Forgot password
 Register account
View 250|Reply 1

[函数] $\frac{\log (1-x)}{\log (1-y)}$的估计

[Copy link]

3200

Threads

7827

Posts

52

Reputation

Show all posts

hbghlyj posted 2023-3-4 03:34 |Read mode
C. V. Durell, A. Robson - Advanced Trigonometry
(page 74) Evercise IV. g.

$$\forall x,y\in(0,1):\quad\frac{x(1-y)}{y}<\frac{\log (1-x)}{\log (1-y)}<\frac{x}{y(1-x)}$$
4ef7ac4bd11373f06ddf0b8bab0f4bfbfaed0472.jpg

3200

Threads

7827

Posts

52

Reputation

Show all posts

original poster hbghlyj posted 2023-3-28 16:53
$\sup_{x\in(0,1]}\frac{\log(1 - x)}x=-1$ at $x=0$
$\inf_{y\in(0,1]}\frac{(1-y)\log(1 - y)}y=-1$ at $y=0$
$$\frac{(1-y)\log(1 - y)}y<-1<\frac{\log(1 - x)}x$$

$$\frac{\log(1 - x)}{\log(1 - y)}>\frac{x(1-y)}{y}$$
把$x,y$交换, 取倒数:
$$\frac{\log(1 - x)}{\log(1 - y)}<\frac{x}{y(1-x)}$$

Quick Reply

Advanced Mode
B Color Image Link Quote Code Smilies
You have to log in before you can reply Login | Register account

$\LaTeX$ formula tutorial

Mobile version

2025-7-15 15:16 GMT+8

Powered by Discuz!

Processed in 0.012569 seconds, 25 queries