Inequality of logarithm function in terms of $L^p$ norm

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I am struggling to show that



for $fin L^q([0,1])$ with $q>0$,



$int_[0,1]log|f|dmleq logleft| f right|_q $.



I even don't know how to start :( Hope I can get some help :)







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  • 1




    Just apply Jensen.
    – Shalop
    Aug 7 at 19:39










  • @Shalop That works! Thank you!
    – Lev Ban
    Aug 7 at 19:40














up vote
1
down vote

favorite












I am struggling to show that



for $fin L^q([0,1])$ with $q>0$,



$int_[0,1]log|f|dmleq logleft| f right|_q $.



I even don't know how to start :( Hope I can get some help :)







share|cite|improve this question

















  • 1




    Just apply Jensen.
    – Shalop
    Aug 7 at 19:39










  • @Shalop That works! Thank you!
    – Lev Ban
    Aug 7 at 19:40












up vote
1
down vote

favorite









up vote
1
down vote

favorite











I am struggling to show that



for $fin L^q([0,1])$ with $q>0$,



$int_[0,1]log|f|dmleq logleft| f right|_q $.



I even don't know how to start :( Hope I can get some help :)







share|cite|improve this question













I am struggling to show that



for $fin L^q([0,1])$ with $q>0$,



$int_[0,1]log|f|dmleq logleft| f right|_q $.



I even don't know how to start :( Hope I can get some help :)









share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 7 at 19:27
























asked Aug 7 at 17:59









Lev Ban

48616




48616







  • 1




    Just apply Jensen.
    – Shalop
    Aug 7 at 19:39










  • @Shalop That works! Thank you!
    – Lev Ban
    Aug 7 at 19:40












  • 1




    Just apply Jensen.
    – Shalop
    Aug 7 at 19:39










  • @Shalop That works! Thank you!
    – Lev Ban
    Aug 7 at 19:40







1




1




Just apply Jensen.
– Shalop
Aug 7 at 19:39




Just apply Jensen.
– Shalop
Aug 7 at 19:39












@Shalop That works! Thank you!
– Lev Ban
Aug 7 at 19:40




@Shalop That works! Thank you!
– Lev Ban
Aug 7 at 19:40










1 Answer
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Hint:



  • $forall p>0, forall qin(p,+infty), |f|_ple|f|_q$;

  • $lim_qrightarrow 0^+ |f|_q=expleft(int_[0,1]log|f(t)|operatornamedtright)$





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    1 Answer
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    up vote
    1
    down vote













    Hint:



    • $forall p>0, forall qin(p,+infty), |f|_ple|f|_q$;

    • $lim_qrightarrow 0^+ |f|_q=expleft(int_[0,1]log|f(t)|operatornamedtright)$





    share|cite|improve this answer

























      up vote
      1
      down vote













      Hint:



      • $forall p>0, forall qin(p,+infty), |f|_ple|f|_q$;

      • $lim_qrightarrow 0^+ |f|_q=expleft(int_[0,1]log|f(t)|operatornamedtright)$





      share|cite|improve this answer























        up vote
        1
        down vote










        up vote
        1
        down vote









        Hint:



        • $forall p>0, forall qin(p,+infty), |f|_ple|f|_q$;

        • $lim_qrightarrow 0^+ |f|_q=expleft(int_[0,1]log|f(t)|operatornamedtright)$





        share|cite|improve this answer













        Hint:



        • $forall p>0, forall qin(p,+infty), |f|_ple|f|_q$;

        • $lim_qrightarrow 0^+ |f|_q=expleft(int_[0,1]log|f(t)|operatornamedtright)$






        share|cite|improve this answer













        share|cite|improve this answer



        share|cite|improve this answer











        answered Aug 7 at 19:36









        Bob

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