On regularly converging series on systems of functions in a disk

Анотація

Let f(z)=∑_k=0^∞f_kz^k be an entire transcendental function, (λ_n) be a sequence of positive numbers increasing to +∞ and the series A(z)=∑_n=1^∞a_nf(λ_nz) be regularly convergent in 𝔻={z:|z|<1}, i.e., ∑_n=1^∞|a_n|M_f(rλ_n)<+∞ for all  r∈[0,1), where M_f(r)=max{|f(z)|:|z|=r}. Suppose that α and β are slowly increasing such that x/β^-1(cα(x))↑+∞, α(x/β^-1(cα(x)))=(1+o(1))α(x) and α(ln x)=o(β(x)) as x₀(c)≤x→+∞ for every c∈(0,+∞). It is proved, for example, that if a_n>0  for all n and α(ln n)=o(β(Γ_f(cλ_n)/ln n)) as n→∞, where Γ_f(r)=(d ln M_f(r))/(d lnr), then       $$
        \varlimsup\limits_{r\uparrow 1}\dfrac{\alpha(\ln\,M_A(r))}{\beta(1/(1-r))}=
        \varlimsup\limits_{k\to\infty}\dfrac{\alpha(\ln^+(|f_k|\mu_D(k))}{\beta(k)},
        $$
where μ_𝔻(σ):=max{|a_n|exp{σln λ_n}:n≥0}.