×

Some families of linear operators associated with certain subclasses of multivalent functions. (English) Zbl 1155.30309

Summary: Making use of a certain linear operator, which is defined here by means of the Hadamard product (or convolution), we introduce two novel subclasses \[ \mathcal P_{a,c}(A,B; p,\lambda)\qquad \text{and}\qquad \mathcal P^+_{a,c}(A,B; p,\lambda) \] of the class \(\mathcal A(p)\) of normalized \(p\)-valent analytic functions in the open unit disk. The main objective of the present paper is to investigate the various important properties and characteristics of each of these subclasses. Furthermore, several properties involving neighborhoods of functions in these subclasses are investigated. We also derive many results for the modified Hadamard products of functions belonging to the class \(\mathcal P^+_{a,c}(A,B;p,\lambda)\). Finally, some applications of fractional calculus operators are considered.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Saitoh, H., A linear operator and its applications of first order differential subordinations, Math. japon., 44, 31-38, (1996) · Zbl 0887.30021
[2] Srivastava, H.M.; Patel, J., Some subclasses of multivalent functions involving a certain linear operator, J. math. anal. appl., 310, 209-228, (2005) · Zbl 1075.30008
[3] Kumar, V.; Shukla, S.L., Multivalent functions defined by Ruscheweyh derivatives, Indian J. pure appl. math., 15, 1216-1227, (1984) · Zbl 0567.30012
[4] Kumar, V.; Shukla, S.L., Multivalent functions defined by Ruscheweyh derivatives. II, Indian J. pure appl. math., 15, 1228-1238, (1984) · Zbl 0567.30013
[5] Aouf, M.K., On certain subclass of analytic \(p\)-valent functions. II, Math. japon., 34, 683-691, (1989) · Zbl 0686.30009
[6] Owa, S., On certain subclass of analytic \(p\)-valent functions, Math. japon., 29, 191-198, (1984) · Zbl 0545.30008
[7] Chen, M.-P., A class of \(p\)-valent functions, Soochow J. math., 8, 15-26, (1982) · Zbl 0526.30022
[8] Goel, R.M.; Sohi, N.S., New criteria for \(p\)-valence, Indian J. pure appl. math., 11, 1356-1360, (1980) · Zbl 0449.30008
[9] Aouf, M.K., A generalization of multivalent functions defined by Ruscheweyh derivatives, Soochow J. math., 17, 83-97, (1991) · Zbl 0734.30015
[10] Mehrok, B.S., A class of univalent functions, Tamkang J. math., 13, 141-155, (1982) · Zbl 0517.30016
[11] Aouf, M.K., A generalization of multivalent functions with negative coefficients. II, Bull. Korean math. soc., 25, 221-232, (1988) · Zbl 0657.30014
[12] Aouf, M.K., Certain classes of \(p\)-valent functions with negative coefficients. II, Indian J. pure appl. math., 19, 761-767, (1988) · Zbl 0674.30013
[13] Shukla, S.L.; Dashrath, On certain classes of multivalent functions with negative coefficients, Soochow J. math., 8, 179-188, (1982) · Zbl 0512.30010
[14] Lee, S.K.; Owa, S.; Srivastava, H.M., Basic properties and characterizations of a certain class of analytic functions with negative coefficients, Utilitas math., 36, 121-128, (1989) · Zbl 0639.30009
[15] Gupta, V.P.; Jain, P.K., Certain classes of univalent functions with negative coefficients. II, Bull. austral. math. soc., 14, 467-473, (1976) · Zbl 0335.30010
[16] Aouf, M.K.; Darwish, H.E., Some classes of multivalent functions with negative coefficients. I, Honam math. J., 16, 119-135, (1994) · Zbl 0976.30005
[17] Uralegaddi, B.A.; Sarangi, S.M., Some classes of univalent functions with negative coefficients, An. ştiinţ. univ. al. I. cuza iaşi seçt. I a mat. (N. S.), 34, 7-11, (1988) · Zbl 0671.30013
[18] Bernardi, S.D., Convex and starlike univalent functions, Trans. amer. math. soc., 135, 429-446, (1969) · Zbl 0172.09703
[19] Libera, R.J., Some classes of regular univalent functions, Proc. amer. math. soc., 16, 755-758, (1969) · Zbl 0158.07702
[20] Livingston, A.E., On the radius of univalence of certain analytic functions, Proc. amer. math. soc., 17, 352-357, (1966) · Zbl 0158.07701
[21] ()
[22] Owa, S., On distortion theorems. I, Kyungpook math. J., 18, 55-59, (1978) · Zbl 0401.30009
[23] (), John Wiley and Sons, New York
[24] Jack, I.S., Functions starlike and convex functions of order \(\alpha\), J. London math. soc. (2), 2, 469-474, (1971) · Zbl 0224.30026
[25] Goodman, A.W., Univalent functions and nonanalytic curves, Proc. amer. math. soc., 8, 598-601, (1957) · Zbl 0166.33002
[26] Ruscheweyh, S., Neighborhoods of univalent functions, Proc. amer. math. soc., 81, 521-527, (1981) · Zbl 0458.30008
[27] Altintaş, O.; Owa, S., Neighborhoods of certain analytic functions with negative coefficients, Internat. J. math. math. sci., 19, 797-800, (1996) · Zbl 0915.30008
[28] Altintaş, O.; Özkan, Ö.; Srivastava, H.M., Neighborhoods of a class of analytic functions with negative coefficients, Appl. math. lett., 13, 3, 63-67, (2000) · Zbl 0955.30015
[29] Altintaş, O.; Özkan, Ö.; Srivastava, H.M., Neighborhoods of a certain family of multivalent functions with negative coefficients, Comput. math. appl., 47, 1667-1672, (2004) · Zbl 1068.30006
[30] Aouf, M.K., Neighborhoods of certain classes of analytic functions with negative coefficients, Internat. J. math. math. sci., 2006, 1-6, (2006) · Zbl 1118.30006
[31] Schild, A.; Silverman, H., Convolution of univalent functions with negative coefficients, Ann. univ. mariae Curie-skłodowska sect. A, 29, 99-107, (1975) · Zbl 0363.30018
[32] Chen, M.-P.; Irmark, H.; Srivastava, H.M., Some families of multivalently analytic functions with negative coefficients, J. math. anal. appl., 214, 674-690, (1997)
[33] Srivastava, H.M.; Aouf, M.K., A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients. I and II, J. math. anal. appl., J. math. anal. appl., 192, 673-688, (1995) · Zbl 0831.30008
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.