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
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