Let $f_1,…, f_m$ be quasiconcave functions on a convex subset X on $\mathbb{R}^n$. Let $p_1,…, p_m$ be nonnegative real numbers.
Then, how can I show that $\sum_{i=1}^mp_if_i $ is also quasi concave function?
Similarly, Let $g_1,…, g_m$ be concave functions on a convex subset X on $\mathbb{R}^n$. Let $p_1,…, p_m$ be nonnegative real numbers.
Then, how can I show that $\sum_{i=1}^mp_ig_i $ is also concave function?
—
If $f_i$ is quasiconcave then, $\{u,v\}\in$ domain of f for $0<a<1$, and for $f_i(u)> f_i(v)$ $f_i(au+(1-a)v)\ge f_i(v)$
Similarly, for concavity, $g_i(au+(1-a)v)\ge ag_i(u)+(1-a)g_i(v)$
And then, how can I proceed the proof?
Thank you. All helps will be appreciated.
Let $f_1(x) = \min(0,1-|x|)$, $f_2(x) = f_1(x-2)$. Both are quasiconcave, but $f={1 \over 2} (f_1 + f_2)$ is not. Note that $f(1) = 0 < \min(f(0), f(2)) = 1$.
It is sufficient to show that if $f,g$ are concave then so is $tf+(1-t)g$ for $t \in [0,1]$. This follows almost immediately from the definition. Let $\mu \in [0,1]$, then \begin{eqnarray} tf(\mu x+(1-\mu)y)+(1-t)g(\mu x+(1-\mu)y) &\ge& t (\mu f(x)+(1-\mu) f(y)) + (1-t) (\mu g(x)+(1-\mu) g(y)) \\ &=& \mu(t f(x)+(1-t)g(x)) + (1-\mu) (t f(y)+(1-t)g(y)) \end{eqnarray} and so $tf+(1-t)g$ is concave.