Properties of solutions of a heterogeneous differential equation of the second order

  • O.M. Mulyava National University of Food Technologies, 68 Volodymyrska str., 01601, Kyiv, Ukraine
  • M.M. Sheremeta Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine
  • Yu.S. Trukhan Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine
Keywords: differential equation, convexity, starlikeness, close-to-convexity, generalized order, convergence class
Published online: 2019-12-31


Suppose that a power series $A(z)=\sum_{n=0}^{\infty}a_n z^{n}$ has the radius of convergence $R[A]\in [1,+\infty]$. For a heterogeneous differential equation $$ z^2 w''+(\beta_0 z^2+\beta_1 z) w'+(\gamma_0 z^2+\gamma_1 z+\gamma_2)w=A(z) $$ with complex parameters geometrical properties of its solutions (convexity, starlikeness and close-to-convexity) in the unit disk are investigated. Two cases are considered: if $\gamma_2\neq0$ and $\gamma_2=0$. We also consider cases when parameters of the equation are real numbers. Also we prove that for a solution $f$ of this equation the radius of convergence $R[f]$ equals to $R[A]$ and the recurrent formulas for the coefficients of the power series of $f(z)$ are found. For entire solutions it is proved that the order of a solution $f$ is not less then the order of $A$ ($\varrho[f]\ge\varrho[A]$) and the estimate is sharp. The same inequality holds for generalized orders ($\varrho_{\alpha\beta}[f]\ge \varrho_{\alpha\beta}[A]$). For entire solutions of this equation the belonging to convergence classes is studied. Finally, we consider a linear differential equation of the endless order $ \sum\limits_{n=0}^{\infty}\dfrac{a_n}{n!}w^{(n)}=\Phi(z), $ and study a possible growth of its solutions.

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How to Cite
MulyavaO., SheremetaM., TrukhanY. Properties of Solutions of a Heterogeneous Differential Equation of the Second Order. Carpathian Math. Publ. 2019, 11 (2), 379-398.

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