Some inequalities related to the Hölder integral inequality and

applications for bounding the Čebyšev functional are given

Cerone, Pietro

Dragomir, Sever S

English

Some inequalities related to the Hölder integral inequality and\ud
applications for bounding the Čebyšev functional are given

Name not available

Integral Inequalities and Applications for Bounding the Čebyšev Functional

Victoria University Eprints Repository

INTEGRAL INEQUALITIES AND APPLICATIONS FORBOUNDING THE µCEBYEV FUNCTIONALP. CERONE AND S.S. DRAGOMIRAbstract. Some inequalities related to the Hölder integral inequality andapplications for bounding the µCebyev functional are given.1. IntroductionThe integral Hölder inequality, namely(1.1)Z baf (t) g (t) dt  Z bajf (t)jp dt! 1p Z bajg (t)jq dt! 1q;plays an important role in Mathematical Analysis and its applications. Here thecomplex-valued functions f; g : [a; b] ! C are p and q integrable respectively on[a; b] ; where p; q > 1 and 1p +1q = 1:In order to provide sharper bounds for the Hölder inequality, Abramovich, Mondand Peµcari´c considered in [1] the function  : [a; b]! R given by(1.2)  (x) :=Z xaf (t) g (t) dt+ Z bxjf (t)jp dt! 1p Z bxjg (t)jq dt! 1qand proved that  () is nondecreasing on [a; b] : As a consequence we can observethatinfx2[a;b] (x) =Z baf (t) g (t) dtandsupx2[a;b] (x) = Z bajf (t)jp dt! 1p Z bajg (t)jq dt! 1q:Using geometrical arguments, G.S. Mahajani [8] obtained the following resultsfor the absolute value of the integralR xaf (t) dt :1. If f has a bounded derivative on [a; b] ; namely jf 0 (t)j  M (M > 0) and ifR baf (t) dt = 0; then Z xaf (t) dt  18 M  (b  a)2 ;for any x 2 [a; b] :Date : 8th January, 2007.2000 Mathematics Subject Classication. 26D15, 26D10.Key words and phrases. Integral inequalities, Hölders integral inequality, µCebyev functional,Grüss inequality.12 P. CERONE AND S.S. DRAGOMIR2. If, additional to the conditions given above, f (a) = f (b) = 0; thenZ xaf (t) dt  116 M  (b  a)2 :Analytic proofs of these results were given by P.R. Beesack, [9, p. 474]. For otherresults related to the Mahajani inequality see Chapter XV of [9].In this paper some similar results are obtained and applied in obtaining boundsfor the quantitiesR xaf (t) dt ;Z baZ xaf (t) dt  x  ab  aZ baf (t) dtrdx; r 2 [1;1)andsupx2[a;b]Z xaf (t) dt  x  ab  aZ baf (t) dt ;under various assumptions for the function f : [a; b]! R:These results are also utilized to provide bounds for the µCebyev functionalT (f; g) :=1b  aZ baf (x) g (x) dx  1b  aZ baf (x) dx  1b  aZ bag (x) dx;where f; g : [a; b] ! C are Lebesgue integrable functions, in terms of the shiftedintegral means:1b  aZ bag (t)  1b  aZ bag (s) dsrdtwith r 2 [1;1): This is possible due to the following representation result obtainedin [2]:T (f; g) =   1b  aZ ba Z xag (t) dt  x  ab  aZ bag (t) dt!f 0 (x) dx;that holds for g Lebesgue integrable and f absolutely continuous on [a; b] :For recent results on bounding the µCebyev functional T (; ) see [2], [3] and [5]where further references are provided.2. The ResultsThe following result may be stated:Theorem 1. Let f : [a; b]! C be a Lebesgue measurable function and p : [a; b]![0;1) a Lebesgue integrable weight with R bap (t) dt = 1: For any r > 1 and x 2 [a; b] ;we have the inequality:(2.1)Z xap (t) jf (t)jr dt+R ba p (t) f (t) dt  R xa p (t) f (t) dtr1  R xap (t) dtr 1  Z bap (t) jf (t)jr dt:In particular,(2.2)Z xajf (t)jr dt+R ba f (t) dt  R xa f (t) dtr(b  x)r 1 Z bajf (t)jr dt:INTEGRAL INEQUALITIES 3Proof. Obviously,(2.3)Z bap (t) f (t) dt Z xap (t) f (t) dt =Z bxp (t) f (t) dtfor any x 2 [a; b] :Utilising the Hölder inequality, we have for r > 1; 1r +1q = 1 thatZ bxp (t) f (t) dt(2.4) Z bxp (t) dt! 1q Z bxp (t) jf (t)jr dt! 1r= Z bap (t) dt Z xap (t) dt! 1q Z bap (t) jf (t)jr dt Z xap (t) jf (t)jr dt! 1r=1 Z xap (t) dt 1q Z bap (t) jf (t)jr dt Z xap (t) jf (t)jr dt! 1rfor each x 2 [a; b] :Utilising (2.3) and (2.4) and taking the power r; we getZ bap (t) f (t) dt Z xap (t) f (t) dtr1 Z xap (t) dt rq Z bap (t) jf (t)jr dt Z xap (t) jf (t)jr dt!;which gives the desired inequality (2.1). Corollary 1. With the assumptions of Theorem 1 and ifR bap (t) f (t) dt = 0; then(2.5)1 + R xap (t) dt1 r 1  R xap (t) dtr 11  R xap (t) dtr 1  Z xap (t) f (t) dtrZ bap (t) jf (t)jr dtfor any x 2 [a; b] :Proof. SinceR bap (t) f (t) dt = 0; then, by (2.1), we have(2.6)Z xap (t) jf (t)jr dt+R xap (t) f (t) dtr1  R xap (t) dtr 1  Z bap (t) jf (t)jr dtfor any x 2 [a; b] :Utilising Hölders inequality for r > 1; 1r +1q = 1; we haveZ xap (t) f (t) dtr  Z xap (t) dt rqZ xap (t) jf (t)jr dt(2.7)=Z xap (t) dtr 1 Z xap (t) jf (t)jr dt:4 P. CERONE AND S.S. DRAGOMIRCombining (2.6) with (2.7), we get the desired result (2.5). Remark 1. IfR baf (t) dt = 0; then from inequality (2.2) we get the following resultas well:(2.8)1 + (x  a)1 r (b  x)r 1(b  x)r 1 Z xaf (t) dtr  Z bajf (t)jr dt;or, equivalently(2.9)Z xaf (t) dt  (b  x)r 11 + (x  a)1 r (b  x)r 1!1=r Z bajf (t)jr dt!1=r;which is a Mahajani type result.The following result is of interest:Corollary 2. Let g : [a; b] ! C be a Lebesgue integrable function and p : [a; b] ![0;1) an integrable weight with R bap (t) dt = 1: Then(2.10)1 + R xap (t) dt1 r 1  R xap (t) dtr 11  R xap (t) dtr 1Z xap (t) g (t) dt Z xap (t) dt Z bxp (t) g (t) dtrZ bap (t)g (t) Z bap (s) g (s) dsrdt;for any x 2 [a; b] :In particular,(2.11)1 + (x  a)1 r (b  x)r 1(b  x)r 1 Z xag (t) dt  x  ab  aZ bag (t) dtrZ bag (t)  1b  aZ bag (s) dsrdt;for each x 2 [a; b] :The proof is by Corollary 1 applied for f (t) = g (t)  R bap (s) g (s) ds ; t 2 [a; b] :Then inequality (2.11) follows by (2.8) on choosing f (t) = g (t)  1b aR bag (s) ds:A similar result concerning the supremum of the weight can be stated as well:Proposition 1. Let p; f be as in Theorem 1. Then we have the inequality(2.12)Z xajf (t)j dt+R ba p (t) f (t) dt  R xa p (t) f (t) dtsupt2[x;b]p (t)Z bajf (t)j dtfor any x 2 [a; b] :INTEGRAL INEQUALITIES 5Proof. We haveZ bap (t) f (t) dt Z xap (t) f (t) dt =Z bxp (t) f (t) dt supt2[x;b]p (t)Z bxjf (t)j dt= supt2[x;b]p (t)"Z bajf (t)j dt Z xajf (t)j dt#;which easily implies (2.12). Corollary 3. If p and f are as in Corollary 1, then we have(2.13)supt2[a;x]p (t) + supt2[x;b]p (t)supt2[x;b]p (t)Z xap (t) f (t) dt  Z bajf (t)j dt;for each x 2 [a; b] :Proof. Since(2.14)Z xap (t) f (t) dt  supt2[a;x]p (t)Z xajf (t)j dtfor any x 2 [a; b] and (2.12) becomes, under the assumption that R bap (t) f (t) dt = 0;(2.15)Z xajf (t)j dt+R xap (t) f (t) dtsupt2[x;b]p (t)Z bajf (t)j dt;hence by (2.14) and (2.4) we deduce the desired result (2.13). Corollary 4. If p; g are as in Corollary 2, then(2.16)supt2[a;x]p (t) + supt2[x;b]p (t)supt2[x;b]p (t)Z xap (t) g (t) dt Z xap (t) dt Z bxp (t) g (t) dtZ bag (t) Z bap (s) g (s) ds dt;for any x 2 [a; b] :The following result holds as well.Proposition 2. With the above assumptions for f and p we have(2.17)Z xap (t) jf (t)j dt+Z bap (t) f (t) dt Z xap (t) f (t) dt Z bap (t) jf (t)j dtfor any x 2 [a; b] :6 P. CERONE AND S.S. DRAGOMIRProof. We haveZ bap (t) f (t) dt Z xap (t) f (t) dt =Z bxp (t) f (t) dtZ bxp (t) jf (t)j dt=Z bap (t) jf (t)j dt Z xap (t) jf (t)j dt;which is clearly equivalent to (2.17). Corollary 5. If p and f are as in Corollary 1, then(2.18) 2Z xap (t) f (t) dt  Z bap (t) jf (t)j dtfor any x 2 [a; b] :Corollary 6. If p; g are as in Corollary 2, then(2.19)Z xap (t) g (t) dt Z xap (t) dt Z bxp (t) g (t) dt 12Z bap (t)g (t) Z bap (s) g (s) ds dt:Remark 2. If in Corollary 6 we choose the uniform weight p (t) = 1b a ; t 2 [a; b] ;then (2.19) becomes:(2.20)Z xag (t) dt  x  ab  aZ bag (t) dt  12Z bag (t)  1b  aZ bag (s) ds dt;for each x 2 [a; b] :The inequality (2.20) can be seen as the limiting case of (2.11) where r ! 1;r > 1:Remark 3. We observe that (2.20) produces the following Mahajani type inequality,which is, in a sense, the limiting case of (2.9) for r ! 1; r > 1 :(2.21)Z xaf (t) dt  12Z bajf (t)j dt;providedR baf (t) dt = 0:3. Applications for Grüss Type InequalitiesFor two Lebesgue integrable functions f; g : [a; b] ! R consider the µCebyevfunctional:(3.1) T (f; g) :=1b  aZ baf (t) g (t) dt  1b  aZ baf (t) dt  1b  aZ bag (t) dt:In 1934, G. Grüss [6] showed that(3.2) jT (f; g)j  14(M  m) (N   n) ;INTEGRAL INEQUALITIES 7provided m;M;n;N are real numbers with the property(3.3)  1 < m  f M <1;  1 < n  g  N <1 a.e. on [a; b] :The constant 14 is best possible in (3.2) in the sense that it cannot be replacedby a smaller one.Another lesser known inequality for T (f; g) was derived in 1882 by µCebyev [4]under the assumption that f 0; g0 exist and are continuous on [a; b] ; and is given by(3.4) jT (f; g)j  112kf 0k1 kg0k1 (b  a)2 ;where kf 0k1 := supt2[a;b] jf 0 (t)j <1:The constant 112 cannot be improved in general.µCebyevs inequality (3.4) also holds if f; g : [a; b] ! R are assumed to be ab-solutely continuous and f 0; g0 2 L1 [a; b] :In 1970, A.M. Ostrowski [11] proved, amongst others, the following result thatis in a sense a combination of the µCebyev and Grüss results:(3.5) jT (f; g)j  18(b  a) (M  m) kg0k1 ;provided f is Lebesgue integrable on [a; b] and satisfying (3.3) while g : [a; b] ! Ris absolutely continuous and g0 2 L1 [a; b] : Here the constant 18 is also sharp.In 1973, A. Lupas¸ [7] (see also [10, p. 210]) obtained the following result as well:(3.6) jT (f; g)j  12kf 0k2 kg0k2 (b  a) ;provided f; g are absolutely continuous and f 0; g0 2 L2 [a; b] :Here the constant 12 is the best possible as well.In [2], P. Cerone and S.S. Dragomir proved the following inequalities:(3.7) jT (f; g)j 8>>>>><>>>>>:inf2Rkg   k1  1b aR ba f (t) dtinf2Rkg   kq  1b aR ba f (t)p dt 1pwhere p > 1; 1p +1q = 1;wheref (t) := f (t)  1b  aZ baf (s) ds; t 2 [a; b] :For  = 0; we get from the rst inequality(3.8) jT (f; g)j  kgk1 1b  aZ ba f (t) dtfor which the constant 1 cannot be replaced by a smaller constant.If m  g  M for a.e. x 2 [a; b] ; then g   m+M2 1  12 (M  m) and by therst inequality in (3.7) we can deduce the following result obtained by Cheng andSun [5](3.9) jT (f; g)j  12(M  m)  1b  aZ ba f (t) dt:The constant 12 is best in (3.9) as shown by Cerone and Dragomir in [3].8 P. CERONE AND S.S. DRAGOMIRFor r > 1; we deneI (r) :=Z ba[(b  x) (x  a)]r 1(b  x)r 1 + (x  a)r 1 dx:For r = 2; we have(3.10) I (2) =1b  aZ ba(b  x) (x  a) dx = (b  a)26:For r > 2; since the inequality(b  x)r 1 + (x  a)r 12(b  x) + (x  a)2r 1=12r 1(b  a)r 1holds, then(b  x)r 1 + (x  a)r 1  22 r (b  a)r 1 ; x 2 [a; b] ;and soI (r)(3.11) 2r 2(b  a)r 1Z ba[(b  x) (x  a)]r 1 dx 2r 2(b  a)r 1Z 10(b  (1  t) a  tb)r 1 ((1  t) a+ tb  a)r 1 (b  a) dt=2r 2(b  a)r 1Z 10(b  a)r 1 (1  t)r 1 (b  a)r 1 tr 1 (b  a) dt= 2r 2 (b  a)r B (r; r) ; r  2;where B (:; :) is the well known Euler beta function.A di¤erent possibility to bound I (r) is by utilising the inequality between theharmonic and geometric means, namely2+ p; ;  > 0:Therefore(b  x)r 1 (x  a)r 1(b  x)r 1 + (x  a)r 1 12q(b  x)r 1 (x  a)r 1; r > 1for x 2 [a; b] ; which gives by integrationI (r)  12Z ba(b  x) r 12 (x  a) r 12 dx(3.12)=12(b  a)rZ 10tr 12 (1  t) r 12 dt=12(b  a)r Br + 12;r + 12for r > 1:INTEGRAL INEQUALITIES 9Figure 1. The plot of 2r 2B (r; r) and 12B r+12 ;r+12for r  2:Remark 4. If we compare 2r 2B (r; r) with 12B r+12 ;r+12for r  2; using theMaple computer package, we observe that the bound provided by (3.11) for I (r) isbetter than the one provided for (3.12). However, the second one is also valid forr 2 (0; 1) : The plot concerning the variations of 2r 2B (r; r) and 12B r+12 ;r+12on[2;1) is depicted in Figure 1. However, we do not have an analytic proof to showthat2r 2B (r; r)  12Br + 12;r + 12for any r 2 [2;1):The following lemma may be stated.Lemma 1. For r > 1 we have the inequality(3.13)Z baZ xag (t) dt  x  ab  aZ bag (t) dtrdt I (r)Z bag (t)  1b  aZ bag (s) dsrdt:In particular:(3.14)Z baZ xag (t) dt  x  ab  aZ bag (t) dt2dt (b  a)26Z bag (t)  1b  aZ bag (s) ds2dt:10 P. CERONE AND S.S. DRAGOMIRThe proof follows by the inequality (2.17) which is equivalent with(3.15)Z baZ xag (t) dt  x  ab  aZ bag (t) dtrdx (b  x)r 1(x  a)r 1(b  x)r 1 + (x  a)r 1Z bag (t)  1b  aZ bag (s) dsrdtfor any x 2 [a; b] :Also, if we take the supremum over x 2 [a; b] in (3.15) for r = 2; then we get(3.16) supx2[a;b]Z xag (t) dt  x  ab  aZ bag (t) dt2 b  a4Z bag (t)  1b  aZ bag (s) ds2dt:Therefore the following lemma may be stated:Lemma 2. With the above assumptions, we have(3.17) supx2[a;b]Z xag (t) dt  x  ab  aZ bag (t) dt (b  a)1220@Z bag (t)  1b  aZ bag (s) ds2dt1A 12 :Also, on utilising the inequality (2.20), we get the following result as well:Lemma 3. With the above assumptions, we have(3.18)Z baZ xag (t) dt  x  ab  aZ bag (t) dt dx 12(b  a)Z bag (t)  1b  aZ bag (s) ds dt:We can now state the following result that provides upper bounds for the absolutevalue of the µCebyev functional T (f; g) :INTEGRAL INEQUALITIES 11Theorem 2. Let f : [a; b]! C be an absolutely continuous function and g : [a; b]!C a Lebesgue integrable function on [a; b] : Then:(3.19) jT (f; g)j 8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:12R bajf 0 (x)j dx1b aR bag (t)  1b a R ba g (s) ds2 dt 1212  supx2[a;b]jf 0 (x)j R bag (t)  1b a R ba g (s) ds dt[I (r)]1r1b aR bag (t)  1b a R ba g (s) dsr dt 1r1b aR bajf 0 (x)jq dx 1qwhere r > 1; 1r +1q = 1:In particular, for r = 2; we have(3.20) jT (f; g)j p66(b  a)0@ 1b  aZ bag (t)  1b  aZ bag (s) ds2dt1A 12 1b  aZ bajf 0 (x)j2 dx! 12:Proof. Utilising the identity (2.13) from [2], namely,T (f; g) =   1b  aZ baG (x) f 0 (x) dx;whereG (x) =Z xag (t) dt  x  ab  aZ bag (t) dt; x 2 [a; b] ;we havejT (f; g)j  1b  aZ ba G (x) jf 0 (x)j dx =: T:Using Lemma 2, we haveT  supx2[a;b] G (x)  1b  aZ bajf 0 (x)j dx (b  a)122 jf 0 (x)j dx0@ 1b  aZ bag (t)  1b  aZ bag (s) ds2dt1A 12=12Z bajf 0 (x)j dx0@ 1b  aZ bag (t)  1b  aZ bag (s) ds2dt1A 12 ;and the rst inequality in (3.19) is proved.12 P. CERONE AND S.S. DRAGOMIRUtilising Lemma 3, we haveT  supx2[a;b]jf 0 (x)j  1b  aZ ba G (x) dx 12supx2[a;b]jf 0 (x)jZ bag (t)  1b  aZ bag (s) ds dt;which proves the second inequality in (3.19).Now, from Hölders inequality and Lemma 1, we also haveT  1b  a Z ba G (x)r dx! 1r  Z bajf 0 (x)jq dx! 1q 1b  a"I (r) Z bag (t)  1b  aZ bag (s) dsrdt# 1r Z bajf 0 (x)jq dx! 1q=1b  a [I (r)]1r"Z bag (t)  1b  aZ bag (s) dsrdt# 1r Z bajf 0 (x)jq dx! 1qand the last part of (3.19) is also proved. Remark 5. It is an open question whether or not the constants 12 in (3.19) andp66 in (3.20) are best possible.References[1] S. ABRAMOVICH, B. MOND and J.E. PE µCARIC´, Sharpening Hölders inequality, J. Math.Anal. Appl., 196 (1995), 1131-1134.[2] P. CERONE and S.S. DRAGOMIR, New bounds for the µCebyev functional, Appl. Math.Lett., 18 (2005), 603-611.[3] P. CERONE and S.S. DRAGOMIR, A renement of the Grüss inequality and applications,Tamkang J. Math. (in press). Preprint available at RGMIA Res. Rep. Coll., 5(2) (2002), Art.14. [ONLINE http://rgmia.vu.edu.au/v5n2.html].[4] P.L. CHEBYSHEV, Sur les expressions approximatives des intègrals dènis par les outresprises entre les même limites, Proc. Math. Soc. Charkov, 2 (1882), 93-98.[5] X.-L. CHENG and J. SUN, Note on the perturbed trapezoid inequality, J. Inequal. Pure Appl.Math., 3(2) (2002), Art. 29. [ONLINE http://jipam.vu.edu.au/article.php?sid=181].[6] G. GRÜSS, Über das maximum des absoluten Betrages von 1b aR ba f (x) g (x) dx  1(b a)2R ba f (x) dx R ba g (x) dx; Math. Z., 39 (1934), 215-226.[7] A. LUPAS¸, The best constant in an integral inequality, Mathematica (Cluj), 15 (38) (1973),No. 2, 219-222.[8] G.S. MAHAJANI, A note on an inequality, Math. Student, 6(1938), 125-128.[9] D.S. MITRINOVIC´, J.E. PE µCARIC´ and A.M. FINK, Inequalities Involving Functions andtheir Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.[10] D.S. MITRINOVIC´, J.E. PE µCARIC´ and A.M. FINK, Classical and New Inequalities inAnalysis, Kluwer Academic Publishers, Dordrecht, 1993.[11] A.M. OSTROWSKI, On an integral inequality, Aequationes Math., 4 (1970), 358-373.INTEGRAL INEQUALITIES 13School of Computer Science and Mathematics, Victoria University, PO Box 14428,Melbourne City, VIC 8001, Australia.E-mail address : pietro.cerone@vu.edu.auURL: http://rgmia.vu.edu.au/ceroneE-mail address : sever.dragomir@vu.edu.auURL: http://rgmia.vu.edu.au/dragomir

Integral Inequalities and Applications for Bounding the Čebyšev Functional

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