Let's assume we have an increasing production function $f:\mathbb{R^+} \to \mathbb{R^+}$
Now, assume this production function is concave and that the price of input z is fixed (this is a single-input and single output case). I want to show this implies the corresponding cost function $C^f(w,q)$ is convex.
My thoughts:
let $z,z' \in \mathbb{R^+}$ where W.L.O.G $z'>z$ and let $f(z)=q$ and $f(z')=q$
Since $f$ is concave, we take $\alpha \in [0,1]$ $s.t$:
$$f(\alpha z +(1- \alpha)z') \geq \alpha f(z) +(1- \alpha)f(z')$$
let $\alpha z +(1- \alpha)z'=z''\in \mathbb{R^+}$ where by necessity $z\leq z'' \leq z'$
Since cost of $z=w$ is fixed at some $w \in \mathbb{R^+}$
we can rewrite our cost function as $C^f(q)$ which is convex if for $\alpha \in [0,1]$:
$$C(\alpha q +(1- \alpha)q') \leq \alpha c(q) +(1- \alpha)c(q')$$
Now, since $f$ is concave, it cannot be the case that $f$ experiences increasing returns to scale. Then $f$ has constant returns to scale or diminishing returns to scale.
Then:
Since $z''$ is clearly feasible then we have that:
$$C(w,q'') \leq w*z''$$ $$=\alpha w*z + (1- \alpha) w*z'$$ $$= \alpha c(w,q) + (1- \alpha) c(w,q')$$
Does this look correct?