Fculty of Sciences nd Mthemtics University of Niš Seri Aville t: http://www.pmf.ni.c.rs/filomt Filomt 25:4 20) 53 63 DOI: 0.2298/FIL0453M INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV FUNCTIONAL Mohmmd Msjed-Jmei Astrct In this pper we introduce two specific clsses of functions in L p - spces tht cn generte new nd known inequlities in the literture. By using some recent results relted to the Cheyshev functionl we then otin upper ounds for the solute vlue of the two introduced functions nd consider three prticulr emples. One of these emples is suitle tool for finding upper nd lower ounds of some incomplete specil functions such s incomplete gmm nd et functions. Introduction Let L p [ ] p < ) denote the spce of p-power integrle functions on the intervl [ ] with the stndrd norm /p f p = ft) dt) p nd L [ ] show the spce of ll essentilly ounded functions on [ ] with the norm f = sup [] f). For two solutely continuous functions f g : [ ] R nd the positive function w : [ ] R + such tht wf wg wfg L [ ] the weighted Cheyshev functionl [] is defined y ) ) T w f g) = w)f) g) d w)f) d w)g) d. ) 200 Mthemtics Suject Clssifictions: 26D5; 26D20. Key words nd Phrses: Cheyshev functionl Upper ounds Gruss type inequlities kernel function Incomplete specil functions. Received: Decemer 0 200 Communicted y Drgn S. Djordjević
54 Mohmmd Msjed-Jmei If w) is uniformly distriuted on [ ] then ) is reduced to the usul Cheyshev functionl T f g) = ) ) f) g) d ) 2 f) d g) d. 2) To dte etensive reserch hs een done on the ounds of Cheyshev functionl see e.g. [25]. The first work dtes ck to 882 when Cheyshev [3] proved tht if f g L [ ] then Lter on in 934 Gruss [9] showed tht T f g) 2 )2 f g. 3) T f g) 4 M m )M 2 m 2 ) 4) where m m 2 M nd M 2 re rel numers stisfying the conditions m f) M nd m 2 g) M 2 for ll [ ]. 5) The constnt /4 is the est possile numer in 4) in the sense tht it cnnot e replced y smller quntity. An inequlity relted to usul Cheyshev functionl is due to Ostrowski [6] in 938. If f : [ ] R is differentile function with ounded derivtive then f) ft) dt ) + )/2)2 + 4 ) 2 ) f 6) for ll [ ]. Tody this inequlity plys key role in numericl qudrture rules [783]. A miture type of inequlities 3) nd 4) ws introduced in [7] s T f g) 8 )M m ) g 7) in which f is Leesgue integrle function stisfying 5) nd g is solutely continuous so tht g L [ ]. The constnt /8 is lso the est possile numer in 7). The following theorem due to Niezgod [5] is proly the most recent work out finding pproprite ounds for the usul Cheyshev functionl... Theorem A. Let f α β L p [ ] nd g L q [ ] /p + /q = p ) e functions such tht αt) + βt) is constnt function nd αt) ft) βt) for ll t [ ]. Then we hve T f g) 2 ) β α p g gt) dt. 8) q
Inequlities for Two Specific Clsses of Functions 55 For p = q = 2 8) leds to the well-known inequlity [4] T f g) 2 M m ) T g g) s.t. m f) M. 9) On the other hnd in 997 Drgomir nd Wng [6] introduced n inequlity of Ostrowski-Grss type ccording to the following theorem..2. Theorem B. If f : [ ] R is differentile function with ounded derivtive nd α 0 f t) β 0 for ll t [ ] then f) ft) dt f) f) + ) 2 4 )β 0 α 0 ). 0) There re mny improvements nd refinements of the right hnd side of inequlity 0) in the literture. See e.g. [40]. In this pper we introduce two specific clsses of functions in L p - spces tht cn generte mny new nd known inequlities in the literture nd otin their upper ounds using theorem A. Hence let us first consider the following kernel defined on [ ] ut) t [ ] K; t u v) = ) vt) t ] where ut) nd vt) re two ritrry integrle functions such tht ut) C [ ] nd vt) C ]. Bsed on kernel ) we now define the two following specific functions nd F ; f u v) = u) v)) f) + v)f) u)f) u t)ft) dt v t)ft) dt F 2 ; f u v) = F ; f u v) f) f) ut) dt + vt) dt ) 2). 3) It cn e verified tht the two functions 2) nd 3) re respectively produced y nd f t) K ; t u v) dt = F ; f u v) 4) ) T f t) K ; t u v)) = F 2 ; f u v) 5) where T.. ) is the sme s usul Cheyshev functionl. 2. Min Theorem. Let f : I R where I is n intervl e function differentile in the interior I 0 of I nd let [ ] I 0. Suppose tht f α β
56 Mohmmd Msjed-Jmei L p [ ] re functions such tht αt) + βt) is constnt function nd αt) f t) βt) for ll t [ ]. Then we respectively hve ut) q dt + /q dt) vt) q f p /p + /q = ) F ; f u v) ut) dt + ) vt) dt f p = ; q = ) m K; t u v) f q = ; p = ) t [] in which } m K; t u v) = m m ut) m vt) t [] [] t [] t ] 6) nd F 2 ; f u v) 2 β α p ut) ut) dt + ) vt) dt q dt + vt) ut) dt + vt) dt ) q dt ) /q 7) where ut) nd vt) re two ritrry integrle functions such tht ut) C [ ] nd vt) C ]. Proof. The proof of 6) is strightforwrd if one pplies the well-known Hlder s inequlity [4] fg f p g q /p + /q = ) 8) for identity 4) nd then refers to the generl kernel ). To prove 7) one should refer to identity 5) nd then use Theorem A so tht we hve F 2 ; f u v) 2 β α p K.) K.) dt 9) q nd since K.) ut) + vt) K.) dt q = ut) dt + ) vt) dt q dt ut) dt + ) vt) dt q ) /q dt 20) the proof is complete. One of the strightforwrd cses of theorem 2. is when p = q = 2. In other words pplying the well known Cuchy-Schwrtz inequlity [2] on 4) nd using the min theorem 2. for 5) respectively yield F ; f u v) /2 u 2 t) dt + v 2 t) dt) f 2) 2
Inequlities for Two Specific Clsses of Functions 57 nd F 2 ; f u v) 2 β α 2 ut) ut) dt + ) 2 vt) dt) dt + vt) ut) dt + vt) dt) ) 2 dt ) /2. 22) Clerly vrious suclsses cn e considered for the min theorem 2. We here study three cses. Other cses cn nturlly e studied seprtely. 2.. Suclss. If u) v) is constnt function If u) v) is constnt numer sy c 0 then the kernel ) nd functions 2) nd 3) re respectively reduced to vt) + c t [ ] K; t v + c v) = vt) t ] 23) nd F ; f v + c v) = c f) + v)f) c + v))f) F 2 ; f v + c v) = F ; f v + c v) In fct reltions 24) nd 25) show tht f) f) c ) + v t)ft) dt 24) vt) dt F ; f v + c v) = c f) + A nd F 2 ; f v + c v) = c f) + A + B + D 26) where c A B nd D re rel constnts. Let us consider prticulr emple of the first suclss here. Emple. Suppose tht [ ] = [0 ] c = nd vt) = t 2. Under these ssumptions reltions 23) 24) nd 25) chnge to ). 25) K; t + t 2 t 2 ) = + t 2 t [0 ] t 2 t ] 27) nd F ; f + t 2 t 2) = f) + f) f0) 2 F 2 ; f + t 2 t 2) = f) + f) f0) 2 t ft) dt 0 f) f0)) + ) 0 t2 ft) dt 0 t ft) dt 28) 29)
58 Mohmmd Msjed-Jmei After some clcultions sustituting the ove reltions in ech two prts of the min theorem respectively yields ) f) 2 /q t ft) dt + f) f0) 0 + t2 ) q dt + 2q+ 2q+ f p + /3) f 0 + 2 ) f 30) nd f) 2 0 t ft) dt + f) f0)) 0 t2 ft) dt) 2 β α p 0 t 2 + 2/3 q dt + t 2 /3 dt) q /q 3) where αt) f t) βt) /p + /q = nd t [0 ]. For instnce p = q = 2 in inequlity 3) gives f) 2 0 t ft) dt + f) f0)) 0 t2 ft) dt) 5 30 303 45 2 + 5 + 4 0 βt) αt))2 dt) /2 for ll [0 ]. 2.2. Suclss 2. If u) nd v) re liner functions Suppose tht ut) = p t + q nd vt) = p 2 t + q 2 where p q nd p 2 q 2 re ll rel prmeters. Therefore we hve 32) p t + q K ; t p t + q p 2 t + q 2 ) = t [ ] p 2 t + q 2 t ] 33) nd F ; f p t + q p 2 t + q 2 ) = p p 2 ) + q q 2 ) f) + p 2 + q 2 )f) p + q )f) p ft) dt p 2 ft) dt 34) F 2 ; f p t + q p 2 t + q 2 ) = F ; f p t + q p 2 t + q 2 ) f) f) p p 2 2 2 + q q 2 ) + 2 p 2 2 p 2 ) + q 2 q ). For the ske of simplicity if we rerrnge 34) nd 35) y tking 35) p p 2 = r q q 2 = r 2 p 2 + q 2 = r 3 nd p q = r 4 36) then these ssumptions would chnge reltions 33) 34) nd 35) s follows K ; t r i } 4 ) r +r 2 +r 3 +r 4 i= = t r +r 2 +r 3 +r 4 r +r 2 +r 3 +r 4 t r +r 2 +r 3 +r 4 t [ ] t ] 37) F ; f ri } i=) 4 = r + r 2 ) f) + r 3 f) + r 4 f) r +r 2 +r 3 +r 4 ft) dt r +r 2 +r 3 +r 4 ft) dt 38)
Inequlities for Two Specific Clsses of Functions 59 nd ) F 2 ; f ri } i=) 4 = F ; f ri } 4 i= f) f) 2 ) r 2 + 2r 2 r + )r 2 + )r 3 r 4 ) ). 39) By noting tht there re four free prmeters r r 2 r 3 nd r 4 in the kernel 37) mny suclsses eist for 38) nd 39). The following emple shows one of them. Emple 2. Let r = 0 in 38) nd 39). Then y referring to the min theorem we hve F ; f ri } i=2) 4 = r 2 f) + r 3 f) + r 4 f) r 2+r 3 +r 4 ft) dt r 2 +r 3 +r 4 q t r2+r3+r4 q ) /q r 2+r 3+r 4 dt + t r2+r3+r4 r 2+r 3+r 4 dt f p r 2 +r 3 +r 4 ) t r2+r3+r4 r 2+r 3+r 4 dt + t r2+r3+r4 r 2+r 3+r 4 dt f m K ; t ri } i=2) 4 f t [] 40) nd F 2 ; f ri } i=2) 4 = ) = F ; f ri } 4 i=2 f) f) 2 ) 2r 2 + )r 2 + )r 3 r 4 )) 2 β α p r 2+r 3 +r 4 t r 2 3+) r 2++)r 3 +r 4 ) 2 ) q dt + r 2+r 3 +r 4 t r 2 +3) r2++)r3+r4) 2 ) q /q dt). 4) Note to compute the integrls of reltions 40) nd 4) we cn use the following generl identity t θ q dt = c d in which c < d nd θ R. d θ) q+ +c θ) q+ q+ if c < θ < d d θ) q+ c θ) q+ q+ if θ < c < d d θ) q+ +c θ) q+ q+ if c < d < θ Remrk. For r r 2 r 3 r 4 ) = 0 0 0) inequlity 4) genertes inequlity 0). Remrk 2. For r = 0 nd r 2 + r 3 + r 4 = 0 since the kernel 37) is reduced to 42) r4 t [ ] K ; t r 4 r 3 ) = t ] r 3 43)
60 Mohmmd Msjed-Jmei so inequlities 38) nd 39) re respectively trnsformed to r 4 q ) + r 3 q )) /q f p r 3 f) + r 4 f) r 3 + r 4 ) f) r 4 ) + r 3 )) f m r 4 r 3 } f 44) nd f) β α p 2 r 3 +r 4 r4 + r 3+r 4 r 3 +r 4 ) q ) + r 3 r 3+r 4 r 3 +r 4 ) q /q )). 45) Remrk 3. For r = 0 r 2 + r 3 + r 4 = 0 nd = 0 since the kernel 37) is reduced to t r4 t [0 ] K ; t t r 4 r 3 ) = 46) r 3 t ] f) f) + f) f) so we hve F ; f t r 4 r 3 ) = r3 + r 4 ))f) + r 3 f) + r 4 f0) 0 ft) dt 0 t r 4 q dt + r 3 q ) ) /q f p 0 t r 4 dt + r 3 ) ) } f m r 3 m t r 4 f [0] t [0] 2.3. Suclss 3. If v) is constnt function If v) is constnt numer sy d 0 then the kernel ) nd functions 2) nd 3) respectively tke the forms 47) ut) t [ ] K; t u d) = d t ] 48) nd F ; f u d) = u) d) f) + df) u)f) F 2 ; f u d) = F ; f u d) f) f) u t)ft) dt 49) ) ut) dt + d ). 50) Hence y pplying the min theorem for the two ove functions 49) nd 50) one cn get ut) q dt + d q ) ) /q f p F ; f u d) ut) dt + d )) } f 5) m d m ut) f [] t []
Inequlities for Two Specific Clsses of Functions 6 nd + F 2 ; f u d) 2 β α p ut) ut) dt + d )) q dt 52) d ut) dt + d )) q ) /q. dt One of the dvntges of inequlities 5) nd 52) is to find two upper ounds for the solute vlue of some incomplete specil functions. In other words since mny incomplete specil functions hve n integrl form s gt) dt the two ltter inequlities cn e used for this purpose see lso [] in this regrd. For emple since the incomplete gmm function cn e represented s Γ; α) = 0 t α ep t) dt α > ) 53) so y choosing ut) = t α /α ft) = ep t) nd = 0 in 49) nd employing the first inequlity of 5) we otin ) α α e Γ; α)+d e e ) αq+ q α q αq + ) +! e p ) p d q ) p 54) where 0 < d 0 nd /p + /q = for p [ ). Also since the incomplete et function cn e represented s B ; α β) = 0 t α t) β dt α β > ) 55) so y choosing ut) = t α /α ft) = t) β nd = 0 in 49) nd employing the first inequlity of 5) we otin α α ) β B; α β) + d ) β ) β ) ) β ) αq+ ) α q αq+) + d q q ) ) +β 2)p p +β 2)p where 0 < d 0 nd /p + /q = for p [ ). 56) Acknowledgements. IPM No. 8933002. This reserch ws in prt supported y grnt from References [] F. Ahmd N.S. Brnett S.S. Drgomir New weighted Ostrowski nd Cheyshev type inequlities Nonliner Anlysis: Theory Methods & Applictions 7 2) 2009) 408-42. [2] P. Cerone S.S. Drgomir New ounds for the Cheyshev functionl Appl. Mth. Lett. 8 2005) 603-6.
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Inequlities for Two Specific Clsses of Functions 63 Mohmmd Msjed-Jmei: Deprtment of Mthemtics K. N. Toosi University of Technology P.O. Bo: 635-68 Tehrn Irn School of Mthemtics Institute for Reserch in Fundmentl Sciences IPM) P.O. Bo: 9395-5746 Tehrn Irn E-mil: mmjmei@kntu.c.ir; mmjmei@yhoo.com.