Boundedness of $l$-index and completely regular growth of entire functions
Abstract
UDC 517.547.22
We study relations between the class of entire functions of order $\rho$ and of completely regular growth and the class of entire functions of bounded $l$-index, where $l(z)=|z|^{\rho-1}+1$ for $|z|\ge 1.$ Possible applications of these functions in the analytic theory of differential equations are considered. We pose three new problems on the existence of functions with given properties which belong to the difference of these classes and, for the fourth problem, we give an affirmative answer. Namely, we suggest sufficient conditions for an infinite product to be an entire function of completely regular growth of order $\rho$ with unbounded $l_{\rho}$-index and its zeros do not satisfy known Levin's conditions (C) and (C$'$). We also construct an entire function of completely regular growth of order $\rho$ with unbounded $l_{\rho}$-index, whose zeros do not satisfy known Levin's conditions (C) and (C$'$).
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