Strict concavity with respect to one argument of a concave homogeneous function?

Question

Let $f(x,y)$ be a concave, strictly increasing function that is also homogenous of degree one. Is it true that $f$ is strictly concave with respect to one of its arguments? I.e. do we have $$ f(x,ay+(1-a)y')>af(x,y)+(1-a)f(x,y') $$ For instance, this is true in the case $f(x,y)=x^by^{1-b}$ with $x\geq 0$, $y\geq 0$ and $0< b< 1$. Is that result more general?