Generalized f-mean

A generalized f-mean is like a secret code that helps us understand how to combine different numbers. It's like when you and your friends play a secret game and agree to add up all the numbers you say, but first, you use a secret rule to change them.

Let's say your secret rule is "f(x) = x^2" (that means you will replace each number with its square, like changing 2 to 4 and 3 to 9). If you want to combine the numbers 1, 2, and 3 using the f-mean, first you need to apply the rule to each number, and then average them.

So, for 1, it becomes 1^2 = 1. For 2, it becomes 2^2 = 4. And for 3, it becomes 3^2 = 9. Now, you add up all these new numbers (1 + 4 + 9 = 14) and divide by how many numbers you started with (3). Therefore, the f-mean of 1, 2, and 3 using f(x)=x^2 is 14/3=4.67.

The great thing about generalized f-mean is that you can use any secret rule (not just f(x) = x^2). You can use f(x) = x^(1/2) which means you take the square root of each number, or f(x) = x^(1/3) which means you take the cube root of each number, and so on.

So next time you want to combine some numbers with your friends, you can use a secret code and the f-mean to make it even more fun and mysterious!