Some properties of the class \(\mathcal{U}\)

Milutin Obradovic, Nikola Tuneski

Abstract


In this paper we study the class \(\mathcal{U}\) of functions that are analytic in the open unit disk \(D =\{z : |z| < 1\}\), normalized such that\(f(0) = f'(0)-1 = 0\) and satisfy \[\left|\left[\frac{z}{f(z)}\right]^2f'(z) - 1\right|< 1\quad  (z\in D).\]
For functions in the class \(\mathcal{U}\) we give sharp estimates of the second and the third Hankel determinant, its relationship with the class of \(\alpha\)-convex functions, as well as certain starlike properties.


Keywords


Analytic; class \(\mathcal{U}\); starlike, \(\alpha\)-convex; Hankel determinant

Full Text:

PDF

References


Goodman, A. W., Univalent Functions. Vol. I, Mariner Publishing Co., Inc., Tampa, FL, 1983.

Jack, I. S., Functions starlike and convex of order (alpha), J. London Math. Soc. 3 (2) (1971), 469–474.

Mocanu, P. T., Une propriete de convexite generalisee dans la theorie de la representation conforme, Mathematica (Cluj) 11 (34) (1969), 127–133.

Miller, S. S., Mocanu, P., Reade, M. O., All (alpha)-convex functions are univalent and starlike, Proc. Amer. Math. Soc. 37 (1973), 553–554.

Obradovic, M., Pascu, N. N., Radomir, I., A class of univalent functions, Math. Japon. 44 (3) (1996), 565–568.

Obradovic, M., Ponnusamy, S., New criteria and distortion theorems for univalent functions, Complex Variables Theory Appl. 44 (3) (2001), 173–191.

Obradovic, M., Ponnusamy, S., On the class U, in: Proc. 21st Annual Conference of the Jammu Math. Soc. and National Seminar on Analysis and its Application, 2011, 11–26.

Prokhorov, D. V., Szynal, J., Inverse coefficients for ((alpha, beta))-convex functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 35 (1981), 125–143 (1984).

Sakaguchi, K., A note on p-valent functions, J. Math. Soc. Japan 14 (1962), 312–321.

Thomas, D. K., Tuneski, N., Vasudevarao, A., Univalent Functions. A Primer, De Gruyter, Berlin, 2018.




DOI: http://dx.doi.org/10.17951/a.2019.73.1.49-56
Date of publication: 2019-12-19 10:33:48
Date of submission: 2019-12-17 11:23:36


Statistics


Total abstract view - 1639
Downloads (from 2020-06-17) - PDF - 834

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2019 Milutin Obradovic, Nikola Tuneski