Is it a concave or convex function?

Question

I want to know whether following function is concave or convex $$(1-x)\left[a-\frac{b}{(x^{m}-cx^{m-1})d}\right]$$ where $a,b,c,d$ are all positive and $m>1$. The range of $x$ over which I am interested is from $0<x<1$. Any insights will greatly appreciated. Thanks in advance.

Answer 1

If $c=1$ then $f(x)=a(1-x)+\dfrac{b}{cx^{m-1}}$ is trivially convex. So, an interesting case is $c\ne 1$.

Of course, the term $a(1-x)$ is an affine function which does not violate convexity, so the question is whether or not the function $$g(x)=\frac{x-1}{x^m-cx^{m-1}}$$ is convex (the constants $b,d>0$ do not matter). So we have $f$ is convex $\iff g$ is convex.

My guess is $g$ is convex for $c\ge 1$ while $g$ is not convex for $0<c<1$. In this case there is an asymptote at $c,$ so our function is necesarrily neither convex nor concave. The graph of $g$ for $c=0.9$ and $m=2$ indicates two pieces: one is convex while the other is concave.

The graph above is for $h$ with $m=1.01$ and $c=1.5$. This function is neither convex nor concave.