$\frac{\log (1-x)}{\log (1-y)}$的估计

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C. V. Durell, A. Robson - Advanced Trigonometry
(page 74) Evercise IV. g.
$$\forall x,y\in(0,1):\quad\frac{x(1-y)}{y}<\frac{\log (1-x)}{\log (1-y)}<\frac{x}{y(1-x)}$$

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$\sup_{x\in(0,1]}\frac{\log(1 - x)}x=-1$ at $x=0$
$\inf_{y\in(0,1]}\frac{(1-y)\log(1 - y)}y=-1$ at $y=0$
$$\frac{(1-y)\log(1 - y)}y<-1<\frac{\log(1 - x)}x$$
即
$$\frac{\log(1 - x)}{\log(1 - y)}>\frac{x(1-y)}{y}$$
把$x,y$交换, 取倒数:
$$\frac{\log(1 - x)}{\log(1 - y)}<\frac{x}{y(1-x)}$$