How to prove that there exist a concave function and γ ∈ [ 0 , 1 ]and some other numbers which satisfy an inequality

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How to prove that there exist a concave function and   γ  ∈  [  0  ,  1  ]and some other numbers which satisfy an inequality

Shyla Maldonado · Accepted Answer

Consider the function   h  (  x  )  =  U  (  x  +  c  )  −  U  (  x  ). Its derivative is   U  ′  (  c  +  x  )  −  U  ′  (  x  ). Because   U  (  x  ) in concave then   U  ′  ′  &amp;lt;  0, which means that   U  ′ is decreasing and hence   h  ′ is negative. Hence   h  (  x  ) is decreasing.Your inequality follows directly:   h  (  a  )  &amp;lt;  h  (  b  ) for   a  &amp;gt;  bWe can also prove a more general result without resorting to derivatives. Let   F be a concave function in an interval and let   x  &amp;lt;  y  ≤  z  &amp;lt;  w be points in this interval. Then the following holds:            F      (      y      )      −      F      (      x      )              y      −      x        ≥            F      (      w      )      −      F      (      z      )              w      −      z      To prove this write   y as a linear combination of   x and   w thus:  y  =            w      −      y              w      −      x        x  +            y      −      x              w      −      x        wThis implies:                    F        (        y        )                            ≥                              w            −            y                                w            −            x                          F        (        x        )        +                              y            −            x                                w            −            x                          F        (        w        )                                          =        F        (        x        )        +                              y            −            x                                w            −            x                          (        F        (        w        )        −        F        (        x        )        )                                          =        F        (        x        )        +                              y            −            x                                w            −            x                          (        F        (        w        )        −        F        (        y        )        +        F        (        y        )        +        F        (        x        )        )            This is equivalent to:            F      (      y      )      −      F      (      x      )              y      −      x        ≥            F      (      w      )      −      F      (      y      )              w      −      y      In the same way we can prove that (note this goes in the other direction):            F      (      w      )      −      F      (      y      )              w      −      y        ≤            F      (      w      )      −      F      (      x      )              w      −      x      Combining the last two inequalities leads to our result:            F      (      y      )      −      F      (      x      )              y      −      x        ≥            F      (      z      )      −      F      (      y      )              z      −      y        ≥            F      (      w      )      −      F      (      z      )              w      −      z

How to prove that there exist a concave function and gamma in [0,1] and some other numbers which satisfy an inequality

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