We showed characterizations of chaotic order via Kantorovich inequality in our previous paper. Recently as a nice application of generalized Furuta inequality, Furuta and Seo showed an extension of one of our results and a related result on operator equations. In this paper, by using essentially the same idea as theirs, we shall show further extensions of both their results and the following our another previous result which is a characterization of chaotic order via Specht's ratio. ``\textit{Let $A$ and $B$ be positive invertible operators satisfying $MI\geq A\geq mI>0$. Then $\log A\geq \log B$ is equivalent to $M_{h}(p)A^p\geq B^p$ holds for all $p>0$, where $h=\frac{M}{m}>1$ and} $$ M_h(p)=\frac{h^{\frac{p}{h^p-1}}}{e\log h^{\frac{p}{h^p-1}}}.\mbox{''} $$