I'm reading the section about analytic solutions in Partial Differetial Equation of Evans. In the book, the theorem of Cauchy–Kowalevski which states the local existence of analytic solutions is proved by analytic majorization. Below is the description of this proof in wikipedia.
Both sides of the partial differential equation can be expanded as formal power series and give recurrence relations for the coefficients of the formal power series for f that uniquely determine the coefficients. The Taylor series coefficients of the Ai's and b are majorized in matrix and vector norm by a simple scalar rational analytic function. The corresponding scalar Cauchy problem involving this function instead of the Ai's and b has an explicit local analytic solution. The absolute values of its coefficients majorize the norms of those of the original problem; so the formal power series solution must converge where the scalar solution converges.
I'm looking for a proof of this theorem by direct estimate rather than analytic majorization, which means we estimate the coefficients of the formal power series and deduce the convergence of the series directly. I think by direct estimate we can understand this theorem deeper and derive more precise results. I have found no such proof on Internet, and I would be very grateful if anyone could give me a reference.