Consider a function f : [ 0 , 1 ] → R + such that f ( 0 ) = 0 and f ( x ) ≤ f ( y ) for all x ≤ y (i.e f is monotone).Is a positive, monotone and sub-additive function concave?

Question

Consider a function   f  :  [  0  ,  1  ]  →            R        +   such that   f  (  0  )  =  0 and   f  (  x  )  ≤  f  (  y  ) for all   x  ≤  y (i.e f is monotone).Is a positive, monotone and sub-additive function concave?

erkvisin7s · Accepted Answer

Subadditivity is implied by a requirement that   g  (  x  )  :=            f      (      x      )        x   be monotonically decreasing. If   g  (  x  )  ≥  g  (  x  +  y  ) and   g  (  y  )  ≥  g  (  x  +  y  ), then  f  (  x  )  +  f  (  y  )  =  x  g  (  x  )  +  y  g  (  y  )  ≥  x  g  (  x  +  y  )  +  y  g  (  x  +  y  )  =  f  (  x  +  y  )  .So any function   f that satisfies the other requirements can be subadditive as long as it does not intersect any line through the origin twice other than at the origin itself (though a single interval of coincidence with a given line through the origin, in addition to the intersection at the origin itself, is licit). This criterion is weaker than concavity (which requires that no line whatsoever intersect   f three times, including at the origin), and a large family of non-concave subadditive functions can be constructed as follows:Take an increasing concave function   g  :  [  0  ,  1  ]  →      R   satisfying   g  (  0  )  =  0.Take some   ξ  ∈  (  0  ,  1  ).Define   f  (  x  )  =  max      {    g    (    x    )    ,                  g        (        ξ        )        x            ξ        }  . That is,   f  =  g on the interval   [  0  ,  ξ  ], and   f follows a continuation of the secant line from   (  0  ,  0  ) to   (  ξ  ,  g  (  ξ  )  ) on the interval   [  ξ  ,  1  ].One function in this family is  f  (  x  )  =      {                                        x                                    x          ≤                      1            4                                                2          x                          x          ≥                      1            4                    .                        More generally, it&#039;s trivial to prove that if   f and   g are two subadditive functions, then   max  {  f  ,  g  } is also subadditive, and not concave anywhere that   f and   g intersect and have different slopes.

Consider a function f:[0,1]->R^+ such that f(0)=0 and f(x)<=f(y) for all x<=y (i.e f is monotone). Additionally, I also restrict f to be a sub-additive function i.e f(x+y)<=f(x)+f(y).

Answered question

Answer & Explanation

New Questions in Calculus 1