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Univalent functions with univalent Gelfond-Leontev derivatives. (English) Zbl 0542.30011

Let \(f(z)=\sum^{\infty}_{n=0}a_ nz^ n\) be analytic in \(| z|<R\) and let \(\{d_ n\}_ 1^{\infty}\) denote a non-decreasing sequence of positive numbers. The Gelfond-Leont’ev derivative of f denoted by Df is defined by \(Df(z)=\sum^{\infty}_{n=1}d_ na_ nz^{n-1}\) and the \(k^{th}\) iterate of D is given by \[ D^ kf(z)=\sum^{\infty}_{n=k}d_ n,...,d_{n-k+1}a_ nz^{n- k}=\sum^{\infty}_{n=k}e_{n-k}e_ n^{-1}a_ nz^{n-k} \] where \(e_ 0=1\) and \(e_ n=(d_ 1,d_ 2,...d_ n)^{-1}, n=1,2,..\).. Set \(p(z)=\sum^{\infty}_{n=0}e_ nz^ n\) and define the p-type of f to be the number \(T_ p(f)=\lim \sup_{n\to \infty}| a_ n/e_ n|^{1/n}.\) In this paper the authors consider analytic functions f such that f along with its Gelfond derivatives is univalent in the unit disk U and show that f must be of finite p-type. It is proved that there exists an f normalised and univalent in U such that f’ is univalent in U but Df is not univalent in U and p-type of f as large as we please. The relation between the growth of \(\rho_ n\), the radius of univalence of \(D^ nf\) and the radius of convergence of f are investigated. Further the relations between the radii of univalence of the Gelfond derivatives of an entire function and its growth constants are also obtained. Another class of functions is defined by a condition of the form \(| a_{n+1}/a_ n| \leq b_{n+1}/d_{n+1},\) where \(\{b_ n\}_ 1^{\infty}\) is a sequence of positive numbers satisfying an inequality and it is shown that all functions in this class together with all their Gelfond derivatives are regular and univalent in U. Finally, an extension of the definition of a linar invariant family is given and some results stated.
Reviewer: K.S.Padmanabhan

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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