Logarithmic derivative and zeros of an entire function of bounded ℓ-index

Some improvements of criteria of $L$-index boundedness in direction

“…As sufficient conditions of boundedness of index or l-index they improve corresponding results of G. H. Fricke (see Theorem 5 in [3] and Theorem 2 in [4]) and M. M. Sheremeta, A. D. Kuzyk (see Theorems 1 and 6 in [5]). …”

“…In the case n = 1 and b = 1 we obtain the definition of an entire function of one variable of bounded l-index (see [5], [10]); in the case n = 1, b = 1 and L(z) ≡ 1 it is reduced to the definition of an function of bounded index, introduced by B. Lepson ([12]).…”

“…[3], [4], [5]). We remind some definitions of the theory of entire functions (more details are in [16]).…”

Composition of entire functions and bounded L-index in direction

“…In the case n = 1 and b = 1 we obtain the definition of entire function of one variable of bounded l-index (see [14,17,18]); in the case n = 1, b = 1 and L(z) ≡ 1 it is reduced to the definition of function of bounded index, supposed by B. Lepson ( [15]).…”

Entire Functions of Bounded L-Index: Its Zeros and Behavior of Partial Logarithmic Derivatives

“…Meanwhile, in corresponding theorem from [26,27] the circle ( 0 , 0 ) is chosen such that ( ) ̸ = 0 for all ∈ ( 0 , 0 ).…”