We study extension of $p$-trigonometric functions ${sin}_p$ and ${cos}_p$ to complex domain. For $p=4,6,8,\cdots$, the function ${sin}_p$ satisfies the initial value problem which is equivalent to (*) $$-{\left(u^{\text{'}}\right)}^{p-2}{u}^{\text{'}\text{'}}-u^{p-1}=0,u\left(0\right)=0,u^{\text{'}}\left(0\right)=1$$ in $ℝ$. In our recent paper, Girg, Kotrla (2014), we showed that ${sin}_p\left(x\right)$ is a real analytic function for $p=4,6,8,\cdots$ on $(-{\pi }_p/2,{\pi }_p/2)$, where ${\pi }_p/2={\int }_0^1{(1-s^p)}^{-1/p}$. This allows us to extend ${sin}_p$ to complex domain by its Maclaurin series convergent on the disc $\{z\in ℂ:|z|<{\pi }_p/2\}$. The question is whether this extensions ${sin}_p\left(z\right)$ satisfies (*) in the sense of differential equations in complex domain. This interesting...

We investigate the growth and fixed points of meromorphic solutions of higher order linear differential equations with meromorphic coefficients and their derivatives. Our results extend the previous results due to Peng and Chen.