In this paper, we investigate the relationship between an entire function of small growth $f$ and its linear differential polynomial $L(f)$ when they share one value by applying value distribution theory and complex oscillation theory. As consequences of the main result we can get the precise form of $f$.

In this paper, we deal with the problem of uniqueness of meromorphic functions that share one finite value CM and obtain some results which improve some theorems given by R. Nevanlinna and R. Bruck and are related to a result of H. X. Yi. Examples are provided to show that our results are sharp.

Let $H(\mathrm{U})$ be the space of analytic functions in the unit disk $\mathrm{U}$ and let $\mathcal{D}=\{\varphi\in H(\mathrm{U}):\varphi(0)=1, \varphi(z)\neq0,z\in\mathrm{U}\}$. For the functions $\phi,\varphi\in\mathcal{D}$ we will determine simple sufficient conditions such that $$\left[\frac{\varphi(z)}{\phi(z)+\frac{1}{\gamma}z\phi'(z)}\right]^{1/\beta} f(z)\prec k(z)\Rightarrow\operatorname{I}_{\phi,\varphi;\beta,\gamma}[f](z) \prec k(z),$$ for all $k\in\mathcal{M'}_{1/\beta}$, where $$\operatorname{I}_{\phi,\varphi;\beta,\gamma}[f](z)= \left[\frac{\gamma}{z^\gamma\phi(z)} \int_0^zf^\beta(t)t^{\gamma-1}\varphi(t) \operatorname{d}t\right]^{1/\beta}$$ and $\mathcal{M'}_{1/\beta}$ represents the class of $1/\beta$-convex functions (not necessarily normalized). In particular, we will give sufficient conditions on $\phi$ and $\varphi$ so that the operators $\operatorname{I}_{\phi,\varphi;\beta,\gamma}$ are averaging operators on certain subsets of $H(\mathrm{U})$. In addition, some particular cases of the main result, obtained for appropriate choices of the $\phi$ and $\varphi$ functions, will also be given.

Using the Newton polygon we prove a factorization theorem for the local polar curves. Then we give some applications to the polar invariants and pencils of plane curve singularities.

The Magnus representation of the Torelli group has been defined in virtue of Fox derivation. The Torelli group is a significant subgroup of the mapping class group of a surface. In this paper, we show some properties of the characteristic polynomials of matrices obtained from the Magnus representation of the Torelli group, which is a class function on the Torelli group.

We give a description of growth at infinity of the gradient of a polynomial in two complex variables near any of its fiber.