Directional logarithmic derivative and the distribution of zeros of an entire function of bounded $L$-index in the direction - Bandura, A. І.; Skaskiv, O. B. (Ukrains’kyi Matematychnyi Zhurnal, 2017)
We establish new criteria of boundedness of the $L$-index in the direction for entire functions in $C^n$. These criteria are formulated as estimate of the maximum modulus via the minimum modulus on a circle and describe the distribution of theirzeros and the behavior of the directional logarithmic derivative. In this way, we prove Hypotheses 1 and 2 from the article[Bandura A. I., Skaskiv O. B. Open problems for entire functions of bounded index in direction // Mat. Stud. – 2015. – 43,№ 1. – P. 103 – 109]. The obtained results are also new for the entire functions of bounded index in $C$. They improve theknown results by M. N. Sheremeta, A. D. Kuzyk, and G. H. Fricke.