In the paper, we take up a new method to prove the following result. Let $f$ be a meromorphic function in the complex plane, all of whose zeros have multiplicity at least $k+1$ ($k\geq 2$) and all of whose poles are multiple. If $T(r,\sin z)=o\{T(r,f(z))\}$ as $n\rightarrow\infty$, then $f^{(k)}(z)-\sin z$ has infinitely many zeros.

Given a set $E=(0, \infty)$, the spherical maximal operator $\mathcal{M}$ associated to the parameter set $E$ is defined as the supremum of the spherical means of a function when the radii of the spheres are in $E$. The aim of this paper is to study the following inequality \begin{equation} ∫_{\mathbf{R}^{n}} (\mathcal{M}f(x))^{p} φ(x) dx ≤ B_{p} ∫_{\mathbf{R}^{n}} |f(x)|^{p} φ(x) dx, \label{Lb1} \end{equation} holds for $p > \frac{2n}{n-1}$ with the continuous spherical maximal operator $\mathcal{M}$ and where the nonnegative function $\phi$ is in some weights obtained from the $A_{p}$ classes. As an application, we will get the boundedness of vector-valued extension of the spherical means.