ON A CLASS OF MEROMORPHIC STARLIKE FUNCTIONS WITH NEGATIVE COEFFICIENTS
Main Article Content
Abstract
Let $T^*_M(A, B, z_0)$ denote the class of functions
\[f(z)=\frac{a}{z}-\sum_{n=1}^\infty a_nz^n, a\ge 1, a_n\ge 0\]
regular and univalent in unit disc $U'=\{z:0<|z|<1\}$,
satisfying the condition
\[-z\frac{f'(z)}{f(z)}=\frac{1+Aw(z)}{1+Bw(z)}, \quad \text{ for } z\in U' \text{ and } w\in E\]
(where $E$ is the class of analytic functions $w$ with $w(0) = 0$ and $|w(z)| \le 1$), where $-1\le A < B \le 1$, $0\le B \le 1$ and $f(z_0) =1/z_0$ ($0<z_0<1$). In this paper sharp coefficient estimates, distortion properties and radius of meromorphic convexity for functions in $T^*_M(A, B, z_0)$ have been obtained. We also study integral transforms of functions in $T^*_M(A, B, z_0)$. In the last, it is proved that the class $T^*_M(A, B, z_0)$ is closed under convex linear combinations.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
References
J. Clunie, "On meromorphic schlicht functions," J. Lond. Math. Soc., 34(1959), 215-216.
O. P. Juneja and T. R. Reddy, "Meromorphic starlike univalent functions with positive coefficients," Anna/es Universitatis Mariae Curie sklodowska Lubin-Polonia, Vol.XXXIX, 9, Section A, (1985), 55-75.
J. Kaczmarski, "On the coefficients of some class of starlike functions," Bull. Acad, Polon, Sci. Ser. Sci. Math. Astronom. Phys., 17(1969), 495-501.
V. Kumar, "Starlike and convex function with negative and missing coefficients," Jnanabha, 13 (1983), 117-134.
V. Kumar and S.L. Shukla., "Certain classes of univalent functions with negative and missing coefficients," Bulletin of the Institute of Mathematics Academia Sinica, Volume 13, Number 2, June 1985.
Z. Nehari, Conformal mapping, Dover Publication INC. New York, (1952), Ex. 8, p.238
Ch. Pommerenke, "On meromorphic starlike functions," Pacific J. Math., 13(1963), 221-235.
W.C. Royster, "Meromorphic starlike multivalent functions," Trans. Amer. Math. Soc., 107(1963), 300-303.
H. Silverman, "Extreme points of univalent functions with two fixed points," Trans. Amer. Math. Soc., 219(1976), 387-395.
H. Silverman and E.M. Silvia, "Fixed coefficients for subclasses of starlike functions," Houston J. Math., 7(1981). 129-136.
H. Silverman, "Univalent functions with negative coefficients," Proc. Amer. Math. Soc., 51(1975), 109-116.
H. Silverman and E. M. Silvia, "Prestarlike functions with negative coefficients," Internat. J. Math. and Math. Sci., 2(1979), 427-439.
E. M. Silvia, Classes related to $alpha$-starlike functions, Ph.D. Dissertation, Clark University, Worcester, Mass., 1972.