How can I prove that this 2D expression is always less than 1 for n≠β?

[Karthiksrao]

Hello,

While analysing the asymptotic value of a ratio of a bessel and a hankel function, I reduced it to something of the form

[(1 + β/n)^ n * (1 + n/β)^ β] / 2^(n+β) ; n and β are integers and greater than 1

how do I show that the above expression is always less than 1, for n≠β. When n=β, the above expression becomes equal to 1.

Or relatedly, if I have to find the line of maximum for a 2D expression given above (for varying n and β), how do I go about ?

Thanks!

 

[verty]

Hello,

While analysing the asymptotic value of a ratio of a bessel and a hankel function, I reduced it to something of the form

[(1 + β/n)^ n * (1 + n/β)^ β] / 2^(n+β) ; n and β are integers and greater than 1

how do I show that the above expression is always less than 1, for n≠β. When n=β, the above expression becomes equal to 1.

Or relatedly, if I have to find the line of maximum for a 2D expression given above (for varying n and β), how do I go about ?

Thanks!

I've played around with this a little. This formula is symmetric in β,n with two variables but as mentioned, β = n is enough to give the maximum value.

Let x = 1 + n/β, y = n + β.
Thus $\frac{y}{x} = β, \frac{y(x-1)}{x} = n$ and x > 1, y ≥ 2.

The formula simplifies to
$[ \frac{1}{2} x (x-1)^{\frac{1}{x} - 1} ]^y$

Y is irrelevant here, it won't affect which x gives the maximum. Discarding y, the derivative of what remains has numerator
$-(x-1)^{\frac{1}{x}} ln(x-1)$

The exponential part is never 0, therefore x = 2 is the only stationary point. I hope this is not the best way to show this.

 

[dextercioby]

Your question is from calculus, or mathematical analysis if you prefer. Functional analysis is built on point set topology and is an abstractization of calculus.