Abstract
We study the interaction between coefficient and solution conditions for complex linear differential equations in the unit disk within the context of normal families and corresponding families of differential equations. In addition, we consider this interaction within the context of normal functions in terms of Noshiro. Consideration of families of differential equations introduces a new perspective for studying normality. Consequently, sharper results are found than in previous studies involving normal functions within the context of one differential equation.
Highlights
IntroductionThe interaction between coefficient conditions and solution conditions for linear differential equations in the unit disk D = {z ∈ C : |z| < 1} has been a topic of many investigations, including [1,2,3,4,5,6,7]
The interaction between coefficient conditions and solution conditions for linear differential equations in the unit disk D = {z ∈ C : |z| < 1} has been a topic of many investigations, including [1,2,3,4,5,6,7]. Instead of studying this interaction within the context of one differential equation as in previous works, we look at this interaction within the setting of a family of differential equations with a corresponding family of coefficients and family of solutions to the differential equations
In order to motivate the meromorphic results involving normal families, we present a sharp improvement of a result by Fowler [14] concerning normal meromorphic functions in the context of one differential equation
Summary
The interaction between coefficient conditions and solution conditions for linear differential equations in the unit disk D = {z ∈ C : |z| < 1} has been a topic of many investigations, including [1,2,3,4,5,6,7]. By looking at the interaction between coefficients and solutions within the setting of families of differential equations in the unit disk, we obtain sharper results than were found in the setting of one differential equation. In order to motivate the meromorphic results involving normal families, we present a sharp improvement of a result by Fowler [14] concerning normal meromorphic functions in the context of one differential equation
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