On regularly converging series on systems of functions in a disk

Myroslav Sheremeta


Let f(z)=∑k=0fkzk be an entire transcendental function, (λn) be a sequence of positive numbers increasing to +∞ and the series A(z)=∑n=1anf(λnz) be regularly convergent in 𝔻={z:|z|<1}, i.e., ∑n=1|an|Mf(rλn)<+∞ for all  r∈[0,1), where Mf(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 an>0  for all n and α(ln n)=o(β(Γf(cλn)/ln n)) as n→∞, where Γf(r)=(d ln Mf(r))/(d lnr), then       $$
        \varlimsup\limits_{r\uparrow 1}\dfrac{\alpha(\ln\,M_A(r))}{\beta(1/(1-r))}=
where μ𝔻(σ):=max{|an|exp{σln λn}:n≥0}.

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DOI: http://dx.doi.org/10.30970/vmm.2022.94.098-108


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