Let $f:R\rightarrow R$ be a concave function with a unique and finite maximum. Let $$g(x, \beta) = \beta f(\alpha \cdot x) + (1-\beta) f((1-\alpha) \cdot x), $$ where $\alpha \in [0,1/2]$ and $\beta \in [0,1/2]$. Furthermore, let $g^\*(\beta) = \max\_{x} g(x,\beta)$. 


I am trying to find conditions for $f$ such that 
$$
\frac{d}{d\beta} g^\*(\beta) \leq 0
$$
for $\alpha \in [0,1/2]$ and $\beta \in [0,1/2]$. Will the inequality above be satisfied if $f$ is concave with a unique and finite maximum? What conditions do I need? Does the inequality hold when the maximizer of $f$, i.e. $\arg \max\_x f(x)$, is positive?



Monotonicity of maximum of convex combination of two scaled concave functions
Let $a:=\alpha\in(0,1/2)$ and $b:=\beta\in(0,1/2)$. Take any positive real $A$ and any real $B$, and let 
$$f(x):=\min[x,(1+A)B-Ax]
$$
for real $x$. Then the function $f$ is concave with a unique and finite maximum (at $x=B$), and 
$$g^\*(b)=g\Big(\frac B{1-a},b\Big)=
\Big(1-\frac{1-2 a}{1-a}\,b\Big) B
$$
if $a$ and $b$ are small enough so that 
$$\frac1{A\_\*(a,b)}<A<A\_\*(a,b):=\frac{(1-a)(1-b)}{ab}, 
$$
and then obviously 
$$\frac{dg^\*(b)}{db}=-\frac{1-2 a}{1-a}\, B>0 
$$
if $B<0$. 
So, your desired inequality does not always hold for concave functions $f$ with a unique and finite maximum. 




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Since, in the above example, the negative slope $-A$ can be any negative real number, it seems unlikely that there exists a simple and good enough sufficient condition for your desired inequality to hold. 


