Some general distortion results for K(α) and S*(α)

Abstract

In this paper we present some general distortion results for the
classes K(α) and S∗(α) of convex, respectively starlike functions of order α
on the unit disc. For this, we start from a classical result for the class S of
univalent and normalized functions on the unit disc. Furthermore, since when
α = 0 these classes reduce to the well-known classes of starlike and convex func-
tions, we obtain also some general distortion results for the classes K and S∗ of
convex, respectively starlike functions on the unit disc.

Authors

Eduard Ștefan Grigoriciuc

Faculty of Mathematics and Computer Science, “Babeș-Bolyai” University, Cluj-Napoca, Romania

Keywords

Univalent functions, starlikeness of order α, convexity of order α, coefficient estimates, distortion results

Paper coordinates

E.S. Grigoriciuc, Some general distortion results for K(α) and S*(α), Mathematica (Cluj), 64 (87) (2022) no. 2, 222-232

DOI: http://dx.doi.org/10.24193/mathcluj.2022.2.07

References

[1]  S. Cobzas, Differential calculus (in Romanian), Cluj University Press, Cluj-Napoca, 1997.
[2] P. L. Duren, Univalent functions, Springer-Verlag Inc., New York, 1973.
[3] N. Ghosh and A. Vasudevarao, Coefficient estimates for certain subclass of analytic functions defined by subordonation, Filomat, 31 (2017), 3307–3318.
[4] A. W. Goodman, Univalent functions, Vols. I and II, Mariner Publ. Co., Tampa, Florida, 1983.
[5] I. Graham and G. Kohr, Geometric function theory in one and higher dimensions, Pure and Applied Mathematics, Marcel Dekker, Vol. 255, Marcel Dekker Inc., New York, 2003.
[6] M. Klein, Functions starlike of order α, Trans. Amer. Math. Soc., 131 (1968), 99–106.
[7] G. Kohr and P. T. Mocanu, Special chapters of complex analysis (in Romanian), Cluj University Press, Cluj-Napoca, 2005.
[8] P. T. Mocanu, T. Bulboaca and G. S, . Salagean, Geometric theory of univalent functions (in Romanian), Casa Cartii de STiinta, Cluj-Napoca, 2006.
[9] B. Pinchuk, On starlike and convex functions of order α, Duke Math. J., 35 (1968), 721–734.
[10] M. S. Robertson, On the theory of univalent functions, Ann. of Math. (2), 37 (1936), 374–408.
[11] A. Schild, On starlike functions of order α, Amer. J. Math., 87 (1965), 65–70.

PDF

About this paper

Journal

Mathematica (Cluj)

Publisher Name

Editura Academiei Române

DOI

10.24193/mathcluj.2022.2.07

Print ISSN

1222-9016

Online ISSN

2601-744X