# zbMATH — the first resource for mathematics

A note on certain classes of starlike functions. (English) Zbl 0596.30018
A number of distortion theorems are proved for the classes $${\mathcal T}_{\alpha}(n)$$ and $${\mathcal C}_{\alpha}(n)$$ of analytic and univalent functions with negative coefficients, studied recently by S. K. Chatterjea [J. Pure Math. 1, 23-26 (1981; Zbl 0514.30010)]. Several interesting results for the convolution product of functions belonging to the classes $${\mathcal T}_{\alpha}(n)$$ and $${\mathcal C}_{\alpha}(n)$$ are also presented rather systematically.

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
##### Keywords:
starlike functions; distortion theorems
Full Text:
##### References:
 [1] S.K. Chatterjea , On starlike functions , J. Pure Math. , 1 ( 1981 ), pp. 23 - 26 . MR 696324 | Zbl 0514.30010 · Zbl 0514.30010 [2] M.-P. Chen , On a class of starlike functions , Nanta Math ., 8 ( 1975 ), pp. 79 - 82 . MR 425096 | Zbl 0321.30009 · Zbl 0321.30009 [3] P.L. Duren , Univalent Functions , Springer-Verlag , New York -Berlin -Heidelberg -Tokyo , 1983 . MR 708494 | Zbl 0514.30001 · Zbl 0514.30001 [4] A.W. Goodman , An invitation to the study of univalent and multivalent functions , Internat. J. Math. and Math. Sci. , 2 ( 1979 ), pp. 163 - 186 . MR 539196 | Zbl 0438.30001 · Zbl 0438.30001 [5] T.H. Macgregor , The radius of convexity for starlike functions of order \frac{1}{2} , Proc. Amer. Math. Soc. , 14 ( 1963 ), pp. 71 - 76 . MR 150282 | Zbl 0113.05505 · Zbl 0113.05505 [6] S. Owa - H.M. Srivastava , Univalent and starlike generalized hypergeometric functions, Univ. of Victoria Report No. DM-336-IR (1984) , pp. 1 - 27 ; Canad. J. Math . (to appear). MR 918587 | Zbl 0611.33007 · Zbl 0611.33007 [7] S. Owa . H.M. Srivastava , Some characterizations of univalent, starlike and convex hypergeometric functions , J. Fac. Sci. and Tech. Kinki Univ. , 21 ( 1985 ), pp. 1 - 9 . MR 912386 | Zbl 0567.30010 · Zbl 0567.30010 [8] B. Pinchuk , On starlike and convex functions of order \alpha , Duke Math. J. , 35 ( 1968 ), pp. 721 - 734 . Article | Zbl 0167.36101 · Zbl 0167.36101 [9] M.S. Robertson , On the theory of univalent functions , Ann. of Math. , 37 ( 1936 ), pp. 374 - 408 . MR 1503286 | Zbl 0014.16505 · Zbl 0014.16505 [10] A. Schild , On starlike functions of order \alpha , Amer. J. Math. , 87 ( 1965 ), pp. 65 - 70 . Zbl 0163.09901 · Zbl 0163.09901 [11] A. Schild and H. Silverman , Convolutions of univalent functions with negative coefficients , Ann. Univ. Mariae Curie-Sklodowska Sect. A , 29 ( 1975 ), pp. 99 - 107 . MR 457698 | Zbl 0363.30018 · Zbl 0363.30018 [12] H. Silverman , Univalent functions with negative coefficients , Proc. Amer. Math. Soc. , 51 ( 1975 ), pp. 109 - 116 . MR 369678 | Zbl 0311.30007 · Zbl 0311.30007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.