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NMinimize--problem with a min-max problem


Hello,

I'm trying to solve numerically the following problem:

min{z}max{y,x} f{x,y,z}

I first do the maximization:

fun1[z_] := NMaximize[{(
  Log[x] + 4/3 z Log[
      y])/(2 (3/4 + z)), x > 0, y > 0, x + y == 1}, {x, y}][[1]]

I can plot fun1[z] (the solution is at z=.75,x=y=1/2) however NMinimize
does accept the definition of fun1[z] as:

NMinimize[fun1[z], z]

yields error. 

Any suggestion?

Thank you in advance,

David


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