# Average of Log of a Function: Bounded by 1 and Convex

• deathprog23
In summary, the conversation discusses the average behavior of the log of a function with known properties, including being convex and bounded from below by 1. The person is interested in finding the circumstances in which the average of the log function would be equal to the log of the average, up to a constant term. They also mention that this may depend on the behavior of the function and ask for clarification on the asymptotic properties.
deathprog23
Hello,

I am interested in the average behaviour of the log of a function.

I know the average of the function over the range of interest: $$F = \frac{1}{(b-a)} \int_a^b f(x) dx.$$

I also know that $$f(x)$$ is convex and bounded from below by $$1.$$

I want to know the average $$\frac{1}{(b-a)} \int_a^b \log( f(x) ) dx.$$

In particular, under what circumstances this would be equal to the log of the average, $$\log(F)$$, up to a constant term, if $$F = \frac{1}{(b-a)}$$ and $$(b-a)$$ tends to zero.

Many thanks for any help.

Last edited:
My first observation is $\lim_{b\rightarrow a}\int_a^b f(x) dx/{(b-a)} = f(a).$

Second, I am not sure how the numerator of F remains constant ($\int_a^b f(x) dx$ = 1) when b--->a. I expect $\lim_{b\rightarrow a}\int_a^b f(x) dx$ = 0. Can you explain?

Last edited:
Ah, sorry - I should have explicitly pointed out that $$f(x)=f(b-a,x).$$

In fact, what I'm looking at is the average slope of a function $$g(x),$$ which has range $$[0,1]$$ and domain $$[a,b].$$

Thus $$f(x)=\frac{dg(x)}{dx}$$ and its integral over the domain must give $$1.$$

The asymptotic properties must depend on the behaviour of $$f(x)$$ I suppose, e.g. if $$\limsup_{(b-a)\to 0}f(b)=K f(a)$$ for a constant $$K,$$ then what I ask for holds.

But what if more generally, as I ask, all I know is that $$f(x)\geq 1$$ and is convex? What about other classes of function?

Thanks for the response!

## 1. What does it mean for a function to be bounded by 1 and convex?

When a function is bounded by 1, it means that the output values of the function are always less than or equal to 1. A convex function is one in which the line connecting any two points on the graph of the function always lies above or on the graph. Therefore, a function bounded by 1 and convex means that the output values are always less than or equal to 1 and the graph of the function curves upwards.

## 2. Why is the average of the logarithm of a function bounded by 1 and convex important?

The average of the logarithm of a function bounded by 1 and convex is important because it is often used in optimization problems. In these problems, we seek to find the maximum or minimum value of a function. The average of the logarithm of a function helps us to determine the optimal value of the function in a more efficient way.

## 3. How is the average of the logarithm of a function bounded by 1 and convex calculated?

To calculate the average of the logarithm of a function bounded by 1 and convex, we first take the logarithm of the function and then find the average of these logarithmic values. This can be done by dividing the sum of the logarithms by the number of values. The resulting value will be the average of the logarithm of the function.

## 4. What are the practical applications of the average of the logarithm of a function bounded by 1 and convex?

The average of the logarithm of a function bounded by 1 and convex has practical applications in various fields such as economics, statistics, and engineering. It is often used in optimization problems to find the maximum or minimum value of a function. It is also used in data analysis and modeling to determine the most representative value of a dataset.

## 5. Can the average of the logarithm of a function bounded by 1 and convex be negative?

Yes, the average of the logarithm of a function bounded by 1 and convex can be negative. This can occur when the values of the function are close to 0, causing the logarithmic values to be negative. However, the average itself will still be bounded by 1 and convex, even if it is a negative value.

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