$a,b,cinBbb R^+, x,y,zin Bbb R, $ show that $fracx^2a+fracy^2b+fracz^2c ge frac(x+y+z)^2a+b+c$Prove $sumlimits_i=1^nfraca_i^2b_i geq frac(sumlimits_i=1^na_i)^2sumlimits_i=1^nb_i$Harmonic mean: show $maxax,by ge frac1a+b(x+y)$, $a,b>1$, $x,yge 0$Proof of analogue of the Cauchy-Schwarz inequality for random variablesOlympiad inequality $fraca2a + b + fracb2b + c + fracc2c + a leq 1$.Prove Inequality with Cauchy SchwarzShow that $frac1x_1(x_1+1)+frac1x_2(x_2+1)+cdots+frac1x_n(x_n+1)gefracn2$ with $x_1x_2cdots x_n=1$Estimating the product of timelike vectors in $Bbb R^n_nu$Showing that two vectors are not orthogonal, using Cauchy-Schwarz$AM$- $QM$ inequalityCauchy-Schwarz inequality to prove $A$ is spd$x,yinBbb R,$ find the maxima of $fracx+2y+3sqrtx^2+y^2+1$

What is the likely impact of grounding an entire aircraft series?

Making a sword in the stone, in a medieval world without magic

How much stiffer are 23c tires over 28c?

How do I express some one as a black person?

Should I tell my boss the work he did was worthless

MTG: Can I kill an opponent in response to lethal activated abilities, and not take the damage?

Virginia employer terminated employee and wants signing bonus returned

Peter's Strange Word

How do anti-virus programs start at Windows boot?

How could our ancestors have domesticated a solitary predator?

Is it true that real estate prices mainly go up?

If the Captain's screens are out, does he switch seats with the co-pilot?

Append a note to one of three files based on user choice

What Happens when Passenger Refuses to Fly Boeing 737 Max?

Why would one plane in this picture not have gear down yet?

Do Bugbears' arms literally get longer when it's their turn?

How to create a hard link to an inode (ext4)?

Why is this plane circling around the Lucknow airport every day?

Why does the negative sign arise in this thermodynamic relation?

PTIJ: How can I halachically kill a vampire?

Good allowance savings plan?

show this identity with trigometric

Do f-stop and exposure time perfectly cancel?

Are the terms "stab" and "staccato" synonyms?



$a,b,cinBbb R^+, x,y,zin Bbb R, $ show that $fracx^2a+fracy^2b+fracz^2c ge frac(x+y+z)^2a+b+c$


Prove $sumlimits_i=1^nfraca_i^2b_i geq frac(sumlimits_i=1^na_i)^2sumlimits_i=1^nb_i$Harmonic mean: show $maxax,by ge frac1a+b(x+y)$, $a,b>1$, $x,yge 0$Proof of analogue of the Cauchy-Schwarz inequality for random variablesOlympiad inequality $fraca2a + b + fracb2b + c + fracc2c + a leq 1$.Prove Inequality with Cauchy SchwarzShow that $frac1x_1(x_1+1)+frac1x_2(x_2+1)+cdots+frac1x_n(x_n+1)gefracn2$ with $x_1x_2cdots x_n=1$Estimating the product of timelike vectors in $Bbb R^n_nu$Showing that two vectors are not orthogonal, using Cauchy-Schwarz$AM$- $QM$ inequalityCauchy-Schwarz inequality to prove $A$ is spd$x,yinBbb R,$ find the maxima of $fracx+2y+3sqrtx^2+y^2+1$













1












$begingroup$


$a,b,cinBbb R^+, x,y,zin Bbb R, $ show that $fracx^2a+fracy^2b+fracz^2c ge frac(x+y+z)^2a+b+c$ (use Cauchy–Schwarz inequality)



I have trouble finding the two vectors. Is it $(x,y,z)$ and $(a,b,c)$?



I need a hint






inequality cauchy-schwarz-inequality







share|cite|improve this question











$endgroup$







  • 2




    $begingroup$
    This is known as Titus Lemma. More details here
    $endgroup$
    – rtybase
    2 days ago










  • $begingroup$
    Try $(x/sqrt a, y/sqrt b, z/sqrt c)$ and $(sqrt a, sqrt b, sqrt c)$…
    $endgroup$
    – Macavity
    2 days ago















1












$begingroup$


$a,b,cinBbb R^+, x,y,zin Bbb R, $ show that $fracx^2a+fracy^2b+fracz^2c ge frac(x+y+z)^2a+b+c$ (use Cauchy–Schwarz inequality)



I have trouble finding the two vectors. Is it $(x,y,z)$ and $(a,b,c)$?



I need a hint






inequality cauchy-schwarz-inequality







share|cite|improve this question











$endgroup$







  • 2




    $begingroup$
    This is known as Titus Lemma. More details here
    $endgroup$
    – rtybase
    2 days ago










  • $begingroup$
    Try $(x/sqrt a, y/sqrt b, z/sqrt c)$ and $(sqrt a, sqrt b, sqrt c)$…
    $endgroup$
    – Macavity
    2 days ago













1












1








1





$begingroup$


$a,b,cinBbb R^+, x,y,zin Bbb R, $ show that $fracx^2a+fracy^2b+fracz^2c ge frac(x+y+z)^2a+b+c$ (use Cauchy–Schwarz inequality)



I have trouble finding the two vectors. Is it $(x,y,z)$ and $(a,b,c)$?



I need a hint






inequality cauchy-schwarz-inequality







share|cite|improve this question











$endgroup$




$a,b,cinBbb R^+, x,y,zin Bbb R, $ show that $fracx^2a+fracy^2b+fracz^2c ge frac(x+y+z)^2a+b+c$ (use Cauchy–Schwarz inequality)



I have trouble finding the two vectors. Is it $(x,y,z)$ and $(a,b,c)$?



I need a hint






inequality cauchy-schwarz-inequality




inequality cauchy-schwarz-inequality






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago









Michael Rozenberg

108k1895200




108k1895200










asked 2 days ago









DavidDavid

454




454







  • 2




    $begingroup$
    This is known as Titus Lemma. More details here
    $endgroup$
    – rtybase
    2 days ago










  • $begingroup$
    Try $(x/sqrt a, y/sqrt b, z/sqrt c)$ and $(sqrt a, sqrt b, sqrt c)$…
    $endgroup$
    – Macavity
    2 days ago












  • 2




    $begingroup$
    This is known as Titus Lemma. More details here
    $endgroup$
    – rtybase
    2 days ago










  • $begingroup$
    Try $(x/sqrt a, y/sqrt b, z/sqrt c)$ and $(sqrt a, sqrt b, sqrt c)$…
    $endgroup$
    – Macavity
    2 days ago







2




2




$begingroup$
This is known as Titus Lemma. More details here
$endgroup$
– rtybase
2 days ago




$begingroup$
This is known as Titus Lemma. More details here
$endgroup$
– rtybase
2 days ago












$begingroup$
Try $(x/sqrt a, y/sqrt b, z/sqrt c)$ and $(sqrt a, sqrt b, sqrt c)$…
$endgroup$
– Macavity
2 days ago




$begingroup$
Try $(x/sqrt a, y/sqrt b, z/sqrt c)$ and $(sqrt a, sqrt b, sqrt c)$…
$endgroup$
– Macavity
2 days ago










2 Answers
2






active

oldest

votes


















2












$begingroup$

Hint: Use Cauchy Schwarz in Engelform:
$$fracx^2a+fracy^2b+fracz^2cgeq frac(x+y+z)^2a+b+c$$
It is equivalent to $$z^2a^2b+y^2a^2c+z^2ab^2-2,abcxy-2,abcxz-2,a
bcyz+y^2ac^2+x^2b^2c+x^2bc^2
geq 0$$
and this is $$a(bz-cy)^2+b(az-xc)^2+c(bx-ay)^2geq 0$$ if $$a,b,c$$ are positive.






share|cite|improve this answer











$endgroup$




















    0












    $begingroup$

    By C-S $$fracx^2a+fracy^2b+fracz^2c=fracleft(fracx^2a+fracy^2b+fracz^2cright)(a+b+c)a+b+cgeq$$
    $$geqfracleft(fracxsqrtacdotsqrta+fracysqrtbcdotsqrtb+fraczsqrtccdotsqrtcright)^2a+b+c=frac(x+y+z)^2a+b+c.$$
    The equality occurs for $$left(fracxsqrta,fracysqrtb,fraczsqrtcright)||(sqrta,sqrtb,sqrtc)$$ or
    $$(x,y,z)||(a,b,c).$$
    Actually, the last writing is very useful.






    share|cite|improve this answer









    $endgroup$












      Your Answer





      StackExchange.ifUsing("editor", function ()
      return StackExchange.using("mathjaxEditing", function ()
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      );
      );
      , "mathjax-editing");

      StackExchange.ready(function()
      var channelOptions =
      tags: "".split(" "),
      id: "69"
      ;
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function()
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled)
      StackExchange.using("snippets", function()
      createEditor();
      );

      else
      createEditor();

      );

      function createEditor()
      StackExchange.prepareEditor(
      heartbeatType: 'answer',
      autoActivateHeartbeat: false,
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader:
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      ,
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      );



      );













      draft saved

      draft discarded


















      StackExchange.ready(
      function ()
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3142310%2fa-b-c-in-bbb-r-x-y-z-in-bbb-r-show-that-fracx2a-fracy2b-fr%23new-answer', 'question_page');

      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2












      $begingroup$

      Hint: Use Cauchy Schwarz in Engelform:
      $$fracx^2a+fracy^2b+fracz^2cgeq frac(x+y+z)^2a+b+c$$
      It is equivalent to $$z^2a^2b+y^2a^2c+z^2ab^2-2,abcxy-2,abcxz-2,a
      bcyz+y^2ac^2+x^2b^2c+x^2bc^2
      geq 0$$
      and this is $$a(bz-cy)^2+b(az-xc)^2+c(bx-ay)^2geq 0$$ if $$a,b,c$$ are positive.






      share|cite|improve this answer











      $endgroup$

















        2












        $begingroup$

        Hint: Use Cauchy Schwarz in Engelform:
        $$fracx^2a+fracy^2b+fracz^2cgeq frac(x+y+z)^2a+b+c$$
        It is equivalent to $$z^2a^2b+y^2a^2c+z^2ab^2-2,abcxy-2,abcxz-2,a
        bcyz+y^2ac^2+x^2b^2c+x^2bc^2
        geq 0$$
        and this is $$a(bz-cy)^2+b(az-xc)^2+c(bx-ay)^2geq 0$$ if $$a,b,c$$ are positive.






        share|cite|improve this answer











        $endgroup$















          2












          2








          2





          $begingroup$

          Hint: Use Cauchy Schwarz in Engelform:
          $$fracx^2a+fracy^2b+fracz^2cgeq frac(x+y+z)^2a+b+c$$
          It is equivalent to $$z^2a^2b+y^2a^2c+z^2ab^2-2,abcxy-2,abcxz-2,a
          bcyz+y^2ac^2+x^2b^2c+x^2bc^2
          geq 0$$
          and this is $$a(bz-cy)^2+b(az-xc)^2+c(bx-ay)^2geq 0$$ if $$a,b,c$$ are positive.






          share|cite|improve this answer











          $endgroup$



          Hint: Use Cauchy Schwarz in Engelform:
          $$fracx^2a+fracy^2b+fracz^2cgeq frac(x+y+z)^2a+b+c$$
          It is equivalent to $$z^2a^2b+y^2a^2c+z^2ab^2-2,abcxy-2,abcxz-2,a
          bcyz+y^2ac^2+x^2b^2c+x^2bc^2
          geq 0$$
          and this is $$a(bz-cy)^2+b(az-xc)^2+c(bx-ay)^2geq 0$$ if $$a,b,c$$ are positive.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 2 days ago

























          answered 2 days ago









          Dr. Sonnhard GraubnerDr. Sonnhard Graubner

          77.7k42866




          77.7k42866





















              0












              $begingroup$

              By C-S $$fracx^2a+fracy^2b+fracz^2c=fracleft(fracx^2a+fracy^2b+fracz^2cright)(a+b+c)a+b+cgeq$$
              $$geqfracleft(fracxsqrtacdotsqrta+fracysqrtbcdotsqrtb+fraczsqrtccdotsqrtcright)^2a+b+c=frac(x+y+z)^2a+b+c.$$
              The equality occurs for $$left(fracxsqrta,fracysqrtb,fraczsqrtcright)||(sqrta,sqrtb,sqrtc)$$ or
              $$(x,y,z)||(a,b,c).$$
              Actually, the last writing is very useful.






              share|cite|improve this answer









              $endgroup$

















                0












                $begingroup$

                By C-S $$fracx^2a+fracy^2b+fracz^2c=fracleft(fracx^2a+fracy^2b+fracz^2cright)(a+b+c)a+b+cgeq$$
                $$geqfracleft(fracxsqrtacdotsqrta+fracysqrtbcdotsqrtb+fraczsqrtccdotsqrtcright)^2a+b+c=frac(x+y+z)^2a+b+c.$$
                The equality occurs for $$left(fracxsqrta,fracysqrtb,fraczsqrtcright)||(sqrta,sqrtb,sqrtc)$$ or
                $$(x,y,z)||(a,b,c).$$
                Actually, the last writing is very useful.






                share|cite|improve this answer









                $endgroup$















                  0












                  0








                  0





                  $begingroup$

                  By C-S $$fracx^2a+fracy^2b+fracz^2c=fracleft(fracx^2a+fracy^2b+fracz^2cright)(a+b+c)a+b+cgeq$$
                  $$geqfracleft(fracxsqrtacdotsqrta+fracysqrtbcdotsqrtb+fraczsqrtccdotsqrtcright)^2a+b+c=frac(x+y+z)^2a+b+c.$$
                  The equality occurs for $$left(fracxsqrta,fracysqrtb,fraczsqrtcright)||(sqrta,sqrtb,sqrtc)$$ or
                  $$(x,y,z)||(a,b,c).$$
                  Actually, the last writing is very useful.






                  share|cite|improve this answer









                  $endgroup$



                  By C-S $$fracx^2a+fracy^2b+fracz^2c=fracleft(fracx^2a+fracy^2b+fracz^2cright)(a+b+c)a+b+cgeq$$
                  $$geqfracleft(fracxsqrtacdotsqrta+fracysqrtbcdotsqrtb+fraczsqrtccdotsqrtcright)^2a+b+c=frac(x+y+z)^2a+b+c.$$
                  The equality occurs for $$left(fracxsqrta,fracysqrtb,fraczsqrtcright)||(sqrta,sqrtb,sqrtc)$$ or
                  $$(x,y,z)||(a,b,c).$$
                  Actually, the last writing is very useful.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 2 days ago









                  Michael RozenbergMichael Rozenberg

                  108k1895200




                  108k1895200



























                      draft saved

                      draft discarded
















































                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid …


                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.

                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function ()
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3142310%2fa-b-c-in-bbb-r-x-y-z-in-bbb-r-show-that-fracx2a-fracy2b-fr%23new-answer', 'question_page');

                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      -

                      Popular posts from this blog

                      Lowndes Grove History Architecture References Navigation menu32°48′6″N 79°57′58″W / 32.80167°N 79.96611°W / 32.80167; -79.9661132°48′6″N 79°57′58″W / 32.80167°N 79.96611°W / 32.80167; -79.9661178002500"National Register Information System"Historic houses of South Carolina"Lowndes Grove""+32° 48' 6.00", −79° 57' 58.00""Lowndes Grove, Charleston County (260 St. Margaret St., Charleston)""Lowndes Grove"The Charleston ExpositionIt Happened in South Carolina"Lowndes Grove (House), Saint Margaret Street & Sixth Avenue, Charleston, Charleston County, SC(Photographs)"Plantations of the Carolina Low Countrye

                      Image
                      Contributing propertyKeeper of the RegisterHistoric districtHistory of the National Register of Historic PlacesNational Park ServiceProperty typesCharlestonColumbiaGreenvilleNorth CharlestonRock Hill Houses on the National Register of Historic Places in South CarolinaGeorgian architecture in South CarolinaHouses completed in 1786Plantation houses in South CarolinaNational Register of Historic Places in Charleston, South CarolinaHouses in Charleston, South Carolina plantation houseAshley RiverCharlestonNational Register of Historic PlacesRevolutionary WarWilliam LowndesU.S. CongressSouth Carolina Inter-State and West Indian ExpositionGeorgianDoricbalustradespedimentoculusentablaturemodilliondentillightship roofterra cottatilescorbelledcornicesRegency stylespiral staircaseskylight Lowndes Grove U.S. National Register of Historic Places Lowndes Grove in 1940 Show map of South Carolina Show map of the United States Location 260 St. Margaret St., Charleston, South Carolina Coordinates 32
                      Read more

                      random experiment with two different functions on unit interval Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Random variable and probability space notionsRandom Walk with EdgesFinding functions where the increase over a random interval is Poisson distributedNumber of days until dayCan an observed event in fact be of zero probability?Unit random processmodels of coins and uniform distributionHow to get the number of successes given $n$ trials , probability $P$ and a random variable $X$Absorbing Markov chain in a computer. Is “almost every” turned into always convergence in computer executions?Stopped random walk is not uniformly integrable

                      Image
                      What would you call this weird metallic apparatus that allows you to lift people? Is it possible to give , in economics, an example of a relation ( set of ordered pairs) that is not a function? Maximum summed subsequences with non-adjacent items How many time has Arya actually used Needle? Is there public access to the Meteor Crater in Arizona? Belief In God or Knowledge Of God. Which is better? What is the meaning of 'breadth' in breadth first search? Why is there Net Work Done on a Pressure/Volume Cycle? How much damage would a cupful of neutron star matter do to the Earth? Sum letters are not two different Why does it sometimes sound good to play a grace note as a lead in to a note in a melody? What initially awakened the Balrog? How to pronounce 伝統色 QGIS virtual layer functionality does not seem to support memory layers AppleTVs create a chatty alternate WiFi network How to write capital alpha? How often does castling occur in grandmaster gam
                      Read more

                      How should I support this large drywall patch? Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?How do I cover large gaps in drywall?How do I keep drywall around a patch from crumbling?Can I glue a second layer of drywall?How to patch long strip on drywall?Large drywall patch: how to avoid bulging seams?Drywall Mesh Patch vs. Bulge? To remove or not to remove?How to fix this drywall job?Prep drywall before backsplashWhat's the best way to fix this horrible drywall patch job?Drywall patching using 3M Patch Plus Primer

                      Image
                      How do living politicians protect their readily obtainable signatures from misuse? Is it possible to force a specific program to remain in memory after closing it? File name problem(?) How to pronounce 伝統色 How does the math work when buying airline miles? Sum letters are not two different An adverb for when you're not exaggerating What does it mean that physics no longer uses mechanical models to describe phenomena? Take 2! Is this homebrew Lady of Pain warlock patron balanced? How many time has Arya actually used Needle? A letter with no particular backstory Converted a Scalar function to a TVF function for parallel execution-Still running in Serial mode Generate an RGB colour grid Co-worker has annoying ringtone Strange behavior of Object.defineProperty() in JavaScript "Lost his faith in humanity in the trenches of Verdun" — last line of an SF story What is this round thing on the pantry door in The Shining What would you call this weir
                      Read more