Kasana, H. S. Asymptotic properties and construction of proximate orders for analytic functions. (English) Zbl 0773.30023 Math., Rev. Anal. Numér. Théor. Approximation, Math. 32(55), No. 1, 39-47 (1990). Let \(f\) be an analytic function in \(U\) having order \(\rho\) and \(\rho(r)\) be a positive function in \(0<r<1\) having (i) \(\rho(r)\to\rho\) as \(r\to 1\), \(0\leq\rho\leq\infty\); \[ -{(1-r)\log(1-r)\rho'(r)\over \rho(r)}\to\alpha-1\quad\text{as }r\to\infty,\quad 0\leq \alpha<\infty.\tag{ii} \] where \(\rho'(r)\) denotes the derivative of \(\rho(r)\) and \(\alpha=1\) for \(0<\rho<\infty\) and \(\alpha\neq 1\) corresponds to \(\rho=0\) or \(\infty\), \(\rho(r)\) is called the proximate order. Further let (iii) \(\rho(r)\) be continuous and piecewise differentiable for \(r>r_ 0\) and \[ \lim\sup{\log M(r,f)\over(1-r)^{- \rho(r)}}=\sigma,\quad m(r,f)=\max_{| z|=r} | f(z)|. \] First the author proves an approximation Theorem on proximable orders and this is used to construct a proximate order for the class of functions \(f\) analytic in \(U\) and satisfy the above mentioned properties. Reviewer: R.Parvatham (Madras) MSC: 30D15 Special classes of entire functions of one complex variable and growth estimates Keywords:approximation theorem; asymptotic properties; proximate order PDFBibTeX XMLCite \textit{H. S. Kasana}, Math. Rev. Anal. Numér. Théor. Approximation, Math. 32(55), No. 1, 39--47 (1990; Zbl 0773.30023)