Exchanging max, log and absolute value

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Let $N in mathbbN$, $f,g: [N] to (0,1]$. It is a consequence of the triangle inequality that :



$$left|max_n in [N] f(n) - max_n in [N] g(n)right| leq max_n in [N] left| f(n) - g(n) right|$$



But is the following also true ?



$$left|lnleft( fracmax_n in [N] f(n) max_n in [N] g(n) right) right| leq max_n in [N] left| ln left( fracf(n)g(n) right) right|$$




inequality logarithms maxima-minima




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  • 1




    What is $[ N ]$?
    – Sobi
    Jul 29 at 15:01










  • The set of integers from 1 to N
    – ippiki-ookami
    Jul 29 at 15:19






  • 1




    Conventionally the two arrows $text“totext''$ and $text“mapstotext''$ have different meanings. The function $wmapsto w^3$ has an output that is the cube of its input. The other arrow is used in things like $f,g: [N]to(0,1].$ I edited accordingly. $qquad$
    – Michael Hardy
    Jul 29 at 15:54















up vote
2
down vote

favorite












Let $N in mathbbN$, $f,g: [N] to (0,1]$. It is a consequence of the triangle inequality that :



$$left|max_n in [N] f(n) - max_n in [N] g(n)right| leq max_n in [N] left| f(n) - g(n) right|$$



But is the following also true ?



$$left|lnleft( fracmax_n in [N] f(n) max_n in [N] g(n) right) right| leq max_n in [N] left| ln left( fracf(n)g(n) right) right|$$




inequality logarithms maxima-minima




share|cite|improve this question

















  • 1




    What is $[ N ]$?
    – Sobi
    Jul 29 at 15:01










  • The set of integers from 1 to N
    – ippiki-ookami
    Jul 29 at 15:19






  • 1




    Conventionally the two arrows $text“totext''$ and $text“mapstotext''$ have different meanings. The function $wmapsto w^3$ has an output that is the cube of its input. The other arrow is used in things like $f,g: [N]to(0,1].$ I edited accordingly. $qquad$
    – Michael Hardy
    Jul 29 at 15:54













up vote
2
down vote

favorite









up vote
2
down vote

favorite











Let $N in mathbbN$, $f,g: [N] to (0,1]$. It is a consequence of the triangle inequality that :



$$left|max_n in [N] f(n) - max_n in [N] g(n)right| leq max_n in [N] left| f(n) - g(n) right|$$



But is the following also true ?



$$left|lnleft( fracmax_n in [N] f(n) max_n in [N] g(n) right) right| leq max_n in [N] left| ln left( fracf(n)g(n) right) right|$$




inequality logarithms maxima-minima




share|cite|improve this question













Let $N in mathbbN$, $f,g: [N] to (0,1]$. It is a consequence of the triangle inequality that :



$$left|max_n in [N] f(n) - max_n in [N] g(n)right| leq max_n in [N] left| f(n) - g(n) right|$$



But is the following also true ?



$$left|lnleft( fracmax_n in [N] f(n) max_n in [N] g(n) right) right| leq max_n in [N] left| ln left( fracf(n)g(n) right) right|$$





inequality logarithms maxima-minima





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 29 at 15:52









Michael Hardy

204k23185461




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asked Jul 29 at 14:57









ippiki-ookami

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303216







  • 1




    What is $[ N ]$?
    – Sobi
    Jul 29 at 15:01










  • The set of integers from 1 to N
    – ippiki-ookami
    Jul 29 at 15:19






  • 1




    Conventionally the two arrows $text“totext''$ and $text“mapstotext''$ have different meanings. The function $wmapsto w^3$ has an output that is the cube of its input. The other arrow is used in things like $f,g: [N]to(0,1].$ I edited accordingly. $qquad$
    – Michael Hardy
    Jul 29 at 15:54













  • 1




    What is $[ N ]$?
    – Sobi
    Jul 29 at 15:01










  • The set of integers from 1 to N
    – ippiki-ookami
    Jul 29 at 15:19






  • 1




    Conventionally the two arrows $text“totext''$ and $text“mapstotext''$ have different meanings. The function $wmapsto w^3$ has an output that is the cube of its input. The other arrow is used in things like $f,g: [N]to(0,1].$ I edited accordingly. $qquad$
    – Michael Hardy
    Jul 29 at 15:54








1




1




What is $[ N ]$?
– Sobi
Jul 29 at 15:01




What is $[ N ]$?
– Sobi
Jul 29 at 15:01












The set of integers from 1 to N
– ippiki-ookami
Jul 29 at 15:19




The set of integers from 1 to N
– ippiki-ookami
Jul 29 at 15:19




1




1




Conventionally the two arrows $text“totext''$ and $text“mapstotext''$ have different meanings. The function $wmapsto w^3$ has an output that is the cube of its input. The other arrow is used in things like $f,g: [N]to(0,1].$ I edited accordingly. $qquad$
– Michael Hardy
Jul 29 at 15:54





Conventionally the two arrows $text“totext''$ and $text“mapstotext''$ have different meanings. The function $wmapsto w^3$ has an output that is the cube of its input. The other arrow is used in things like $f,g: [N]to(0,1].$ I edited accordingly. $qquad$
– Michael Hardy
Jul 29 at 15:54











1 Answer
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Logarithm is an increasing function, hence



$$ln (max_n in [N] f(n))=max_n in [N] ln (f(n))$$



Your problem is equivalent to



$$left|max_n in [N] ln (f(n)) - max_n in [N] ln (g(n)) right| le max_n in [N] left|ln (f(n)) - ln (g(n)) right|$$



Hence, if the proof of the first inequality doesn't depend on the range of $f$ or $g$ and if it is true for all real number, then the second inequalities hold.






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    1 Answer
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    active

    oldest

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    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

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    active

    oldest

    votes








    up vote
    2
    down vote



    accepted










    Logarithm is an increasing function, hence



    $$ln (max_n in [N] f(n))=max_n in [N] ln (f(n))$$



    Your problem is equivalent to



    $$left|max_n in [N] ln (f(n)) - max_n in [N] ln (g(n)) right| le max_n in [N] left|ln (f(n)) - ln (g(n)) right|$$



    Hence, if the proof of the first inequality doesn't depend on the range of $f$ or $g$ and if it is true for all real number, then the second inequalities hold.






    share|cite|improve this answer

























      up vote
      2
      down vote



      accepted










      Logarithm is an increasing function, hence



      $$ln (max_n in [N] f(n))=max_n in [N] ln (f(n))$$



      Your problem is equivalent to



      $$left|max_n in [N] ln (f(n)) - max_n in [N] ln (g(n)) right| le max_n in [N] left|ln (f(n)) - ln (g(n)) right|$$



      Hence, if the proof of the first inequality doesn't depend on the range of $f$ or $g$ and if it is true for all real number, then the second inequalities hold.






      share|cite|improve this answer























        up vote
        2
        down vote



        accepted







        up vote
        2
        down vote



        accepted






        Logarithm is an increasing function, hence



        $$ln (max_n in [N] f(n))=max_n in [N] ln (f(n))$$



        Your problem is equivalent to



        $$left|max_n in [N] ln (f(n)) - max_n in [N] ln (g(n)) right| le max_n in [N] left|ln (f(n)) - ln (g(n)) right|$$



        Hence, if the proof of the first inequality doesn't depend on the range of $f$ or $g$ and if it is true for all real number, then the second inequalities hold.






        share|cite|improve this answer













        Logarithm is an increasing function, hence



        $$ln (max_n in [N] f(n))=max_n in [N] ln (f(n))$$



        Your problem is equivalent to



        $$left|max_n in [N] ln (f(n)) - max_n in [N] ln (g(n)) right| le max_n in [N] left|ln (f(n)) - ln (g(n)) right|$$



        Hence, if the proof of the first inequality doesn't depend on the range of $f$ or $g$ and if it is true for all real number, then the second inequalities hold.







        share|cite|improve this answer













        share|cite|improve this answer



        share|cite|improve this answer











        answered Jul 29 at 15:03









        Siong Thye Goh

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        76.9k134794






















             

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