Growth of entire functions with some univalent Gelfond-Leontev derivatives.

*(English)*Zbl 0664.30013Let f(z) be analytic and univalent in the unit disc \(\Delta:z:<1\). If f and each successive derivative \(f^{(k)}\) are univalent in \(\Delta\), then S. M. Shah and S. Y. Trimble showed that f must be an entire function of exponential type (1969; see also A. Sathaye and S. M. Shah, 1982). This hypothesis that each \(f^{(k)}\) univalent was improved to the condition that each \(f^{(n_ k)}\) \((k=1,2,...)\), where \(\{n_ k\}\) is an increasing sequence of positive integers such that
\[
(1)\quad n_{k+2}-2n_{k+1}+n_ k=o(n_ k)
\]
is univalent in \(\Delta\) ” implies that f is entire.

Let \(\{d_ n\}\) be a nondecreasing sequence of positive numbers. The Gelfond-Leontev derivative of \(f(z)=\sum^{\infty}_{0}a_ nz^ n\), \(| z| <R\) is defined by \(Df(z)=\sum^{\infty}_{n=1}d_ na_ nz^{n-1}\) and the kth iterative in D is given by \(D^ kf(z)=\sum^{\infty}_{n=k}d_ n...d_{n-k+1}a_ nz^{n-k}\). O. P. Juneja and S. M. Shah (1984) considered the Gelfond-Leontev derivative for univalence and growth properties, and extended some results of Shah and Trimble. In the present paper the authors prove three theorems. In Theorem 1, f(z) is assumed to be entire and the condition analogues to (1) is assumed for univalence of \(D^{np}f\) and a bound on the growth of f is obtained. In Theorem 2, log d(n) is assumed to be the restriction of a slowly oscillating function on integers.

Let \(\{d_ n\}\) be a nondecreasing sequence of positive numbers. The Gelfond-Leontev derivative of \(f(z)=\sum^{\infty}_{0}a_ nz^ n\), \(| z| <R\) is defined by \(Df(z)=\sum^{\infty}_{n=1}d_ na_ nz^{n-1}\) and the kth iterative in D is given by \(D^ kf(z)=\sum^{\infty}_{n=k}d_ n...d_{n-k+1}a_ nz^{n-k}\). O. P. Juneja and S. M. Shah (1984) considered the Gelfond-Leontev derivative for univalence and growth properties, and extended some results of Shah and Trimble. In the present paper the authors prove three theorems. In Theorem 1, f(z) is assumed to be entire and the condition analogues to (1) is assumed for univalence of \(D^{np}f\) and a bound on the growth of f is obtained. In Theorem 2, log d(n) is assumed to be the restriction of a slowly oscillating function on integers.

Reviewer: S.M.Shah

##### MSC:

30C50 | Coefficient problems for univalent and multivalent functions of one complex variable |

30D45 | Normal functions of one complex variable, normal families |