On entire solutions with a two-member recurrent formula for Taylor’s coefficients
of linear differential equations

Ya. S. Mahola

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	On entire solutions with a two-member recurrent formula for Taylor’s coefficients of linear differential equations
Author	Ya. S. Mahola mahola@ukr.net Ivan Franko National University of Lviv
Abstract	It is proved that the differential equation $z^nw^{(n)}+(a_1^{(n-1)}z+a_2^{(n-1)})z^{n-1}w^{(n-1)}+ \sum_{k=0}^{n-2}{(a_{n-1-k}^{(k)}z^2+a_{n-k}^{(k)}z+a_{n+1-k}^{(k)})z^kw^{(k)}}=0$ has an entire solution $f$ with a two-member recurrent formula for its Taylor's coefficients. The growth of such function $f$ is studied. The conditions for coefficients $a_k^{(j)}$ are obtained, under which the solution $f$ is convex or close-to-convex in $\mathbb{D}=\{z:\|z\|<1\}$ .
Keywords	entire function; linear differential equations; convexity; close-to-convexity; regular growth
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Pages	133-141
Volume	36
Issue	2
Year	2011
Journal	Matematychni Studii
Full text of paper	PDF
Table of content of issue	HTML

On entire solutions with a two-member recurrent formula for Taylor’s coefficients of linear differential equations