  We obtain a generalization of the Picone inequality which, in combination
with the classical Picone inequality, appears to be useful for problems with
the $(p,q)$-Laplace type operators. With its help, as well as with the help of
several other known generalized Picone inequalities, we provide some nontrivial
facts on the existence and nonexistence of positive solutions to the zero
Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = f_\mu(x,u,\nabla
u)$ in a bounded domain $\Omega \subset \mathbb{R}^N$ under certain assumptions
on the nonlinearity and with a special attention to the resonance case
$f_\mu(x,u,\nabla u) = \lambda_1(p) |u|^{p-2} u + \mu |u|^{q-2} u$, where
$\lambda_1(p)$ is the first eigenvalue of the $p$-Laplacian.
