For entire functions () = ∑︀ +∞ =0 , ∈ C, P. Lévy (1929) established that in the classical Wiman's inequality () ()(ln ()) 1/2+ , > 0, which holds outside a set of finite logarithmic measure, the constant 1/2 can be replaced almost surely in some sense by 1/4; here () = max{| ()| : | | = }, () = max{| | : 0}, > 0. In this paper we prove that the phenomenon discovered by P. Lévy holds also in the case of Wiman's inequality for entire functions of several variables, which gives an affirmative answer to the question of A. A. Goldberg and M. M. Sheremeta (1996) on the possibility of this phenomenon.

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We find unimprovable conditions for the Laplace-Stieltjes type integrals under which the Wiman-type inequality holds.

In the paper we prove for the first time an analogue of the Wiman inequality in the class of analytic functions f∈A0p(G) in an arbitrary complete Reinhard domain G⊂Cp, p∈N represented by the power series of the form f(z)=f(z1,⋯,zp)=∑‖n‖=0+∞anzn with the domain of convergence G. We have proven the following statement: If f∈Ap(G) and h∈Hp, then for a given ε=(ε1,…,εp)∈R+p and arbitrary δ>0 there exists a set E⊂|G| such that ∫E∩Δεh(r)dr1⋯drpr1⋯rp<+∞ and for all r∈Δε∖E we have Mf(r)≤μf(r)(h(r))p+12lnp2+δh(r)lnp2+δ{μf(r)h(r)}∏j=1p(lnerjεj)p−12+δ. Note, that this assertion at p=1,G=C,h(r)≡const implies the classical Wiman–Valiron theorem for entire functions and at p=1, the G=D:={z∈C:|z|<1},h(r)≡1/(1−r) theorem about the Kővari-type inequality for analytic functions in the unit disc D; p>1 implies some Wiman’s type inequalities for analytic functions of several variables in Cn×Dk, n,k∈Z+,n+k∈N.

Abstract:For the Dirichlet series of the formwith pairwise independent real exponents (λ k (ω)) on probability space (Ω, A, P ) an estimates of abscissas convergence and absolutely convergence are established.