Another view of the Weierstrass theorem

Abstract:

We present two theorems which conclude that polynomials in $z$ and a given continuous function $f$ are dense in all continuous complex valued functions on the closed unit disk. The first theorem requires that $f$ be differentiable and satisfy $\operatorname {Re} {f_{\overline z }} \geqslant |{f_z}|$ in the open disk and also that ${f^{ - 1}}(f(a))$ be countable for each $a$ in $D$. The second theorem requires that $f$ be a class ${C^1}$-function in a neighborhood of the disk satisfying $|{f_{\overline z }}| > |{f_z}|$ almost everywhere and $\operatorname {Re} {f_{\overline z }} \geqslant |{f_z}|$ everywhere inside the disk.