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

shiya43

shiya43

Answered question

2022-10-28

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

Answer & Explanation

Shyla Maldonado

Shyla Maldonado

Beginner2022-10-29Added 15 answers

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 < 0, which means that U is decreasing and hence h is negative. Hence h ( x ) is decreasing.
Your inequality follows directly: h ( a ) < h ( b ) for a > b
We can also prove a more general result without resorting to derivatives. Let F be a concave function in an interval and let x < y z < 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 w
This 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

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