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

Abstract

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.

GROWTH OF \(\varphi\)–ORDER SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS ON THE COMPLEX PLANE

“…Many previous results due to Chyzhykov-Semochko, Belaïdi, Cao-Xu-Chen, Kinnunen have been extended. Now, it is interesting to study the growth of meromorphic solutions of such equations by using the concept of (α, β)-order called the generalized order introduced by Sheremeta [20], see the recent paper of Mulyava-Sheremeta-Trukhan [17].…”

Growth properties of solutions of complex differential equations with entire coefficients of finite (alpha,beta,gamma)-order

“…In [25], Mulyava et al have used the concept of (α, β)-order or generalized order of an entire function in order to investigate the properties of solutions of a heterogeneous differential equation of the second order and obtained several interesting results. For details about (α, β)-order one may see [25,28].…”

Closeness-to-convexity of solutions of a second order nonhomogeneous differential equation

“…Íàéïåðøå çàóâàaeèìî, ùî [6] àíàëiòè÷íà â äåÿêîìó îêîëi ïî÷àòêó êîîðäèíàò ôóíêöiÿ (1) ¹ ðîçâ'ÿçêîì äèôåðåíöiàëüíîãî ðiâíÿííÿ (3) òîäi i òiëüêè òîäi, êîëè (4)…”