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Suppose that $\gamma$ is a Jordan analytic curve on the Riemann sphere, and there exist two rational functions $f$ and $g$ such that $f$ maps $\gamma$ into a circle, and $g$ maps a circle into $\gamma …

Everyone knows that analytic functions $f_1,\ldots,f_n$ are linearly dependent if and only if their Wronski determinant is identically equal to zero. There are several proofs of this, one in Polya-Sze …

Let $\mathbb C$ be the complex plane, $H(\mathbb C)$ the set of all entire functions, and $D(\mathbb C)$ the set of all non-negative divisors in $\mathbb C$. Consider the map $Z:H(\mathbb C)\to D(\ma …

Consider the set $X$ of all analytic functions $f$ in the unit disk $U$ satisfying $f(0)=0, f'(0)\neq 0$. We say that $f\prec g$ if there exists $\phi\in X$ which maps $U$ into itself, and $f=g\circ\p …

Do there exist Jordan analytic curves $J$ in the complex plane $C$, other than circles, with the following property: There exists a harmonic function $u$ in the unbounded component of $C\backslash J$ …

Let $D$ be a circular quadrilateral (that is a Jordan region whose boundary consists of 4 arcs of circles all orthogonal to the unit circle) whose interior angles are all equal to 0, the vertices lie …

Let $\gamma$ be a Jordan analytic curve on the Riemann sphere, and $f$ a rational function of degree at least 2 which maps $\gamma$ onto itself homeomorphically. The following examples of such situat …

Is anything non-trivial known about univalent functions with non-negative coefficients? Let $U$ be the unit disc, and $f$ a univalent (=injective) holomorphic function, $f(0)=0$, $f'(0)>0$. There …

Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$. Everyone who worked with this set knows that there is only one reasonable topology on it: the uniform convergence on …

Recently I was talking to an alien who does not know complex function theory. I was trying to convince her that the set of conformal equivalence classes of smooth embedded tori in $R^3$ is two paramet …

Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$, …

A friend recently told me the following two facts, for which he cannot recall a proof or a reference (but he remembers seeing them in the literature): Let $f$ be a holomorphic function mapping the u …

Let $f$ and $g$ be analytic functions in the unit disk $D$, continuous in the closed disk and locally univalent, $f'(z)\neq 0,\; g'(z)\neq 0,\; z\in D$. Assume that each has only finitely many singula …

Is the following theorem known, or can be easily derived from known results? Consider the differential equation $$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$ where $k>0$ is fixed, $\lambda$ is a large (compl …