## 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 $${\mathcal H}_{a+1,c}(p;A,B)\subset{\mathcal H}_{a,c}(p;A,B)$$.

### MSC:

 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

### Keywords:

 [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