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There are very many different proofs of Picard's theorem, and some of them are really "simple". (Picard's original proof occupies about 2 lines, using the things already known at that time). None of t …

"The number of zeros of identity" is the strange expression: this number is always $0$ or $1$. Assuming that $f$ is analytic in an open set containing $C$ and its interior region, and has no zeros on …

An explicit (but quite complicated) answer can be found here: MR0224803 Kuzʹmina, G. V. Estimates of the transfinite diameter of a certain family of continua and covering theorems for schlicht functio …

One motivation is mentioned in Yamanoi's paper where the second main theorem for small functions is proved: a theorem of Picard says that if we have meromorphic solutions $x(z),y(z)$ of $F(x,y)=0$, wh …

Every closed set $F$ on the unit circle is the set of singularities of some analytic function. Take a countable dense subset $z_k$ of $F$ and then choose positive $a_k$ so small that the series $$f(z) …

To complement the answer of Alex Degtyarev, the answer to the original question is "no", and a necessary and sufficient condition is quite complicated. It can be written in various forms, see, for exa …

$L(u)$ can be the whole domain. In dimension $1$, take a dense countable set $\{ z_k\}$ and consider the (pluri) subharmonic function $\sum_k a_k\log|z-z_k|$, where $a_k>0$ tend to zero sufficiently …

Yes you can. First of all the condition $|f(0)|\geq 1$ is irrelevant: you can always multiply your function on a positive constant. Take an entire function $F$ for which the set $\{ z:|F(z)|>1\}$ is …

Under just slightly stronger condition, namely that $\log|\phi(re^{i\theta})|\to 0$ in $L^1$, as $r\to 2$ and $r\to 1/2$, the answer is "no". One (real) condition must be satisfied, and this conditio …

The book of Goldberg and Ostrovskii MR2435270 contains several examples of functions whose deficient value is not asymptotic. And in fact there are such functions without asymptotic values at all. But …

$V$ is pluripolar. Let $v$ be a plurisubharmonic function which is $-\infty$ on $V$. Then $\phi+\epsilon v$ is plurisubharmonic for $\epsilon>0$ (the definition of plurisubharmonic function is easily …

Another exposition of Ostrowski's result in English is here: arXiv:0710.1281 and here arXiv:1208.0779.

This function is only piecewise real-analytic. It is not difficult to construct examples when it is discontinuous, no matter how you choose it, and each interval of analyticity is bounded. All details …

The simplest example is constructed using Mittag-Leffler functions which are bounded outside a the sector $|\arg z|<\alpha$. They are defined by the integral $$f(z)=\int_\gamma\frac{e^{\zeta^{\alpha}} …

Describing the set $\{ e^f+e^g:f,g\in H(C)\}$ is difficult, and probably it cannot be described in any reasonable form. However many conditions can be given for a function $h$ not to be of this form. …