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An existence theorem for the refined order of analytic functions. (Russian) Zbl 0645.30027

Let \(L_ 0\) be the class of positive continuous functions strongly increasing on [a,\(\infty)\) with properties \[ \lim_{x\to \infty}h(x)=\infty,\quad \lim_{x\to \infty}h(x(1+g(x)))/h(x)=1 \] for each function g(x) with \(\lim_{x\to \infty}g(x)=0.\)
Let f(z) be an analytic function in \(| z| <1\). For \(\alpha (e^ x)\), \(\beta (x)\in L_ 0\) define \[ \rho_ f(\alpha,\beta)=\limsup_{r\to 1}\alpha (\log \max_{| z| =r}| f(z)|)/\beta (1/1-r)). \] The main result is that for a given function \(f(z)\in A(| z| <1)\) there exists a proximate order \(\rho (r)\to \rho_ f(\alpha,\beta)\).
Reviewer: A.M.Russakovskij

MSC:

30E15 Asymptotic representations in the complex plane

Keywords:

proximate order
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