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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.)

Citations:

Zbl 0514.30010
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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 · doi:10.1155/S016117127900017X
[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 · doi:10.2307/2033957
[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 · doi:10.4153/CJM-1987-054-3
[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 · doi:10.1215/S0012-7094-68-03575-8
[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 · doi:10.2307/1968451
[10] A. Schild , On starlike functions of order \alpha , Amer. J. Math. , 87 ( 1965 ), pp. 65 - 70 . Zbl 0163.09901 · Zbl 0163.09901 · doi:10.2307/2373224
[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 · doi:10.2307/2039855
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