteditegrl,dtherighthd ide for the correpodig q-derivtive d q-retricted itegrl, we obti )+t ) b +t b q) Dq f x) ) +t dq x f x) gx) t d q x + t Dq gx) ) ) +t dq x. After iterchgig the boudrie i the right-hd ide itegrl d multiplyig both ide of the lt iequlity by b q), we obti )+t f x) gx) t d q x + t q Sice q) +t +t q ) +t,wefid b )+t b q) ) +t q Dq gx) ) ) +t dq x. b Dq f x) ) +t dq x f x) gx) t dq x b+t q ) +t Dq f x) ) +t dq x + t Dq gx) ) ) +t dq x, d we filly rrive t the iequlity f x) gx) t dq x b )+t Dq f x) ) +t dq x + t Dq gx) ) ] +t dq x, wherebywecompletetheproof. Remrk 3.5 We ote tht, i the pecil ce whe = t = r d f x) =gx) =hx), the iequlity etblihed i 7) reduce to the followig q-wirtiger iequlity: ) hx) r b r dq x Dq hx) ) r dq x. 4 Cocluio I thi pper we hve etblihed ew geerl Opil type itegrl iequlity i q-clculu. Further, we ivetigted the Opil iequlitie i q-clculu ivolvig two fuctio d their firt order derivtive. We lo dicued everl prticulr ce. The method we ued to etblih our reult i quite elemetry d bed o ome imple obervtio d pplictio of ome fudmetl iequlitie. Ackowledgemet The uthor re thkful to the editor d oymou referee for their helpful commet d uggetio. Fudig Thi reerch w doe without y upport.

8 Mirkovićetl.Jourl of Iequlitie d Applictio 8) 8:347 Pge 8 of 8 Competig iteret The uthor declre tht they hve o competig iteret. Author cotributio The uthor cotributed eqully to thi work. All uthor hve red d pproved the mucript. Author detil College of Applied Profeiol Studie, Vrje, Serbi. Deprtmet of Mthemtic, Fculty of Civil Egieerig, Uiverity of Niš, Niš, Serbi. 3 Mthemticl Ititute, Serbi Acdemy of Sciece d Art, Beogrd, Serbi. Publiher Note Spriger Nture remi eutrl with regrd to juridictiol clim i publihed mp d ititutiol ffilitio. Received: 8 Februry 8 Accepted: December 8 Referece. Agrwl, R., Lkhmikthm, V.: Uiquee d Nouiquee Criteri for Ordiry Differetil Equtio. World Sci, Sigpore 993). Agrwl, R., Pg, P.: Opil Iequlitie with Applictio i Differetil d Differece Equtio. Kluwer Acd. Publ., Dordrecht 995) 3. Atiou, G.: Blced Cvti type frctiol Opil iequlitie. J. Appl. Fuct. Al. 9/),3 38 4) 4. Biov, D., Simeoov, P.: Itegrl Iequlitie d Applictio. Kluwer Acd. Publ., Dordrecht 99) 5. Cputo, M., Fbrizio, M.: A ew defiitio of frctiol derivtive without igulr kerel. Prog. Frct. Differ. Appl., ) 6. Guchm, H.: Itegrl iequlitie i q-clculu. Comput. Mth. Appl. 47, 8 3 4) 7. Jcko, M.: O q-defiite itegrl. Qurt. J. Pure d Appl. Mth. 4, ) 8. Kc, V., Cheug, P.: Qutum Clculu. Spriger, New York ) 9. Mriković, S., Rjković, P., Stković, M.: The iequlitie for ome type of q-itegrl. Comput. Mth. Appl. 56, ). Opil, Z.: Sur ue ieglite. A. Pol. Mth. 8, ). Rjković, P., Mriković, S., Stković, M.: Diferecijlo-itegrli rču bzičih hipergeometrijkih fukcij. Mšiki fkultet Niš 8). Shum, D.: O cl of ew iequlitie. Tr. Am. Mth. Soc. 4, ) 3. Triboo, J., Ntouy, S.: Qutum itegrl iequlitie o fiite itervl. J. Iequl. Appl. 4, 4) 4. Yg, G.: O certi reult of Z. Opil. Proc. Jp. Acd. 4, )