This post concerns the monotonicity of

on with . The two cases and are of special because these are always mentioned in many textbooks as

Clearly, is monotone increasing with respect to . Hence we are left with the monotonicity of with respect to . From the above discussion, the function is monotone increasing with respect to when . When , as the function is monotone increasing with respect to , we deduce that the function is monotone increasing with respect to . Hence, we are left with the case . To study this problem, we examine with respect to .

**Derivative of **. It is easy to get

Hence the sign of is determined by the sign of

on .

Obviously, when we have

and when we have

Consequently, we recover the monotonicity of in the two cases .

**Derivative of **. It is easy to get

Hence the sign of is determined by the sign of

on . Clearly we have three possible cases: , , and . For the case , this is easy to handle because which immediately implies

everywhere on . For the case , we note that if . Hence putting the above two cases together we obtain

Once we know precisely the sign of , we can examine the sign of . This part is a little bit sticky as we are dealing with both the log and fractional functions. Again, we separate the two cases and .

**Estimate of when **. Via Taylor expansion up to two terms we obtain

Hence for large enough, we easily get

It is crucial to realize that the coefficient of the leading term does not vanish. Hence together with the monotonicity of we have just shown that

- if , then is monotone increasing with and
- if , then is monotone decreasing on with .

**Estimate of when **. The above argument does not work since . Hence, in this case, we need an extra work. Still by Taylor expansion up to three terms we obtain

Hence for large enough, we easily get .

**Conclusion**. From the above computation, we know that

- if , then is monotone decreasing on ;
- if , then is monotone increasing on ;
- if , then is monotone increasing on .

In fact, the last two conclusions can be combined together.

**Application**. Using the monotonicity in , we easily get

In addition, using the monotone decreasing in and the fact that

provided , we always have

Notice that generally the inequities

do not hold if is small. For example

However, the above inequalities hold starting from large .

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