Filevych P. V.

For entire functions with Borel exceptional values, we establish the relationship between the rate of approaching $\infty$ for the sequence of their $a$-points and the rate of approaching 0 for the sequence of their Taylor coefficients.

For the entire Dirichlet series $f(z) = ∑_{n = 0}${∞$ a_n e^{zλn}$, we establish necessary and sufficient conditions on the coefficients $a_n$ and exponents $λ_n$ under which the function $f$ has the Paley effect, i.e., the condition $$\underset{r\to +\infty }{ \lim \sup}\frac{ \ln {M}_f(r)}{T_f(r)}=+\infty$$ is satisfied, where $M_f (r)$ and $T_f (r)$ are the maximum modulus and the Nevanlinna characteristic of the function $f$, respectively.

We establish a necessary and sufficient condition for the coefficients a _n of an entire function \(f(z) = \sum {_{n = 0}^\infty } {\text{ }}a_n z^n \) under which its central index and the logarithms of the maximum of the modulus and the maximum term are regularly varying functions. We construct an entire function the logarithm of the maximum of whose modulus is a regularly varying function, whereas the Nevanlinna characteristic function is not a regularly varying function.

Let M _f(r) and μ_f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let Φ be a continuously differentiable function convex on (−∞, +∞) and such that x = o(Φ(x)) as x → +∞. We establish that, in order that the equality \(\lim \inf \limits_{r \to + \infty} \frac{\ln M_f (r)}{\Phi (\ln r)} = \lim \inf \limits_{r \to + \infty} \frac{\ln \mu_f (r)}{\Phi (\ln r)}\) be true for any entire function f, it is necessary and sufficient that ln Φ′(x) = o(Φ(x)) as x → +∞.

Let M _f(r) and μ _f (r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let l(r) be a continuously differentiable function convex with respect to ln r. We establish that, in order that ln M _f(r) ∼ ln μ _f (r), r → +∞, for every entire function f such that μ _f (r) ∼ l(r), r → +∞, it is necessary and sufficient that ln (rl′(r)) = o(l(r)), r → +∞.

We obtain an exact estimate for the measure of the exceptional set in the Borel relation for entire functions.

We establish that, for the “majority” of entire functions of finite order, their generalized Phragmén–Lindelöf indicators are identically equal to constants.

An estimate exact in a certain sense is obtained for the value of the exceptional set in the Borel relation for entire functions