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Convex functions

  1. Nov 6, 2009 #1
    1. The problem statement, all variables and given/known data
    How do I show that if [tex]f\in C^2 \text{(}\mathbb{R}\text{)}[/tex] is convex then the function [tex]yf(y^{-1}\textbf{x})[/tex] is convex on (x,y):y>0?


    2. Relevant equations

    I know the standard definitions and whatnot about convexity, but I tried chugging through the algebra and didn't have any luck, can anyone show me a nice way to solve this?

    Thanks!

    -Mathmos6
     
  2. jcsd
  3. Nov 6, 2009 #2

    Mark44

    Staff: Mentor

    Show us what you tried...
     
  4. Nov 7, 2009 #3
    I figured [tex]yf(y^{-1}\textbf{x})=yf(\frac{x}{y},1)[/tex], so set [tex]f_x=\frac{\partial}{\partial{x}}f(\frac{x}{y},1), f_{yy}=\frac{\partial^2}{\partial{y^2}}f(\frac{x}{y},1)[/tex] and so on:

    then we get

    [tex]\frac{\partial{}}{\partial{x}}(yf(\frac{x}{y},1))=yf_x \Rightarrow \frac{\partial{}^2}{\partial{x^2}}=yf_{xx}[/tex]

    and

    [tex]\frac{\partial{}^2}{\partial{x}\partial{y}} (yf(\frac{x}{y},1))=f_x+yf_{xy}[/tex]

    Also, W.R.T y:

    [tex] \frac{\partial{}}{\partial{y}}=yf_y+f(\frac{x}{y},1) \Rightarrow \frac{\partial{}^2}{\partial{y}^2} = yf_{yy}+2f_y[/tex] - right?

    Then looking at the Hessian, we know the first principal minor (=yfxx) is >=0 if y is, because f is convex so its corresponding first principal minor must also be >=0. With regards to the second principal minor though, i.e. the determinant of the Hessian, we get

    [tex] \frac{\partial{}^2}{\partial{x}^2} \frac{\partial{}^2}{\partial{y}^2} - (\frac{\partial{}^2}{\partial{x}\partial{y}})^2=y^2(f_{xx}f_{yy}-f_{xy}^2)+2y(f_{xx}f_y-f_{xy}f_x)-f_x^2[/tex]

    if my algebra is correct. We want to show this >=0 - the first term (the thing in the brackets multiplied by y2) is >=0 because it corresponds to the determinant of the Hessian for f - however, my concern is that I've gone wrong because something in the derivative of f(x/y,1) means this wouldn't work the same as for f(x,y) and so I'm not sure how to proceed... thanks :)
     
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