What are some characterizations of concave functions of real functions of a single real variable?

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Zackary Hatfield · Accepted Answer

One very important characterization is the dual characterization of concave (and convex) functions: every concave function   f is the minimum of all linear functions   φ such that   φ  ≥  f.

perlejatyh8 · Accepted Answer

A function   f  :  D  →      R  , where   D is an interval, is concave if  f  (  a      x    1    +  (  1  −  a  )      x    2    )  ≥  a  f  (      x    1    )  +  (  1  −  a  )  f  (      x    2    )for all       x    1    ,      x    2    ∈  D and all   a  ∈  (  0  ,  1  ).A simple characterisation is that the hypograph  H  (  f  )  =  {  (  x  ,  y  )  :    y  ≤  f  (  x  )  }is a convex set.If   f is differentiable,   f is concave if and only if  f  (  x  )  ≤  f  (      x    0    )  +  (  x  −      x    0    )      f    ′    (      x    0    )for all   x  ,      x    0    ∈  D.If   f is twice differentiable,   f is concave if and only if      f          ′      ′        (  x  )  ≤  0for all   x  ∈  D.

What are some characterizations of concave functions of real functions of a single real variable?

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