The $n$th partial sum of an analytic function $f(z)=z+\sum_{k=2}^\infty a_k z^k$ is the polynomial $f_n(z):=z+\sum_{k=2}^n a_k z^k$. A survey of the univalence and other geometric properties of the $n$th partial sum of univalent functions as well as other related functions including those of starlike, convex and close-to-convex functions are presented.
Accumulation points of the sets of real parts of zeros of the partial sums of the Riemann zeta function
It is shown that, for every integer n>2, there exists δ_{n}>0 such that the closure of the set of the real parts of the zeros of the nth partial sum of the Riemann zeta function ζ_{n} contains to the interval [-δ_{n},b^{n}]. b^{n} is the supremum of the real parts of the zeros of ζ_{n}. It is also demonstrated that b^{n} is positive for all n>2. It is also shown that 0 is an accumulation point common to all the sets P_{ζ_{n}} wich are the sets of the real parts of the zeros of ζ_{n}.
Universal partial sums of Taylor series as functions of the centre of expansion
V. Nestorian证明，如果$Ω$是$\mathbb{C}$的一个不包含$0$的单连通子集，并且$S(Ω)$是所有函数$f\in \mathcal{H}(Ω)$的集合，具有集合$\left\{T_N(f)(z)\coloneqq\sum_{n=0}^N\dfrac{f^{(n)}(z)}{n!} (-z)^n : N = 0,1,2,\dots \right\}$在$\mathcal{H}(Ω)$中稠密的性质，则$S(Ω)$是$\mathcal{H}(Ω)$中稠密的$G_δ$集合。在$Ω$是一个不含$0$的开放圆盘$D(z_0,r)$的特殊情况下，我们肯定地回答这个猜想。
