Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-17T17:05:19.775Z Has data issue: false hasContentIssue false
  • English
  • Français

On the generalised Ruscheweyh class of analytic functions of complex order

Published online by Cambridge University Press:  17 April 2009

O.P. Ahuja
O.P. Ahuja
Affiliation:
Department of Mathematics, The University of Papua, New Guinea Box 320, University Post Office Papua, New Guinea
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The main object of this paper is to identify various classes of analytic functions which are starlike, convex, pre-starlike, Ruscheweyh class of order α, β-spiral-like, β-convex-spiral-like, starlike of complex order, complex of complex order, and others as special cases of a family of analytic functions of complex order in the unit disk. This makes a uniform treatment possible. Finally, we derive sharp estimates for all coefficients of the functions from the family.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 47 , Issue 2 , April 1993 , pp. 247 - 257
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Ahuja, O.P., ‘Integral operators of certain univalent functions’, Internal. J. Math. Math. Sci. 8 (1985), 653–662.Google Scholar
[2]Ahuja, O.P., ‘Certain generalizations of the Robertson functions’, Yokohama Math. J. 31 (1983), 5–11.Google Scholar
[3]Ahuja, O.P. and Silverman, H., ‘Function classes related to Ruscheweyh derivatives’, J. Austral. Math. Soc. Ser. A 47 (1989), 438–444.CrossRefGoogle Scholar
[4]Ahuja, O.P. and Silverman, H., ‘A survey on spiral-like and related function classes’, Math. Chronicle 20 (1991), 39–66.Google Scholar
[5]Ahuja, O.P. and Silverman, H., ‘Convolutions of prestarlike functions’, Internat. J. Math. Math. Sci. 6 (1983), 59–68.Google Scholar
[6]Al-Amiri, H.S., ‘On Ruscheweyh derivatives’, Ann. Polon. Math. 38 (1980), 87–94.Google Scholar
[7]Al-Amiri, H.S., ‘Certain generalizations of prestarlike functions’, J. Austral. Math. Soc. Ser. A 28 (1979), 325–334.Google Scholar
[8]Aouf, M.K. and Nasr, M.A., ‘Starlike functions of complex order’, J. Natur. Sci. Math. 25 (1985), 1–12.Google Scholar
[9]Chichra, P.N., ‘Regular functions f(z) for which zf′(z) is α-spiral-like’, Proc. Amer. Math. Soc. 49 (1975), 151–160.Google Scholar
[10]Dasrath, and Shukla, S.L., ‘Coefficient estimates for a subclass of spiral-like functions’, Indian J. Pure Appl. Math. 14 (1983), 431–439.Google Scholar
[11]Goel, R.M., ‘A subclass of α-spiral functions’, Publ. Math. Debrecen 23 (1976), 79–84.CrossRefGoogle Scholar
[12]Goel, R.M. and Mehrok, B.S., ‘On the coefficients of a subclass of starlike functions’, Indian J. Pure Appl. Math. 12 (1981), 634–647.Google Scholar
[13]Janowski, W., ‘Some extremal problems for certain families of analytic functions’, Bull. de. L'Acad. Pol. des Sci. 21 (1973), 17–25.Google Scholar
[14]Janowski, W., ‘Some extremal problems for certain families of analytic functions’, Ann. Poln. Math. 28 (1973), 297–326.CrossRefGoogle Scholar
[15]Kulshrestha, P.K., ‘Bounded Robertson functions’, Rend. Mat. 6 (1976), 137–150.Google Scholar
[16]Libera, R.J., ‘Univalent α-spiral functions’, Canad. J. Math. 19 (1967), 449–456.Google Scholar
[17]Makowka, B., ‘On some subclasses of univalent functions’, Zeszyty Nauk. Politech Lódz. Math. 254 (1976), 71–76.Google Scholar
[18]Mogra, M.L. and Ahuja, O.P., ‘On spiral-like functions of order α and type β’, Yokohama Math. J. 29 (1981), 145–156.Google Scholar
[19]Padmanabhan, K.S., ‘On certain classes of starlike functions in the unit disk’, J. Indian Math. Soc. 32 (1968), 89–103.Google Scholar
[20]Robertson, M.S., ‘Quasi subordinate functions’, in Mathematical Essays dedicated to A.J. MacIntyre Volume (Ohio University, 1967), pp. 311–340.Google Scholar
[21]Robertson, M.S., ‘Univalent functions for which zf′(z) is spiral-like’, Michigan Math. J. 16 (1969), 97–101.CrossRefGoogle Scholar
[22]Ruscheweyh, S., ‘New criteria for univalent functions’, Proc. Amer. Math. Soc. 49 (1975), 109–115.CrossRefGoogle Scholar
[23]Ruscheweyh, S., ‘Linear operators between classes of prestarlike functions’, Comment. Math. Helv. 52 (1977), 497–509.Google Scholar
[24]Schild, A. and Silverman, H., ‘Convolutions of univalent functions with negative coefficients’, Ann. Univ. Marie Curie-Sklodowska Sect. A 29 (1975), 99–107.Google Scholar
[25]Sheil-Small, T., Silverman, H. and Silvia, E.M., ‘Convolution multipliers and starlike functions’, J. Analyse Math. 41 (1982), 181–192.Google Scholar
[26]Silverman, H., ‘Subclasses of starlike functions’, Rev. Roumaine Math. Pures Appl. 23 (1978), 1093–1099.Google Scholar
[27]Silverman, H. and Silvia, E.M., ‘Subclasses of starlike functions subordinate to convex functions’, Canad. J. Math. 37 (1985), 48–61.CrossRefGoogle Scholar
[28]Silverman, H. and Silvia, E.M., ‘Prestarlike functions with negative coefficients’, Internat. J. Math. Math. Sci. 2 (1979), 427–439.CrossRefGoogle Scholar
[29]Silvia, E.M., ‘Subclasses of spiral-like functions’, Tamkang J. Math. 14 (1983), 161–169.Google Scholar
[30]Singh, R., ‘On a class of starlike functions II’, Ganita 19 (1968), 103–110.Google Scholar
[31]Singh, R. and Singh, S., ‘Integrals of certain univalent functions’, Proc. Amer. Math. Soc. 77 (1979), 336–343.Google Scholar