Certain subclasses of starlike functions of complex order involving the Hurwitz-Lerch Zeta function
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 54 (2010) no. 2.

Voir la notice de l'article provenant de la source Library of Science

Making use of the Hurwitz-Lerch Zeta function, we define a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients of complex order denoted by TS^μ_b(α, β, γ) and obtain coefficient estimates, extreme points, the radii of close to convexity, starlikeness and convexity and neighbourhood results for the class TS^μ_b(α, β, γ). In particular, we obtain integral means inequalities for the function f(z) belongs to the class TS^μ_b(α, β, γ) in the unit disc.
Keywords: Univalent, starlike, convex, uniformly convex, uniformly starlike, Hadamard product, integral means, Hurwitz-Lerch Zeta function
@article{AUM_2010_54_2_a1,
     author = {Murugusundaramoorthy, G. and Uma, K.},
     title = {Certain subclasses of starlike functions of complex order involving the {Hurwitz-Lerch} {Zeta} function},
     journal = {Annales Universitatis Mariae Curie-Sk{\l}odowska. Mathematica },
     publisher = {mathdoc},
     volume = {54},
     number = {2},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUM_2010_54_2_a1/}
}
TY  - JOUR
AU  - Murugusundaramoorthy, G.
AU  - Uma, K.
TI  - Certain subclasses of starlike functions of complex order involving the Hurwitz-Lerch Zeta function
JO  - Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
PY  - 2010
VL  - 54
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AUM_2010_54_2_a1/
LA  - en
ID  - AUM_2010_54_2_a1
ER  - 
%0 Journal Article
%A Murugusundaramoorthy, G.
%A Uma, K.
%T Certain subclasses of starlike functions of complex order involving the Hurwitz-Lerch Zeta function
%J Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
%D 2010
%V 54
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AUM_2010_54_2_a1/
%G en
%F AUM_2010_54_2_a1
Murugusundaramoorthy, G.; Uma, K. Certain subclasses of starlike functions of complex order involving the Hurwitz-Lerch Zeta function. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 54 (2010) no. 2. http://geodesic.mathdoc.fr/item/AUM_2010_54_2_a1/

[1] Alexander, J. W., Functions which map the interior of the unit circle upon simple regions, Ann. of Math. 17 (1915), 12-22.

[2] Altintas, O., Ozkan, O. and Srivastava, H. M., Neighborhoods of a class of analytic functions with negative coefficients, Appl. Math. Lett. 13 (2000), 63-67.

[3] Bernardi, S. D., Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969), 429-446.

[4] Choi, J., Srivastava, H. M., Certain families of series associated with the Hurwitz-Lerch Zeta function, Appl. Math. Comput. 170 (2005), 399-409.

[5] Ferreira, C., López, J. L., Asymptotic expansions of the Hurwitz-Lerch Zeta function, J. Math. Anal. Appl. 298 (2004), 210-224.

[6] Flet, T. M., The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 746-765

[7] Garg, M., Jain, K. and Srivastava, H. M., Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch Zeta functions, Integral Transform. Spec. Funct. 17 (2006), 803-815.

[8] Goodman, A. W., Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc. 8 (1957), 598-601.

[9] Goodman, A. W., On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87-92.

[10] Goodman, A. W., On uniformly starlike functions, J. Math. Anal. Appl. 155 (1991), 364-370.

[11] Jung, I. B., Kim, Y. C. and Srivastava, H. M., The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math. Anal.Appl. 176 (1993), 138-147.

[12] Kanas, S., Wiśniowska, A., Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105 (1999), 327-336.

[13] Kanas, S., Wiśniowska, A., Conic domains and starlike functions, Rev. Roumaine Math. Pures Appl. 45(4) (2000), 647-657.

[14] Kanas, S., Srivastava, H. M., Linear operators associated with k-uniformly convex functions, Integral Transform. Spec. Funct. 9(2) (2000), 121-132.

[15] Kanas, S., Yaguchi, T., Subclasses of k-uniformly convex and starlike functions defined by generalized derivative. II, Publ. Inst. Math. (Beograd) (N.S.) 69(83) (2001), 91-100.

[16] Lin, S.-D., Srivastava, H. M., Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154 (2004), 725-733.

[17] Lin, S.-D., Srivastava, H. M. and Wang, P.-Y., Some expansion formulas for a class of generalized Hurwitz-Lerch Zeta functions, Integral Transform. Spec. Funct. 17 (2006), 817-827.

[18] Littlewood, J. E., On inequalities in theory of functions, Proc. London Math. Soc. 23 (1925), 481-519.

[19] Murugusundaramoorthy, G., Srivastava H .M., Neighborhoods of certain classes of analytic functions of complex order, J. Inequal. Pure Appl. Math. 5(2) (2004), Art. 24, 1-8.

[20] Prajapat, J. K., Goyal, S. P., Applications of Srivastava-Attiya operator to the classes of strongly starlike and strongly convex functions, J. Math. Inequal. 3 (2009), 129-137.

[21] Raducanu, D., Srivastava, H. M., A new class of analytic functions defined by means of a convolution operator involving the Hurwitz-Lerch Zeta function, Integral Transform. Spec. Funct. 18 (2007), 933-943.

[22] Rønning, F., Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189-196.

[23] Rønning, F., Integral representations for bounded starlike functions, Ann. Polon. Math. 60 (1995), 289-297.

[24] Ruscheweyh, S., Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81 (1981), 521-527.

[25] Silverman, H., Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), 109-116.

[26] Silverman, H., Integral means for univalent functions with negative coefficients, Houston J. Math. 23 (1997), 169-174.

[27] Srivastava, H. M., Attiya, A. A., An integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination, Integral Transform. Spec. Funct. 18 (2007), 207-216.

[28] Srivastava, H. M., Choi, J., Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.