TY - JOUR
AU - Mulyava, O.M.
AU - Sheremeta, M.M.
AU - Trukhan, Yu.S.
PY - 2019/12/31
Y2 - 2024/04/22
TI - Properties of solutions of a heterogeneous differential equation of the second order
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 11
IS - 2
SE - Scientific articles
DO - 10.15330/cmp.11.2.379-398
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/2116
SP - 379-398
AB - <p>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 $\gamma_2
eq0$ 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.</p>
ER -