Conjectured decreasing of sequence of integrals of products of Fourier transformations Let $f$ be a real function with the following properties:


*

*$$f(x) \geq 0$$

*$$f(x) = 0 \textrm{ for } x < 0 $$

*$$ \int_0^\infty dx f(x) = 1$$
and define $$\hat{f}(k) = \int_{-\infty}^{\infty}dx \, e^{ikx} f(x) = \int_{0}^{\infty}dx \, e^{ikx} f(x)$$ and $$\hat{g}(k) = \int_0^\infty dx \, e^{ikx} \int_x^\infty dy f(y).$$
Finally, define for $n\geq 2$
$$H_n(f) = \frac{n}{2 \pi}\int_{-\infty}^{\infty} dk \, \hat{g}(-k) (\hat{f}(k))^{n-1}$$
and note $H_n$ is real: complex conjugating its defining equation will change the sign of the arguments of $f$ and $g$, and the original signs return via substitution of $k \to -k$. 

Is $$H_{n+1}(f) < H_n(f) \textrm{ for } n \geq 2 ?$$

That is, I want to check whether $H_n(f)$ is a decreasing function of $n$ for any fixed $f$ of the above properties. For example, one can check with a quick contour integral in the upper half plane that for $f(x) = \lambda e^{-\lambda x}$ that $H_n = \frac{n}{2^{n-1}}$, which enjoys $H_{n+1} < H_{n}$ for $n \geq 2$.

$H_n$ has a nice interpretation as the probability that for $n$ i.i.d. non-negative random variables with probability density distribution function $f$ that the largest is larger than the sum of the rest. Indeed, for $n=2$, for which the probability the largest of the two is larger than the smaller of the two is just 1, we enjoy $H_2 =1$ for any $f$ of the properties specified above, which can be seen by inserting the definition of $g$ in terms of $f$ and integrating on $k$ first. 
I've tested the gamma probability distribution at integer shape parameters and found healthy evidence of $H_{n+1}(f) < H_{n}(f)$ regardless of shape parameter.

I am happy with proofs based off any method, such as equalities and inequalities in Fourier transforms as well as with probabilistic proofs. Counter-examples are also welcome, but I'd appreciate at least a rough statement of additional properties $f$ should hold such that $H_{n+1}(f) < H_n(f)$ is guaranteed. 
 A: Counter-example to the combinatorial interpretation.  
I am very rusty with transforms, so this counter-example assumes the OP's interpretation that "$H_n$ has a nice interpretation as the probability that for $n$ i.i.d. non-negative random variables with probability density distribution function $f$ that the largest is larger than the sum of the rest."
The random variable $X$ is Bernoulli, and $P(X=1) = p, P(X=1000) = 1-p = q$.


*

*$H_3 = Prob(\text{one $1000$ and two $1$'s}) = {3 \choose 1} p^2 q$

*$H_4 = Prob(\text{one $1000$ and three $1$'s}) = {4 \choose 1} p^3 q$
$$H_4 > H_3 \iff 4 p^3 q > 3 p^2 q \iff p > 3/4$$
This example is discrete, so its $f$ requires Dirac deltas, but you can easily replace each Dirac delta with a very tall and narrow peak to make $f$ a function.  (E.g. just add $Uniform(-0.01, +0.01)$ to $X$.)

As to "additional requirements on $f$"...  Here are some pure guesses: 


*

*Perhaps given finite variance, the statement $H_{n+1} \le H_n$ would be true for large enough $n$?  

*Maybe given finite variance, and additionally $f(x)$ decreasing (in $x \ge 0$), the statement $H_{n+1} \le H_n$ would be true for all $n$?

*I wouldn't be surprised if there are examples with infinite variance where the statement is always false...
