  We improve and generalize in several accounts the recent rigorous proof of
convergence of delta expansion - order dependent mappings (variational
perturbation expansion) for the energy eigenvalues of anharmonic oscillator.
For the single-well anharmonic oscillator the uniformity of convergence in
$g\in[0,\infty]$ is proven. The convergence proof is extended also to complex
values of $g$ lying on a wide domain of the Riemann surface of $E(g)$. Via the
scaling relation \`a la Symanzik, this proves the convergence of delta
expansion for the double well in the strong coupling regime (where the standard
perturbation series is non Borel summable), as well as for the complex ``energy
eigenvalues'' in certain metastable potentials. Sufficient conditions for the
convergence of delta expansion are summarized in the form of three theorems,
which should apply to a wide class of quantum mechanical and higher dimensional
field theoretic systems.
