TY - JOUR AU - Trukhan, Yu.S. AU - Mulyava, O.M. PY - 2018/01/02 Y2 - 2024/03/29 TI - On meromorphically starlike functions of order $\alpha$ and type $\beta$, which satisfy Shah's differential equation JF - Carpathian Mathematical Publications JA - Carpathian Math. Publ. VL - 9 IS - 2 SE - Scientific articles DO - 10.15330/cmp.9.2.154-162 UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/1459 SP - 154-162 AB - <p>According to M.L. Mogra, T.R. Reddy and O.P. Juneja an analytic in ${\mathbb <span lang="en-US">D_0}=\{z: 0&lt;|z|&lt;1\}$</span> function $f(z)=\frac{1}{z}+\sum_{n=1}^{\infty}f_n z^{n}$ is said to be meromorphically starlike of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$ if $|zf'(z)+f(z)|&lt;\beta|zf'(z)+(2\alpha-1)f(z)|, \, z\in {\mathbb D_0}. $ Here we investigate conditions on complex parameters $\beta_0,\,\beta_1,\,\gamma_0,\,\gamma_1,\,\gamma_2$, under which the differential equation of S. Shah $z^2 w''+(\beta_0 z^2+\beta_1 z) w'+(\gamma_0 z^2+\gamma_1 z+\gamma_2)w=0$ has meromorphically starlike solutions of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$. Beside the main case $n+\gamma_2 ot=0, \, n\ge 1,$ cases $\gamma_2=-1$ and $\gamma_2=-2$ are considered. Also the possibility of the existence of the solutions of the form $f(z)=\frac{1}{z}+\sum_{n=1}^{m}f_n z^{n}, \, m\ge 2,$ is studied. In addition we call an analytic in ${\mathbb D_0}$ function $f(z)=\frac{1}{z}+\sum_{n=1}^{\infty}f_n z^{n}$ meromorphically convex of order $\alpha\in [0,1)$ and type $\beta\in (0,1]$ if $|zf''(z)+2f'(z)|&lt;\beta|zf''(z)+2\alpha f'(z)|, \, z\in {\mathbb D_0}$ and investigate sufficient conditions on parameters $\beta_0,\,\beta_1,\,\gamma_0,$ $\gamma_1,\,\gamma_2$ under which the differential equation of S. Shah has meromorphically convex solutions of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$. The same cases as for the meromorphically starlike solutions are considered.</p><p> </p> ER -