researcharticle a new generalization on cauchy-schwarz inequality · 2019. 7. 30. · the...

Research ArticleA New Generalization on Cauchy-Schwarz Inequality

Songting Yin

Department of Mathematics and Computer Science Tongling University Tongling Anhui 244000 China

Correspondence should be addressed to Songting Yin yst419163com

Received 7 April 2017 Accepted 23 May 2017 Published 13 June 2017

Academic Editor Yoshihiro Sawano

Copyright copy 2017 Songting YinThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We extend the well-knownCauchy-Schwarz inequality involving any number of real or complex functions and also give a necessaryand sufficient condition for the equality This is another generalized version of the Cauchy-Schwarz inequality

1 Introduction

Let 119886 = (1198861 1198862 119886119899) and 119887 = (1198871 1198872 119887119899) be two vec-tors in R119899 Then the discrete version of Cauchy-Schwarzinequality is (see [1 2])

119899sum119894=1

1198862119894 119899sum119894=1

1198872119894 ge ( 119899sum119894=1

119886119894119887119894)2 (1)

and its integral representation in the space of the continuousreal-valued functions 119862([119886 119887]R) reads (see [2 3])int119887119886

119891 (119909)2 119889119909int119887119886

119892 (119909)2 119889119909 ge (int119887119886

119891 (119909) 119892 (119909) 119889119909)2 (2)

In undergraduate teaching material the above twoinequalities are presented Some other forms such as matrixform and determinant form are shown in many literaturesIt is well known that the Cauchy-Schwarz inequality plays animportant role in different branches of modern mathematicssuch as Hilbert space theory probability and statistics classi-cal real and complex analysis numerical analysis qualitativetheory of differential equations and their applications Upto now a large number of generalizations and refinementsof the Cauchy-Schwarz inequality have been investigated inthe literatures (see [4 5]) In [6] Harvey generalized it to aninequality involving four vectors Namely for any 119886 119887 119888 119889 isinR119899 it holds that1198862 1198872 + 1198882 1198892 ge 2119886119879119888119887119879119889 + (119886119879119887)2 + (119888119879119889)2minus (119886119879119889)2 minus (119887119879119888)2 (3)

It is a new generalized version of the Cauchy-Schwarzinequality Recently the result was refined by Choi [7] to astronger one

119898sum119896=1

1003817100381710038171003817119886(119896)10038171003817100381710038172 1003817100381710038171003817119887(119896)10038171003817100381710038172ge 119898sum119896=1

(119886119879(119896)119887(119896))2+ 2119898 minus 1 sum

1le119896lt119897le119898

10038161003816100381610038161003816119886119879(119896)119886(119897)119887119879(119896)119887(119897) minus 119886119879(119896)119887(119897)119887119879(119896)119886(119897)10038161003816100381610038161003816(4)

for any 119886(119896) 119887(119896) isin R119899 119896 = 1 119898 and119898 ge 2 Furthermorehe also gained the complex version of the inequality Sincethe articles are not so difficult to understand they are moreimportant for undergraduate students to study

To meet the need for the teaching we will consider inthis paper the counterparts of (4) and extend them into thefollowing

119898sum119896=1

100381710038171003817100381711989111989610038171003817100381710038172 1003817100381710038171003817ℎ11989610038171003817100381710038172 ge 119898sum119896=1

⟨119891119896 ℎ119896⟩2+ 2119898 minus 1 sum

1le119896lt119897le119898

1003816100381610038161003816⟨119891119896 119891119897⟩ ⟨ℎ119896 ℎ119897⟩ minus ⟨119891119896 ℎ119897⟩ ⟨ℎ119896 119891119897⟩1003816100381610038161003816 (5)

for 2119898 real-valued functions 119891119896 ℎ119896 isin 1198620([119886 119887]R) | 119896 =1 119898 119898 ge 2 (Theorem 2) Instead of the Euclidean norm

HindawiJournal of Function SpacesVolume 2017 Article ID 9576375 4 pageshttpsdoiorg10115520179576375

2 Journal of Function Spaces

on R119899 or C119899 used in [6 7] here the norm and inner productof functions are defined by100381710038171003817100381711989110038171003817100381710038172 š int119887

119886

119891 (119909)2 119889119909⟨119891 ℎ⟩ š int119887

119886

119891 (119909) ℎ (119909) 119889119909 (6)

In this work we further give a necessary and sufficientcondition if the equality holds in (5) The complex version ofinequality (5) is also obtained (Theorem 4) The whole proofis direct as in [6 7] but the calculations have to be made withsome adjustments in the integral case

2 The Real Case ofthe Cauchy-Schwarz Inequality

Theorem 1 Let e(x) f(x) g(x) and h(x) be four continuousreal-valued functions on [ab] Then1198902 100381710038171003817100381711989110038171003817100381710038172 + 100381710038171003817100381711989210038171003817100381710038172 ℎ2ge ⟨119890 119891⟩2 + ⟨119892 ℎ⟩2+ 2 1003816100381610038161003816⟨119890 119892⟩ ⟨119891 ℎ⟩ minus ⟨119890 ℎ⟩ ⟨119891 119892⟩1003816100381610038161003816 (7)

where the equality holds if and only if 119890(119909)119891(119910) minus 119890(119910)119891(119909) =plusmn(119892(119909)ℎ(119910) minus 119892(119910)ℎ(119909)) for all 119909 119910 isin [119886 119887]Proof Notice that 1198902 = int119887

119886119890(119909)2119889119909 1198912 = int119887

119886119891(119909)2119889119909 and⟨119890 119891⟩ = int119887

119886119890(119909)119891(119909)119889119909 Then we have1198902 100381710038171003817100381711989110038171003817100381710038172 minus ⟨119890 119891⟩2= int119887

119886

119890 (119909)2 119889119909 times int119887119886

119891 (119909)2 119889119909minus (int119887119886

119890 (119909) 119891 (119909) 119889119909)2= int119887119886

119890 (119909)2 119889119909int119887119886

119891 (119910)2 119889119910minus int119887119886

119890 (119909) 119891 (119909) 119889119909int119887119886

119890 (119910) 119891 (119910) 119889119910= ∬119863

[119890 (119909)2 119891 (119910)2 minus 119890 (119909) 119891 (119909) 119890 (119910) 119891 (119910)] 119889119909 119889119910= 12 ∬119863 [119890 (119909) 119891 (119910) minus 119890 (119910) 119891 (119909)]2 119889119909 119889119910

(8)

where 119863 = (119909 119910) isin R2 | 119886 le 119909 119910 le 119887 Using the formulaabove we obtain1198902 100381710038171003817100381711989110038171003817100381710038172 + 100381710038171003817100381711989210038171003817100381710038172 ℎ2 minus ⟨119890 119891⟩2 minus ⟨119892 ℎ⟩2 = 12sdot ∬

119863

[(119890 (119909) 119891 (119910) minus 119890 (119910) 119891 (119909))2 + (119892 (119909) ℎ (119910)minus 119892 (119910) ℎ (119909))2] 119889119909 119889y = 12

sdot ∬119863

[(119890 (119909) 119891 (119910) minus 119890 (119910) 119891 (119909))plusmn (119892 (119909) ℎ (119910) minus 119892 (119910) ℎ (119909))]2 ∓ 2 (119890 (119909) 119891 (119910)minus 119890 (119910) 119891 (119909)) (119892 (119909) ℎ (119910)minus 119892 (119910) ℎ (119909)) 119889119909 119889119910 ge ∓∬119863

(119890 (119909)sdot 119891 (119910) minus 119890 (119910) 119891 (119909)) (119892 (119909) ℎ (119910) minus 119892 (119910)sdot ℎ (119909)) 119889119909 119889119910 = ∓2(int119887119886

119890 (119909) 119892 (119909) 119889119909sdot int119887119886

119891 (119910) ℎ (119910) 119889119910 minus int119887119886

119890 (119909) ℎ (119909) 119889119909int119887119886

119891 (119910)sdot 119892 (119910) 119889119910) = ∓2 (⟨119890 119892⟩ ⟨119891 ℎ⟩ minus ⟨119890 ℎ⟩ ⟨119891119892⟩)

(9)

which gives the desired inequality The condition for theequality is obvious This finishes the proof

Theorem 2 Let1198911 119891119898 ℎ1 ℎ119898 be 2m continuous real-valued functions Then for119898 ge 2 one has119898sum119896=1

100381710038171003817100381711989111989610038171003817100381710038172 1003817100381710038171003817ℎ11989610038171003817100381710038172 ge 119898sum119896=1

⟨119891119896 ℎ119896⟩2 + 2119898 minus 1sdot sum1le119896lt119897le119898

1003816100381610038161003816⟨119891119896 119891119897⟩ ⟨ℎ119896 ℎ119897⟩ minus ⟨119891119896 ℎ119897⟩ ⟨ℎ119896 119891119897⟩1003816100381610038161003816 (10)

where the equality holds if and only if 119891119896(119909)ℎ119896(119910) minus119891119896(119910)ℎ119896(119909) = plusmn(119891119897(119909)ℎ119897(119910) minus 119891119897(119910)ℎ119897(119909)) for all 119896 119897 = 1 119898and all 119909 119910 isin [119886 119887]Proof It follows fromTheorem 1 that100381710038171003817100381711989111989610038171003817100381710038172 1003817100381710038171003817ℎ11989610038171003817100381710038172 + 100381710038171003817100381711989111989710038171003817100381710038172 1003817100381710038171003817ℎ11989710038171003817100381710038172 minus ⟨119891119896 ℎ119896⟩2 minus ⟨119891119897 ℎ119897⟩2ge 2 1003816100381610038161003816⟨119891119896 119891119897⟩ ⟨ℎ119896 ℎ119897⟩ minus ⟨119891119896 ℎ119897⟩ ⟨ℎ119896 119891119897⟩1003816100381610038161003816 (11)

for 1 le 119896 lt 119897 le 119898 Thus we have

119898sum119896=1

[100381710038171003817100381711989111989610038171003817100381710038172 1003817100381710038171003817ℎ11989610038171003817100381710038172 minus ⟨119891119896 ℎ119896⟩2] = 1119898 minus 1sdot sum1le119896lt119897le119898

[100381710038171003817100381711989111989610038171003817100381710038172 1003817100381710038171003817ℎ11989610038171003817100381710038172 + 100381710038171003817100381711989111989710038171003817100381710038172 1003817100381710038171003817ℎ11989710038171003817100381710038172 minus ⟨119891119896 ℎ119896⟩2minus ⟨119891119897 ℎ119897⟩2] ge 2119898 minus 1 sum

1le119896lt119897le119898

1003816100381610038161003816⟨119891119896 119891119897⟩ ⟨ℎ119896 ℎ119897⟩minus ⟨119891119896 ℎ119897⟩ ⟨ℎ119896 119891119897⟩1003816100381610038161003816 (12)

Journal of Function Spaces 3

3 The Complex Case ofthe Cauchy-Schwarz Inequality

In this section we consider the complex case The norm andthe inner product of complex-valued functions are defined by100381710038171003817100381711989110038171003817100381710038172 š int119887

119886

119891 (119909) 119891 (119909)119889119909⟨119891 ℎ⟩ š int119887

119886

119891 (119909) ℎ (119909)119889119909119891 ℎ isin 1198620 ([119886 119887] C) (13)

First we give the following

Theorem 3 Let 119890 119891 119892 and ℎ be four continuous complex-valued functions on [119886 119887] Then1198902 100381710038171003817100381711989110038171003817100381710038172 + 100381710038171003817100381711989210038171003817100381710038172 ℎ2ge 1003816100381610038161003816⟨119890 119891⟩10038161003816100381610038162 + 1003816100381610038161003816⟨119892 ℎ⟩10038161003816100381610038162+ 2 1003816100381610038161003816Re (⟨119890 119892⟩ ⟨119891 ℎ⟩ minus ⟨119890 ℎ⟩ ⟨119891 119892⟩)1003816100381610038161003816 (14)

where the equality holds if and only if 119890(119909)119891(119910) minus 119890(119910)119891(119909) =plusmn(119892(119909)ℎ(119910) minus 119892(119910)ℎ(119909)) for all 119909 119910 isin [119886 119887]Proof Note that 119890 and 119891 are complex-valued functions Byusing the above definition on 1198902 1198912 and ⟨119890 119891⟩ we have

1198902 100381710038171003817100381711989110038171003817100381710038172 minus 1003816100381610038161003816⟨119890 119891⟩10038161003816100381610038162 = int119887119886

119890 (119909) 119890 (119909)119889119909 times int119887119886

119891 (119909)sdot 119891 (119909)119889119909 minus 100381610038161003816100381610038161003816100381610038161003816int119887119886 119890 (119909) 119891 (119909)1198891199091003816100381610038161003816100381610038161003816100381610038162= ∬119863

(119890 (119909) 119890 (119909)119891 (119910) 119891 (119910)minus 119890 (119909) 119891 (119909)119890 (119910) 119891 (119910)) 119889119909 119889119910 = 12sdot ∬119863

1003816100381610038161003816119890 (119909) 119891 (119910) minus 119890 (119910) 119891 (119909)10038161003816100381610038162 119889119909 119889119910(15)

where119863 = (119909 119910) isin R2 | 119886 le 119909 119910 le 119887 Therefore

1198902 100381710038171003817100381711989110038171003817100381710038172 + 100381710038171003817100381711989210038171003817100381710038172 ℎ2 minus 1003816100381610038161003816⟨119890 119891⟩10038161003816100381610038162 minus 1003816100381610038161003816⟨119892 ℎ⟩10038161003816100381610038162 = 12sdot ∬119863

[1003816100381610038161003816119890 (119909) 119891 (119910) minus 119890 (119910) 119891 (119909)10038161003816100381610038162+ 1003816100381610038161003816119892 (119909) ℎ (119910) minus 119892 (119910) ℎ (119909)10038161003816100381610038162] 119889119909 119889119910= 12 ∬119863 1003816100381610038161003816(119890 (119909) 119891 (119910) minus 119890 (119910) 119891 (119909))plusmn (119892 (119909) ℎ (119910) minus 119892 (119910) ℎ (119909))10038161003816100381610038162

∓ 2Re [(119890 (119909) 119891 (119910) minus 119890 (119910) 119891 (119909))sdot (119892 (119909) ℎ (119910) minus 119892 (119910) ℎ (119909))] 119889119909 119889119910ge ∓∬119863

Re [(119890 (119909) 119891 (119910) minus 119890 (119910) 119891 (119909))sdot (119892 (119909) ℎ (119910) minus 119892 (119910) ℎ (119909))] 119889119909 119889119910= ∓2Re(int119887

119886

119890 (119909) 119892 (119909)119889119909int119887119886

119891 (119910) ℎ (119910)119889119910minus int119887119886

119890 (119909) ℎ (119909)119889119909int119887119886

119891 (119910) 119892 (119910)119889119910)= ∓2Re (⟨119890 119892⟩ ⟨119891 ℎ⟩ minus ⟨119890 ℎ⟩ ⟨119891 119892⟩) (16)

Then the inequality follows as required

By a similar argument we further obtain the following

Theorem 4 Let1198911 119891119898 ℎ1 ℎ119898 be 2m continuous com-plex-valued functions on [119886 119887] Then for119898 ge 2 one has119898sum119896=1

100381710038171003817100381711989111989610038171003817100381710038172 1003817100381710038171003817ℎ11989610038171003817100381710038172 ge 119898sum119896=1

1003816100381610038161003816⟨119891119896 ℎ119896⟩10038161003816100381610038162 + 2119898 minus 1sdot sum1le119896lt119897le119898

1003816100381610038161003816Re (⟨119891119896 119891119897⟩ ⟨ℎ119896 ℎ119897⟩ minus ⟨119891119896 ℎ119897⟩ ⟨ℎ119896 119891119897⟩)1003816100381610038161003816 (17)

where the equality holds if and only if 119891119896(119909)ℎ119896(119910)minus119891119896(119910)ℎ119896(119909)= plusmn(119891119897(119909)ℎ119897(119910) minus 119891119897(119910)ℎ119897(119909)) for all 119896 119897 = 1 119898 and all119909 119910 isin [119886 119887]Remark 5 (i) In this paper we study the Cauchy-Schwarzinequality by using the real or complex valued functionsdefined on an interval [119886 119887] Indeed it holds for any othercontinuous functions defined on a measurable set

(ii) To prove the results we follow the arguments from[6 7]We remark that by using theGramianmatrix119866definedby

119866 =(⟨119890 119890⟩ ⟨119890 119891⟩ ⟨119890 119892⟩ ⟨119890 ℎ⟩⟨119891 119890⟩ ⟨119891 119891⟩ ⟨119891 119892⟩ ⟨119891 ℎ⟩⟨119892 119890⟩ ⟨119892 119891⟩ ⟨119892 119892⟩ ⟨119892 ℎ⟩⟨ℎ 119890⟩ ⟨ℎ 119891⟩ ⟨ℎ 119892⟩ ⟨ℎ ℎ⟩) (18)

we can also prove Theorem 1 In fact since 119866 = ( 119860 119861119861 119862 ) ge 0it holds that ( det119860 det119861

det119861 det119862 ) ge 0 from which Theorem 1 followsFurther using 2119899 times 2119899 Gramian matrix we generalized theresult to Theorem 2 Complex cases are also derived in thesame way

Conflicts of Interest

The author declares that they have no conflicts of interest


Acknowledgments

This work is supported by AHNSF (1608085MA03) andTLXYRC (2015tlxyrc09)

References

[1] A-L Cauchy Cours drsquoAnalyse de lrsquoEcole Royale Polytechnique IPartie Analyse Algebrique Paris France 1821

[2] D S Mitrinovic J E Pecaric and A M Fink Classical andNew Inequalities in Analysis Kluwer Academic Dordrecht TheNetherlands 1993

[3] V Y Buniakowski ldquoSur quelques inegalites concernant les inte-grales aux differences finiesrdquoMemoires de lrsquoAcademie imperialedes sciences de St Petersbourg vol 1 no 9 pp 1ndash18 1859

[4] S S Dragomir ldquoA survey on Cauchy-Bunyakovsky-Schwarztype discrete inequalitiesrdquo JIPAM Journal of Inequalities in Pureand Applied Mathematics vol 4 no 3 article 63 142 pages2003

[5] J C Kuang Applied Inequalities ShanDong Science and Tech-nology Press Jinan China 2004

[6] N J Harvey ldquoA generalization of the Cauchy-Schwarz inequal-ity involving four vectorsrdquo Journal of Mathematical Inequalitiesvol 9 no 2 pp 489ndash491 2015

[7] D Choi ldquoA generalization of the Cauchy-Schwarz inequalityrdquoJournal of Mathematical Inequalities vol 10 no 4 pp 1009ndash1012 2016

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on R119899 or C119899 used in [6 7] here the norm and inner productof functions are defined by100381710038171003817100381711989110038171003817100381710038172 š int119887

119886

119891 (119909)2 119889119909⟨119891 ℎ⟩ š int119887

119886

119891 (119909) ℎ (119909) 119889119909 (6)

In this work we further give a necessary and sufficientcondition if the equality holds in (5) The complex version ofinequality (5) is also obtained (Theorem 4) The whole proofis direct as in [6 7] but the calculations have to be made withsome adjustments in the integral case

2 The Real Case ofthe Cauchy-Schwarz Inequality

Theorem 1 Let e(x) f(x) g(x) and h(x) be four continuousreal-valued functions on [ab] Then1198902 100381710038171003817100381711989110038171003817100381710038172 + 100381710038171003817100381711989210038171003817100381710038172 ℎ2ge ⟨119890 119891⟩2 + ⟨119892 ℎ⟩2+ 2 1003816100381610038161003816⟨119890 119892⟩ ⟨119891 ℎ⟩ minus ⟨119890 ℎ⟩ ⟨119891 119892⟩1003816100381610038161003816 (7)

where the equality holds if and only if 119890(119909)119891(119910) minus 119890(119910)119891(119909) =plusmn(119892(119909)ℎ(119910) minus 119892(119910)ℎ(119909)) for all 119909 119910 isin [119886 119887]Proof Notice that 1198902 = int119887

119886119890(119909)2119889119909 1198912 = int119887

119886119891(119909)2119889119909 and⟨119890 119891⟩ = int119887

119886119890(119909)119891(119909)119889119909 Then we have1198902 100381710038171003817100381711989110038171003817100381710038172 minus ⟨119890 119891⟩2= int119887

119886

119890 (119909)2 119889119909 times int119887119886

119891 (119909)2 119889119909minus (int119887119886

119890 (119909) 119891 (119909) 119889119909)2= int119887119886

119890 (119909)2 119889119909int119887119886

119891 (119910)2 119889119910minus int119887119886

119890 (119909) 119891 (119909) 119889119909int119887119886

119890 (119910) 119891 (119910) 119889119910= ∬119863

[119890 (119909)2 119891 (119910)2 minus 119890 (119909) 119891 (119909) 119890 (119910) 119891 (119910)] 119889119909 119889119910= 12 ∬119863 [119890 (119909) 119891 (119910) minus 119890 (119910) 119891 (119909)]2 119889119909 119889119910

(8)

where 119863 = (119909 119910) isin R2 | 119886 le 119909 119910 le 119887 Using the formulaabove we obtain1198902 100381710038171003817100381711989110038171003817100381710038172 + 100381710038171003817100381711989210038171003817100381710038172 ℎ2 minus ⟨119890 119891⟩2 minus ⟨119892 ℎ⟩2 = 12sdot ∬

119863

[(119890 (119909) 119891 (119910) minus 119890 (119910) 119891 (119909))2 + (119892 (119909) ℎ (119910)minus 119892 (119910) ℎ (119909))2] 119889119909 119889y = 12

sdot ∬119863

[(119890 (119909) 119891 (119910) minus 119890 (119910) 119891 (119909))plusmn (119892 (119909) ℎ (119910) minus 119892 (119910) ℎ (119909))]2 ∓ 2 (119890 (119909) 119891 (119910)minus 119890 (119910) 119891 (119909)) (119892 (119909) ℎ (119910)minus 119892 (119910) ℎ (119909)) 119889119909 119889119910 ge ∓∬119863

(119890 (119909)sdot 119891 (119910) minus 119890 (119910) 119891 (119909)) (119892 (119909) ℎ (119910) minus 119892 (119910)sdot ℎ (119909)) 119889119909 119889119910 = ∓2(int119887119886

119890 (119909) 119892 (119909) 119889119909sdot int119887119886

119891 (119910) ℎ (119910) 119889119910 minus int119887119886

119890 (119909) ℎ (119909) 119889119909int119887119886

119891 (119910)sdot 119892 (119910) 119889119910) = ∓2 (⟨119890 119892⟩ ⟨119891 ℎ⟩ minus ⟨119890 ℎ⟩ ⟨119891119892⟩)

(9)

which gives the desired inequality The condition for theequality is obvious This finishes the proof

Theorem 2 Let1198911 119891119898 ℎ1 ℎ119898 be 2m continuous real-valued functions Then for119898 ge 2 one has119898sum119896=1

100381710038171003817100381711989111989610038171003817100381710038172 1003817100381710038171003817ℎ11989610038171003817100381710038172 ge 119898sum119896=1

⟨119891119896 ℎ119896⟩2 + 2119898 minus 1sdot sum1le119896lt119897le119898

1003816100381610038161003816⟨119891119896 119891119897⟩ ⟨ℎ119896 ℎ119897⟩ minus ⟨119891119896 ℎ119897⟩ ⟨ℎ119896 119891119897⟩1003816100381610038161003816 (10)

where the equality holds if and only if 119891119896(119909)ℎ119896(119910) minus119891119896(119910)ℎ119896(119909) = plusmn(119891119897(119909)ℎ119897(119910) minus 119891119897(119910)ℎ119897(119909)) for all 119896 119897 = 1 119898and all 119909 119910 isin [119886 119887]Proof It follows fromTheorem 1 that100381710038171003817100381711989111989610038171003817100381710038172 1003817100381710038171003817ℎ11989610038171003817100381710038172 + 100381710038171003817100381711989111989710038171003817100381710038172 1003817100381710038171003817ℎ11989710038171003817100381710038172 minus ⟨119891119896 ℎ119896⟩2 minus ⟨119891119897 ℎ119897⟩2ge 2 1003816100381610038161003816⟨119891119896 119891119897⟩ ⟨ℎ119896 ℎ119897⟩ minus ⟨119891119896 ℎ119897⟩ ⟨ℎ119896 119891119897⟩1003816100381610038161003816 (11)

for 1 le 119896 lt 119897 le 119898 Thus we have

119898sum119896=1

[100381710038171003817100381711989111989610038171003817100381710038172 1003817100381710038171003817ℎ11989610038171003817100381710038172 minus ⟨119891119896 ℎ119896⟩2] = 1119898 minus 1sdot sum1le119896lt119897le119898

[100381710038171003817100381711989111989610038171003817100381710038172 1003817100381710038171003817ℎ11989610038171003817100381710038172 + 100381710038171003817100381711989111989710038171003817100381710038172 1003817100381710038171003817ℎ11989710038171003817100381710038172 minus ⟨119891119896 ℎ119896⟩2minus ⟨119891119897 ℎ119897⟩2] ge 2119898 minus 1 sum

1le119896lt119897le119898

1003816100381610038161003816⟨119891119896 119891119897⟩ ⟨ℎ119896 ℎ119897⟩minus ⟨119891119896 ℎ119897⟩ ⟨ℎ119896 119891119897⟩1003816100381610038161003816 (12)




119886

119891 (119909) 119891 (119909)119889119909⟨119891 ℎ⟩ š int119887

119886

119891 (119909) ℎ (119909)119889119909119891 ℎ isin 1198620 ([119886 119887] C) (13)




1198902 100381710038171003817100381711989110038171003817100381710038172 minus 1003816100381610038161003816⟨119890 119891⟩10038161003816100381610038162 = int119887119886

119890 (119909) 119890 (119909)119889119909 times int119887119886

119891 (119909)sdot 119891 (119909)119889119909 minus 100381610038161003816100381610038161003816100381610038161003816int119887119886 119890 (119909) 119891 (119909)1198891199091003816100381610038161003816100381610038161003816100381610038162= ∬119863

(119890 (119909) 119890 (119909)119891 (119910) 119891 (119910)minus 119890 (119909) 119891 (119909)119890 (119910) 119891 (119910)) 119889119909 119889119910 = 12sdot ∬119863

1003816100381610038161003816119890 (119909) 119891 (119910) minus 119890 (119910) 119891 (119909)10038161003816100381610038162 119889119909 119889119910(15)






119886

119890 (119909) 119892 (119909)119889119909int119887119886

119891 (119910) ℎ (119910)119889119910minus int119887119886

119890 (119909) ℎ (119909)119889119909int119887119886

119891 (119910) 119892 (119910)119889119910)= ∓2Re (⟨119890 119892⟩ ⟨119891 ℎ⟩ minus ⟨119890 ℎ⟩ ⟨119891 119892⟩) (16)




100381710038171003817100381711989111989610038171003817100381710038172 1003817100381710038171003817ℎ11989610038171003817100381710038172 ge 119898sum119896=1


1003816100381610038161003816Re (⟨119891119896 119891119897⟩ ⟨ℎ119896 ℎ119897⟩ minus ⟨119891119896 ℎ119897⟩ ⟨ℎ119896 119891119897⟩)1003816100381610038161003816 (17)



119866 =(⟨119890 119890⟩ ⟨119890 119891⟩ ⟨119890 119892⟩ ⟨119890 ℎ⟩⟨119891 119890⟩ ⟨119891 119891⟩ ⟨119891 119892⟩ ⟨119891 ℎ⟩⟨119892 119890⟩ ⟨119892 119891⟩ ⟨119892 119892⟩ ⟨119892 ℎ⟩⟨ℎ 119890⟩ ⟨ℎ 119891⟩ ⟨ℎ 119892⟩ ⟨ℎ ℎ⟩) (18)






Acknowledgments


References















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







119886

119891 (119909) 119891 (119909)119889119909⟨119891 ℎ⟩ š int119887

119886

119891 (119909) ℎ (119909)119889119909119891 ℎ isin 1198620 ([119886 119887] C) (13)




1198902 100381710038171003817100381711989110038171003817100381710038172 minus 1003816100381610038161003816⟨119890 119891⟩10038161003816100381610038162 = int119887119886

119890 (119909) 119890 (119909)119889119909 times int119887119886

119891 (119909)sdot 119891 (119909)119889119909 minus 100381610038161003816100381610038161003816100381610038161003816int119887119886 119890 (119909) 119891 (119909)1198891199091003816100381610038161003816100381610038161003816100381610038162= ∬119863

(119890 (119909) 119890 (119909)119891 (119910) 119891 (119910)minus 119890 (119909) 119891 (119909)119890 (119910) 119891 (119910)) 119889119909 119889119910 = 12sdot ∬119863

1003816100381610038161003816119890 (119909) 119891 (119910) minus 119890 (119910) 119891 (119909)10038161003816100381610038162 119889119909 119889119910(15)






119886

119890 (119909) 119892 (119909)119889119909int119887119886

119891 (119910) ℎ (119910)119889119910minus int119887119886

119890 (119909) ℎ (119909)119889119909int119887119886

119891 (119910) 119892 (119910)119889119910)= ∓2Re (⟨119890 119892⟩ ⟨119891 ℎ⟩ minus ⟨119890 ℎ⟩ ⟨119891 119892⟩) (16)




100381710038171003817100381711989111989610038171003817100381710038172 1003817100381710038171003817ℎ11989610038171003817100381710038172 ge 119898sum119896=1


1003816100381610038161003816Re (⟨119891119896 119891119897⟩ ⟨ℎ119896 ℎ119897⟩ minus ⟨119891119896 ℎ119897⟩ ⟨ℎ119896 119891119897⟩)1003816100381610038161003816 (17)



119866 =(⟨119890 119890⟩ ⟨119890 119891⟩ ⟨119890 119892⟩ ⟨119890 ℎ⟩⟨119891 119890⟩ ⟨119891 119891⟩ ⟨119891 119892⟩ ⟨119891 ℎ⟩⟨119892 119890⟩ ⟨119892 119891⟩ ⟨119892 119892⟩ ⟨119892 ℎ⟩⟨ℎ 119890⟩ ⟨ℎ 119891⟩ ⟨ℎ 119892⟩ ⟨ℎ ℎ⟩) (18)






Acknowledgments


References















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Acknowledgments


References















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




researcharticle a new generalization on cauchy-schwarz inequality · 2019. 7. 30. · the...

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