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Added the very last phrase about the case when the maximizer of f is positive.
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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?

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?

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?

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ACopt
ACopt
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Source Link
ACopt
ACopt
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Monotonicity of maximum of convex combination of two scaled concave functions

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?