@article{Bandura_Skaskiv_2020, title={Boundedness of $l$-index and completely regular growth of entire functions}, volume={72}, url={http://umj.imath.kiev.ua/index.php/umj/article/view/1048}, DOI={10.37863/umzh.v72i3.1048}, abstractNote={<p>UDC 517.547.22 </p> <p>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.<span class="Apple-converted-space"> </span>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.<span class="Apple-converted-space"> </span>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$’$).<span class="Apple-converted-space"> </span>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$’$).</p>}, number={3}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Bandura, A. I. and Skaskiv, O. B.}, year={2020}, month={Mar.}, pages={316-325} }