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The following classical result of Privalov (1919) gives a partial answer to OP's original question : Let $E$ be a zero measure subset of $\mathbb{T}$. Then there exists a nonzero bounded analytic fu …

A detailed study of Cauchy integrals and a proof of the result are in Complex Variables, M.J. Ablowitz, A.T. Fokas, Cambridge University Press, Chapter 7 Riemann-Hilbert problems, Section 7.2 p.518 …

For unrestricted polynomials of a given degree $n$, there is no lower bound. Indeed, consider $$ P(z)=cz^n+1, $$ with $|c|$ small. Then $$ \frac{\|P'\|_\infty}{\|P\|_\infty}=\frac{|c|n}{1+|c|}, $$ whi …

Concerning the other coefficients $c_k$, Branan [1] showed that $(k+1)c_{k+1}=(n-k)\bar c_{n-k}$ is a necessary condition for univalence in $|z|<1$ if $c_1=1$ and $c_n=1/n$. For univalent polynomials …

E.C. Titchmarsh, The theory of functions, Oxford University Press Section 4.6 Hadamard multiplication theorem, p.158

Since the Cantor set $K$ has Hausdorff dimension $\log2/\log 3<1$, it is a removable set for bounded analytic functions, and so, as you say, there is no bounded analytic function outside of $K$. But i …

Consider the function $$ f(z)=\exp\left(-\frac{1}{(1-z)^{2}}\right), $$ which is a classical example of a function analytic in the unit disk $\mathbb{D}$, with radial limits everywhere on the unit cir …

Yes, indeed, a multi-dimensional version of Pringsheim Theorem holds true. If a power series $\sum_\alpha c_\alpha z^\alpha$ has real, nonnegative coefficients $c_\alpha$, then the series is singular …

There is indeed a relation between $\rho$ and the modulus of continuity $\omega_{f}$ of $f$ on $[-1,1]$ which is obtained via the rate of polynomial approximation to $f$ on $[-1,1]$. Denote by $E_{n}( …

1) No, in general, convergence does not hold because of the presence of spurious poles. A typical result, see [1], is Theorem. Let $(n_{\nu})_{1}^{\infty}$ be a sequence of positive integers satisfyi …

The property that the zeros of the derivative of a polynomial $P$ lie in the convex hull of the zeros of $P$ is usually called the Gauss-Lucas theorem. About question 2), the algebra of entire functi …

This is the main result in a paper by J.L. Walsh from 1927 (in german): J.L. Walsh, Über die Entwicklung einer Funktion einer komplexen Veränderlichen nach Polynomen. Math. Ann. 96 (1927), no. 1, 437– …

An elementary proof (~ 1 page) for the more general case of Riesz potentials in $\mathbb{R}^n$ (hence for the logarithmic potential or Green potential in $\mathbb{C}$), and for more general exponents …

The answer is yes : If the Lelong number of a plurisubharmonic function $\varphi$ at a point $a$ satisfies the condition $\nu(\varphi,a) < 2$, then the function $e^{-\varphi}$ is locally integrable w …

Yes, indeed there is. Explicit expressions for multipoint Padé approximants to the exponential (and power) function at points $z=0,\ldots,m+n$, were given in A. Zhedanov, Explicit multipoint rati …