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Univalent functions with univalent Gelfond-Leontev derivatives

Published online by Cambridge University Press:  17 April 2009

O.P. Juneja  and
S.M. Shah
O.P. Juneja
Affiliation:
Department of Mathematics, Indian Institute of Technology, IIT Post Office, Kanpur 208016, U.P., India
S.M. Shah
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506, USA.
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Abstract

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Let be a nondecreasing sequence of positive numbers. We consider Gelfond-Leontev derivative Df(z), of a function , defined by for univalence and growth properties, and extend some results of Shah and Trimble. Set en = {d1d2 … dn), n≥l, e0 = 1, . Let r be the radius of convergence of p(z). We state parts of Theorem 1 and Corollaries. Let f and all Dkf, k = 1, 2,…, be analytic and univalent in the unit disk U. Then

(iii) if p is entire and of growth (ρ, T) then f must be entire and of growth not exceeding (ρ, 2d2T),

(iv) if D corresponds to the shift operator (dn ≡ l), then .

Another class of functions is defined by a condition of the form |an+1/an| ≤ bn+1/dn+1, where is a sequence of positive numbers satisfying and inequality, and it is shown that all functions in this class along with all their Gelfond–Leontev successive derivatives are regular and univalent in U. An extension of the definition of a linear invariant family is given and results analogous to (i) and (ii) are stated.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 29 , Issue 3 , June 1984 , pp. 329 - 348
Copyright
Copyright © Australian Mathematical Society 1984

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