Cauchy estimates for mixed derivatives on a polydisc

Given: A holomorphic $f$ on ${\mathrm{\Delta }}_{\rho }(a)$ and a polyradius $r$ with $r_k<{\rho }_k$; ${\mathbb{C}}^m$ is read through Complex $m$-space and its real coordinate dictionary.

For $f$ continuous and separately holomorphic on ${\mathrm{\Delta }}_{\rho }(a)$ and $r_k<{\rho }_k$, the iterated-integral coefficients satisfy $\mathrm{\mid }{c}_{\alpha }\mathrm{\mid }\le M{\prod }_{k<m}{r}_k^{-{\alpha }_k}$ with $M={\sup }_{{\mathrm{\Gamma }}_r(a)}\mathrm{\mid }f\mathrm{\mid }$, and $f={\sum }_{\alpha }{c}_{\alpha }(z-a{)}^{\alpha }$ on ${\mathrm{\Delta }}_r(a)$ (A continuous separately holomorphic function is the sum of an absolutely convergent power series with Cauchy-integral coefficients on every smaller polydisc).

Every iterated complex partial derivative of a holomorphic function exists and is holomorphic (Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic). For coefficient families satisfying the geometric polyradius bound, the power-series representation about a fixed centre is unique and its coefficients are $c_{\alpha }={\mathrm{\partial }}_z^{\alpha }f(a)\mathrm{/}\alpha !$ (The coefficients of a convergent multi-indexed power series are its derivative coefficients, hence unique).

A holomorphic function of several variables is continuous and separately holomorphic (A holomorphic function of several variables is continuous and separately holomorphic).

If $\mathrm{\mid }g\mathrm{\mid }\le K$ on the trace of a rectifiable contour $\gamma$, then $\mathrm{\mid }{\int }_{\gamma }gdz\mathrm{\mid }\le KL(\gamma )$ (ML estimate: a contour integral is bounded by a supremum bound times path length), and the once-traversed circle of radius $r^{\mathrm{\prime }}>0$ has length $2\pi r^{\mathrm{\prime }}$ (Every circle has circumference 2 pi r and circumference-to-diameter ratio pi).

For $f$ holomorphic on $D(a^{\mathrm{\prime }},R)$, $0<r^{\mathrm{\prime }}<R$ and $\mathrm{\mid }f\mathrm{\mid }\le K$ on the circle $\mathrm{\mid }\zeta -a^{\mathrm{\prime }}\mathrm{\mid }=r^{\mathrm{\prime }}$, one has $\mathrm{\mid }{f}^{(n)}(a^{\mathrm{\prime }})\mathrm{\mid }\le n!K\mathrm{/}{r}^{\mathrm{\prime }n}$ (Cauchy's inequalities bound every derivative by a boundary bound on a compactly contained circle).

${\mathrm{\Gamma }}_r(a)$ is the set of points with $\mathrm{\mid }{z}_k-a_k\mathrm{\mid }=r_k$ for every $k<m$, and for $m\ge 2$ it is a proper subset of the topological boundary of ${\bar{\mathrm{\Delta }}}_r(a)$ (Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$).

$\alpha !={\prod }_{k<m}{\alpha }_k!$ ($C^k$ maps and multi-index derivative notation in Euclidean space) and $\alpha !\mathrm{\ne }0$ (The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$); negative integer powers need a nonzero base (Integer powers in the complex field).