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Functional inequality

  1. Oct 5, 2015 #1
    1. The problem statement, all variables and given/known data
    Let ##f,g## be two real valued functions, defined on the segment ##[a,b]## and continuous on ##[a,b]##, such that ## 0 < g < f ##. Show there exist ##\lambda > 0 ## such that ## (1+\lambda) g \le f ##

    2. Relevant equations

    3. The attempt at a solution

    Set ##h = f/g##. Since ##g\neq 0##, ##h## is continuous on ##[a,b]##.
    Therefore, ##h## is bounded on ##[a,b]## and reaches its bounds. Call ##m = h(x_0)## its lower bound. By construction, ##1 < m \le h##, so ## \lambda = m-1 > 0 ## and ## (1+\lambda) g \le f ##. Is it correct ?
     
  2. jcsd
  3. Oct 5, 2015 #2

    RUber

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    Looks good to me. Then ##(1+\lambda)g(x_0) = f(x_0)## and is less than or equal to at all other points in the domain.
     
  4. Oct 5, 2015 #3
    Thank you !
     
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