How can I find the eigenfunction of the power mean ?

$\displaystyle k(x,y) = \left(\frac{x^m+y^m}{2}\right)^{1/m}, m<0$

it is positive definite for m<0.

more précisely i want to find the serie of $\displaystyle f_i(x)$ such that :

$\displaystyle \int_0^1 k(x,y)f_i(y)dy = \lambda_i f_i(x)$