On axisymmetric Helfrich surfaces

In this paper we study axisymmetric Helfrich surfaces. We prove the convergence of the formal power series solution of the Euler–Lagrange equation for the Helfrich functional in a neighborhood of its singular point. We also prove the following inequality
$$ \lambda_v R^3+ (c^2+2\lambda_a)R^2-2cR+1\geqslant 0, $$
for a smooth axisymmetric Helfrich surfaces, that homeomorphic to a sphere, where $c$ is the spontaneous curvature of the surface, $\lambda_a$ and $\lambda_v$ are Lagrange multipliers, $R$ is the maximum distance between the axis of rotational symmetry and surface.