A linear operator and associated families of meromorphically multivalent functions. (English) Zbl 0997.30009

Let \(\Sigma_p\) denote the class of functions \(f(z)\) which are analytic and \(p\)-valent in the punctured unit disk \[ {\mathcal U}^*={\mathcal U}\setminus\{0\},\quad {\mathcal U}= \{z:|z|< 1\}. \] For the given real numbers \(a,c-c\not\in\mathbb{N}\) we can define a linear operator \[ {\mathcal L}_p(a, c) f(z):= \phi_p(a,c;z)* f(z),\quad(f\in\Sigma_p) \] where \(*\) is a convolution (Hadamard product) and \(\phi_p(a,c;z)\) is a special function defined as follows \[ \phi_p(a,c;z):= z^{-p}+ \sum^\infty_{k=1} {(a)_k\over (c)_k} z^{k- p}. \] For the given fixed parameters \(p\), \(a\), \(c\), \(A\), \(B\), \(-1\leq B< A\leq 1\), we say that a function \(f\in\Sigma_p\) is in the class \({\mathcal H}_{a,c}(p;A,B)\) if it also satisfies the inequality \[ \Biggl|{z({\mathcal L}_p(a,c) f(z))'+ p{\mathcal L}_p(a,c) f(z)\over Bz({\mathcal L}_p(a, c) f(z))'+ Ap{\mathcal L}_p(a, c) f(z)}\Biggr|< 1\quad\text{for }z\in{\mathcal U}. \] In this paper some properties of the classes \({\mathcal H}_{a,c}(p;A,B)\) and the operators \({\mathcal L}_p(a,c)f\) are investigated. Among others it is proved: Theorem. If \(a\geq {p(A-B)\over B+1}\), then \({\mathcal H}_{a+1,c}(p;A,B)\subset{\mathcal H}_{a,c}(p;A,B)\).


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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[1] Altintaş, O.; Owa, S., Neighborhoods of certain analytic functions with negative coefficients, Int. J. Math. Math. Sci., 19, 797-800 (1996) · Zbl 0915.30008
[2] Altintaş, O.; Özkan, Ö.; Srivastava, H. M., Neighborhoods of a class of analytic functions with negative coefficients, Appl. Math. Lett., 13, 63-67 (2000) · Zbl 0955.30015
[3] Carlson, B. C.; Shaffer, D. B., Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15, 737-745 (1984) · Zbl 0567.30009
[4] Dziok, J.; Srivastava, H. M., Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput., 103, 1-13 (1999) · Zbl 0937.30010
[5] Goodman, A. W., Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8, 598-601 (1957) · Zbl 0166.33002
[6] Jack, I. S., Functions starlike and convex of order α, J. London Math. Soc. (2), 3, 469-474 (1971) · Zbl 0224.30026
[7] Joshi, S. B.; Srivastava, H. M., A certain family of meromorphically multivalent functions, Comput. Math. Appl., 38, 201-211 (1999) · Zbl 0959.30010
[8] Mogra, M. L., Meromorphic multivalent functions with positive coefficients, I and II, Math. Japon., 35, 1-11 (1990) · Zbl 0705.30019
[9] Ruscheweyh, S., Neighborhoods of univalent functions, Proc. Amer. Math. Soc., 81, 521-527 (1981) · Zbl 0458.30008
[10] Saitoh, H., A linear operator and its applications of first order differential subordinations, Math. Japon., 44, 31-38 (1996) · Zbl 0887.30021
[11] Silverman, H., Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51, 109-116 (1975) · Zbl 0311.30007
[12] Srivastava, H. M.; Hossen, H. M.; Aouf, M. K., A unified presentation of some classes of meromorphically multivalent functions, Comput. Math. Appl., 38, 63-70 (1999) · Zbl 0978.30011
[13] Srivastava, H. M.; Owa, S., Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions, Nagoya Math. J., 106, 1-28 (1987) · Zbl 0607.30014
[14] Srivastava, H. M.; Owa, S., Current Topics in Analytic Function Theory (1992), World Scientific: World Scientific Singapore · Zbl 0976.00007
[15] Yang, D.-G., On new subclasses of meromorphic \(p\)-valent functions, J. Math. Res. Exposition, 15, 7-13 (1995) · Zbl 1108.30309
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