Abstract
In this paper we introduce some new subclasses of the p-valent analytic functions with higher-order derivatives that generalize some related subclasses of starlike and convex functions of a positive order. We found the order of (p,q)-valent starlikeness and convexity for the products of functions that belong to these classes. The order of (p,q)-valent starlikeness and convexity of certain integral operators for the product of functions of these classes were also obtained.
Acknowledgement
The authors are grateful to the reviewers of this article who gave valuable remarks, comments, and advices, in order to revise and improve the results of the paper.
(Communicated by Stanisława Kanas )
References
[1] AGHALARY, R.—JAHANGIRI, J. M.: Inclusion relations fork-uniformly starlike and related functions under certain integral operators, Bull. Korean Math. Soc. 42(3) (2005), 623–629.10.4134/BKMS.2005.42.3.623Search in Google Scholar
[2] ALI, R. M.—RAVICHANDRAN, V.: Integral operators on Ma–Minda type starlike and convex functions, Math. Comput. Model. 53(5–6) (2011), 581–586.10.1016/j.mcm.2010.09.007Search in Google Scholar
[3] AOUF, M. K.: On a class ofp-valent starlike functions of orderα, Int. J. Math. Math. Sci. 10(4) (1987), 733-744.10.1155/S0161171287000838Search in Google Scholar
[4] AOUF, M. K.: A generalization of mulivalent functions with negative coefficients, J. Korean Math. Soc. 25(1) (1988), 53–66.Search in Google Scholar
[5] AOUF, M. K.: Certain classes of multivalent functions with negative coefficients defined by using a differential operator, J. Math. Anal. 30 (2008), 5–21.Search in Google Scholar
[6] AOUF, M. K.: Some families ofp-valent functions with negative coefficients, Acta Math. Univ. Comenian. (N.S.) 78(1) (2009), 121–135.Search in Google Scholar
[7] AOUF, M. K.: Boundedp-valent Robertson functions defined by using a differential operator, J. Franklin Inst. 347 (2010), 1927–1941.10.1016/j.jfranklin.2010.10.012Search in Google Scholar
[8] BREAZ, D.: A convexity properties for an integral operator on the classSp(β), Gen. Math. 15(2–3) (2007), 177–183.10.1155/2008/143869Search in Google Scholar
[9] BREAZ, D.: Certain integral operators on the classesM(βi) andN(βi), J. Inequal. Appl. (2008), Art. ID 719354.Search in Google Scholar
[10] BREAZ, D.—AOUF, M. K.—BREAZ, N.: Some properties for integral operators on some analytic functions with complex order, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 25 (2009), 39–43.Search in Google Scholar
[11] BREAZ, D.—BREAZ, N.: Two integral operators, Stud. Univ. Babeş-Bolyai Math. 47(3) (2002), 13–21.Search in Google Scholar
[12] BREAZ, D.—BREAZ, N.: Some convexity properties for a general operator, J. Inequal. Pure Appl. Math. 7(5) (2006), Art. 177.Search in Google Scholar
[13] BREAZ, D.—GüNEY, H. Ö.: The integral operator on the classes
[14] BREAZ, D.—OWA, S.—BREAZ, N.: A new integral univalent operator, Acta Univ. Apulensis Math. Inform. 16 (2008), 11–16.Search in Google Scholar
[15] BULUT, S.: A note on the paper of Breaz and Güney, J. Math. Inequal. 2(4) (2008), 549–553.10.7153/jmi-02-49Search in Google Scholar
[16] FRASIN, B. A.: New integral operators ofp-valent functions, J. Ineq. Pure Appl. Math. 10(4) (2009), Art. 109.Search in Google Scholar
[17] FRASIN, B. A.: Convexity of integral operators ofp-valent functions, Math. Comput. Model. 51(5–6) (2010), 601–605.10.1016/j.mcm.2009.10.045Search in Google Scholar
[18] GANGADHARAN, A.—SHANMUGAM, T. N.—SRIVASTAVA, H. M.: Generalized hypergeometric functions associated withk-uniformly convex functions, Comput. Math. Appl. 44(12) (2002), 1515–1526.10.1016/S0898-1221(02)00275-4Search in Google Scholar
[19] GOODMAN, A. W.: On uniformly starlike functions, J. Math. Anal. Appl. 155(2) (1991), 364–370.10.1016/0022-247X(91)90006-LSearch in Google Scholar
[20] KANAS, S.: Norm of pre-Schwarzian derivative for the class ofk-uniformly convex andk-starlike functions, Appl. Math. Comput. 215(6) (2009), 2275–2282.10.1016/j.amc.2009.08.021Search in Google Scholar
[21] KANAS, S.—WISNIOWSKA, A.: Conic regions andk-uniform convexity, J. Comput. Appl. Math. 105(1–2) (1999), 327–336.10.1016/S0377-0427(99)00018-7Search in Google Scholar
[22] KANAS, S.—WISNIOWSKA, A.: Conic domains starlike functions, Rev. Roum. Math. Pures Appl. 45(4) (2000), 647–657.Search in Google Scholar
[23] MA, W—MINDA, D.: Uniformly convex functions, Ann. Polon. Math. 57(2) (1992), 165–175.10.4064/ap-57-2-165-175Search in Google Scholar
[24] MERKES, P.—ROBERTSON, M. S.—SCOTT, W. T.: On products of starlike functions, Proc. Amer. Math. Soc. 13(6) (1962), 960–964.10.1090/S0002-9939-1962-0142747-7Search in Google Scholar
[25] NISHIWAKI, J.—OWA, S: Coefficient inequalities for analytic functions, Int. J. Math. Math. Sci. 29(5) (2002), 285–290.10.1155/S0161171202006890Search in Google Scholar
[26] OWA, S: On certain classes ofp-valent functions with negative coefficients, Bull. Belg. Math. Soc. Simon Stevin 25(4) (1985), 385–402.Search in Google Scholar
[27] OWA, S—NISHIWAKI, J.: Coefficient estimates for certain classes of analytic functions, J. Inequal. Pure Appl. Math. 3(5) (2002), Art. 72.Search in Google Scholar
[28] OWA, S.—SRIVASTAVA, H. M.: Some generalized convolution properties associated with certain subclasses of analytic functions, J. Inequal. Pure Appl. Math. 3(3) (2002), Art. 42.Search in Google Scholar
[29] URALEGADDI, B. A.—GANIGI, M. D.—SARANGI, S. M.: Univalent functions with positive coefficients, Tamkang J. Math. 25(3) (1994), 225–230.10.5556/j.tkjm.25.1994.4448Search in Google Scholar
[30] YANG, D.—OWA, S.: Properties of certainp-valently convex functions, Int. J. Math. Math. Sci. 41 (2003), 2603–2608.10.1155/S0161171203209327Search in Google Scholar
© 2021 Mathematical Institute Slovak Academy of Sciences
