Positive majorants dominate the Cauchy recursion

Expanding each denominator of G gives a product of two multinomial series. A monomial of positive total degree q has coefficient $M{r}^{-q}{\rho }^{-\mathrm{\ell }}$ times two positive integer multinomial factors, where $\mathrm{\ell }$ is its t-degree. This is at least $M{r}^{-q}$. The constant coefficient is zero, matching F(0)=0. Thus F is majorised by G.

To compute ${\mathrm{\partial }}_x^{\alpha }{\mathrm{\partial }}_t^{q+1}{u}_j(0)$, apply ${\mathrm{\partial }}_x^{\alpha }{\mathrm{\partial }}_t^q$ to $F_j(t,x,u,D_{x}u)$. Iterating the product and chain rules produces a finite sum of products of derivatives of F at zero and derivatives ${\mathrm{\partial }}_x^{\beta }{\mathrm{\partial }}_t^{\mathrm{\ell }}{u}_k(0)$ with $\mathrm{\ell }\le q$, multiplied by nonnegative integers. Indeed a spatial differentiation only increases beta; a time differentiation can increase ell at most by one, and there are only q such differentiations. The inner arguments at the origin are zero, since u and its initial spatial gradient vanish. The identical polynomial computation for U evaluates derivatives of G at the same zero argument by its stipulated origin conditions.

For normal order zero, all derivatives of the zero trace of u vanish, whereas the corresponding trace derivatives of U are nonnegative. Suppose the comparison holds at every normal order at most q, for all tangential indices and all components. In each finite product from step 1.2 take absolute values, apply those bounds and the derivative bounds from step 1.1, and then sum. Nonnegative polynomial coefficients give the comparison at normal order q+1 and simultaneously its nonnegativity for U. Induction proves every derivative bound; division by the same positive factorials gives $u_j\ll U_j$.