The second conjecture in this blog is related to my recent work on Lévy processes.

Suppose that x and y are positive reals, and is an increasing function. Then, whenever the integral

makes sense, it is equal to .

Although the statement is quite elementary, I fail to find any elementary proof. Even for simple , except and perhaps , the conjecture seems to be highly non-trivial.

I showed this conjecture to several people. Let me cite here two comments: ‘*It is the worst formula for I have ever seen!*’, and ‘*Come on, this must be either elementary or false*’.

The conjecture originated in my recent preprint, while I was studying spectral theory of the transition semigroup of a subordinate Brownian motion killed at the time of first exit from the half-line. Every subordinate Brownian motion corresponds to a Bernstein function . For the theory developed in the article, I need to be a *complete* Bernstein function and satisfy the above conjecture.

In my preprint I prove the conjecture for , where , and a class of complete Bernstein functions, including for example and for . Unfortunately, the argument is rather involved. As far as I know, the problem is open for all other functions, even for with .

One can easily verify the conjecture numerically for various . Try playing around with the following *Mathematica* code:

psi[t_] = t^2 + Exp[-t] + Sin[Sqrt[t]];
dpsi[t_] = D[psi[t], t];
x = 3;
y = 2;
2/Pi NIntegrate[
Hold[
t^2 (x + y)/(x^2 + t^2)/(y^2 + t^2)
Exp[1/Pi
NIntegrate[
(x/(x^2 + s^2) + y/(y^2 + s^2))
Log[dpsi[t^2] (t^2 - s^2)/(psi[t^2] - psi[s^2])],
{s, 0, Infinity}
]
]
],
{t, 0, Infinity}, WorkingPrecision -> 50]

I used this code to convince myself that the conjecture is true for all complete Bernstein functions. In fact I tried some other functions (that is, not complete Bernstein ones) only to check if the code works correctly. I was quite surprised to see that it works in the general case. Or perhaps I was just not smart enough to find a good counterexample?

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