Let $f$ and $g$ be two strictly increasing functions such that $%
f:[0,1]\rightarrow \mathbb{R}_{+}$ (with $f(0)=0$) and $g:[2,\infty )\rightarrow \mathbb{R}_{+}$.
Can anyone come up with sufficient conditions that will ensure (i) and (ii)
always hold whenever $0<x_{1}$, $0<x_{2}$, and
$x_{1}+x_{2}\leq 1$, respectively?
\begin{equation*}
\text{(i) }f(x_{1})g(y_{1})+f(x_{2})g(y_{2})\leq f(x_{1}+x_{2})g(y_{1}+y_{2})%
\text{.}
\end{equation*}
\begin{equation*}
\text{(ii) }f(x_{1})g(y_{1})+f(x_{2})g(y_{2})\geq
f(x_{1}+x_{2})g(y_{1}+y_{2})\text{.}
\end{equation*}
One thing I can see is that since both functions are strictly increasing,
(i) is always true if
\begin{equation*}
\text{(a) }f(x_{1})+f(x_{2})\leq f(x_{1}+x_{2})
\end{equation*}
or
\begin{equation*}
\text{(b) }g(y_{1})+g(y_{2})\leq g(y_{1}+y_{2})
\end{equation*}
Is always true. But, this begs the question of what conditions guarantee (a)
and (b)?
2026-03-27 16:28:10.1774628890