Binomial inequality

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I have two binomial expressions.



beginalign
mm_1binomN(d+1)+nntag1\
binomN+n+1n+1tag2
endalign



wher $n,m,m_1,d$ are constants and only $N$ varies.



Now we have the following inequalities that are



beginalign
mm_1binomN(d+1)+nn&leq c_1 N^n\
binomN+n+1n+1&>N^n+1\
endalign



Hence it is suggested that for large enough $N$ it implies that $(2)>(1)$.



I want to find the smallest $N$ for which this happens in terms of $n,d,m,m_1$. Any method how to approach the problem would be really appreciated.




inequality optimization binomial-coefficients nonlinear-optimization




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    I have two binomial expressions.



    beginalign
    mm_1binomN(d+1)+nntag1\
    binomN+n+1n+1tag2
    endalign



    wher $n,m,m_1,d$ are constants and only $N$ varies.



    Now we have the following inequalities that are



    beginalign
    mm_1binomN(d+1)+nn&leq c_1 N^n\
    binomN+n+1n+1&>N^n+1\
    endalign



    Hence it is suggested that for large enough $N$ it implies that $(2)>(1)$.



    I want to find the smallest $N$ for which this happens in terms of $n,d,m,m_1$. Any method how to approach the problem would be really appreciated.




    inequality optimization binomial-coefficients nonlinear-optimization




    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I have two binomial expressions.



      beginalign
      mm_1binomN(d+1)+nntag1\
      binomN+n+1n+1tag2
      endalign



      wher $n,m,m_1,d$ are constants and only $N$ varies.



      Now we have the following inequalities that are



      beginalign
      mm_1binomN(d+1)+nn&leq c_1 N^n\
      binomN+n+1n+1&>N^n+1\
      endalign



      Hence it is suggested that for large enough $N$ it implies that $(2)>(1)$.



      I want to find the smallest $N$ for which this happens in terms of $n,d,m,m_1$. Any method how to approach the problem would be really appreciated.




      inequality optimization binomial-coefficients nonlinear-optimization




      share|cite|improve this question













      I have two binomial expressions.



      beginalign
      mm_1binomN(d+1)+nntag1\
      binomN+n+1n+1tag2
      endalign



      wher $n,m,m_1,d$ are constants and only $N$ varies.



      Now we have the following inequalities that are



      beginalign
      mm_1binomN(d+1)+nn&leq c_1 N^n\
      binomN+n+1n+1&>N^n+1\
      endalign



      Hence it is suggested that for large enough $N$ it implies that $(2)>(1)$.



      I want to find the smallest $N$ for which this happens in terms of $n,d,m,m_1$. Any method how to approach the problem would be really appreciated.





      inequality optimization binomial-coefficients nonlinear-optimization





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Jul 17 at 4:15
























      asked Jul 16 at 3:13









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