On derivatives and higher-order derivatives of chromatic polynomials

Let $G$ be a graph of order $n$ with maximum degree $\Delta$, and let $P(G,x)$ denote its chromatic polynomial. We investigate several properties of $P(G,x)$ related to its derivatives and higher-order derivatives. First, we study the monotonicity of $P(G,x)/x^n$. Dong proved that $(x-1)^nP(G,x)\geq x^nP(G,x-1)$ for all real $x\geq n$. In particular, taking $x=n$ establishes the Bartels-Welsh ``shameful conjecture" that $P(G,n)/P(G,n-1)>e$. Fadnavis later showed that the same inequality holds for all real $x\geq 36\Delta^{3/2}$. We improve this bound by proving that it also holds for all real $x\geq 10\Delta^{3/2}$. We then consider a conjecture of Dong, Ge, Gong, Ning, Ouyang, and Tay asserting that $\frac{d^k}{dx^k} \bigl( \ln[(-1)^n P(G, x)] \bigr) < 0$ for all $k \geq 2$ and $x \in (-\infty, 0)$. We establish this conjecture for all $k \geq 2$ and $x\leq -3.01\Delta k$.