Summary
The present paper is concerned with functions analytic in the unit disc having rapidly increasing maximum moduli. To study the precise rates of growth of such functions the concept of index is introduced. Several growth parameters in terms of the index are defined for a function analytic in the unit disc and their characterizations in terms of the Taylorseries development of the function are obtained. The results in the present paper improve and refine the earlier results ofSons (J. Math. Anal. Appl,24 (1968), pp. 296–306),MacLane (Asymptotic values of [holomorphic functions,Rice University Studies, Houston, 1963), andKapoor andJuneja (Indian J. Pura Appl. Kath.,7 (3) (1976), pp. 241–248).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
F. Beuermann, Wachstumsordnung, koeffizientenwachstum und nullstellendichte bei potenzreihen mit endlichem konvergenzkeis, Math. Z.,33 (1931), pp. 98–108.
G. P. Kapoor,On the lower order of functions analytic in the unit disc, Math. Japon,17 (1) (1972), pp. 49–54.
G. P. Kapoor,A study in the growth properties and coefficients of analytic functions, Dissertation, Indian Institute of Technology, August 1972.
G. P. Kapoor -O. P. Juneja,On the lower order of functions analytic in the unit disc II, Indian J. Pure Appl. Math.,7 (3) (1976), pp. 241–248.
G. R. Machane,Asymptotic values of holmorphic functions, Rice University Studies, Houston, 1963.
L. R. Sons,Regularity of growth and gaps, J. Math. Anal. Appl.,24 (1968), pp. 296–306.
G. Valiron,Fonctions Analytiques, Paris Presses Universitaires de France, 1954.
S. K. Bajpai -J. Tanne -D. Whittier,A decomposition theorem for an analytic function, J. Math. Anal. Appl.,48 (1974), pp. 736–742.
Author information
Authors and Affiliations
Additional information
Entrata in Redazione il 7 giugno 1978.
Rights and permissions
About this article
Cite this article
Kapoor, G.P., Gopal, K. On the coefficients of functions analytic in the unit disc having fast rates of growth. Annali di Matematica 121, 337–349 (1979). https://doi.org/10.1007/BF02412011
Issue Date:
DOI: https://doi.org/10.1007/BF02412011