Abstract
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.
Highlights
Notations and Preliminaries2021, 10, 348. https://doi.org/Let C, R, Z, N be sets of complex numbers, real numbers, integers, and positive p integers, respectively, and Z+ = N ∪ {0}
We denote by A0 (G), p ∈ N, the class of an analytic functions f in a complete Reinhardt domain G ⊂ C p, represented by the power series of the form
We say that a domain G ⊂ C p is the complete Reinhardt domain if: ( a) z = (z1, . . . , z p ) ∈ G =⇒ (∀ R = ( R1, . . . , R p ) ∈ [0, 1] p ) : Rz = ( R1 z1, . . . , R p z p ) ∈ G
Summary
We denote by A0 (G), p ∈ N, the class of an analytic functions f in a complete Reinhardt domain G ⊂ C p , represented by the power series of the form. Reinhardt domain G with a center at z = 0 can be represented in G by the series of form (1). The domain of convergence of each series of form (1) is the logarithmically-convex complete Reinhardt domain with the center z = 0. We say that a domain G ⊂ C p is the complete Reinhardt domain if:. The Reinhardt domain G is called logarithmically-convex if the image of the set Reinhardt domain is a disc. 1) ∈ R p , B p (r ), Π p (r ) (r > 0) are the logarithmically-convex complete Reinhardt domains.
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