On meromorphically starlike functions of order $\alpha$ and type $\beta$, which satisfy Shah's differential equation

  • Yu.S. Trukhan Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine
  • O.M. Mulyava National University of Food Technologies, 68 Volodymyrska str., 01601, Kyiv, Ukraine
Keywords: meromorphically starlike function of order $\alpha$ and type $\beta$, meromorphically convex function of order $\alpha$ and type $\beta$, Shah's differential equation
Published online: 2018-01-02

Abstract


According to M.L. Mogra, T.R. Reddy and O.P. Juneja an analytic in ${\mathbb D_0}=\{z: 0<|z|<1\}$ 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)|<\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 $n+\gamma_2\not=0, \, n\ge 1,$ 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 $|zf''(z)+2f'(z)|<\beta|zf''(z)+2\alpha f'(z)|, \, z\in {\mathbb D_0}$ 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.

 

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How to Cite
(1)
TrukhanY., MulyavaO. On Meromorphically Starlike Functions of Order $\alpha$ and Type $\beta$, Which Satisfy Shah’s Differential Equation. Carpathian Math. Publ. 2018, 9 (2), 154-162.