Accessible Requires Authentication Pre-published online by De Gruyter September 25, 2021

Certain classes of analytic functions defined by Hurwitz–Lerch zeta function

Bolineni Venkateswarlu ORCID logo, Pinninti Thirupathi Reddy ORCID logo, Galla Swapna ORCID logo and Rompilli Madhuri Shilpa

Abstract

In this work, we introduce and investigate a new class k-US~s(b,μ,γ,t) of analytic functions in the open unit disk U with negative coefficients. The object of the present paper is to determine coefficient estimates, neighborhoods and partial sums for functions f belonging to this class.

MSC 2010: 30C45

Acknowledgements

The authors would like to express sincere thanks to the esteemed referee(s) for their careful readings, valuable suggestions and comments, which helped to improve the presentation of this paper.

References

[1] J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. of Math. (2) 17 (1915), no. 1, 12–22. Search in Google Scholar

[2] E. Aqlan, J. M. Jahangiri and S. R. Kulkarni, New classes of k-uniformly convex and starlike functions, Tamkang J. Math. 35 (2004), no. 3, 261–266. Search in Google Scholar

[3] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969), 429–446. Search in Google Scholar

[4] J. Choi and H. M. Srivastava, Certain families of series associated with the Hurwitz–Lerch zeta function, Appl. Math. Comput. 170 (2005), no. 1, 399–409. Search in Google Scholar

[5] C. Ferreira and J. L. López, Asymptotic expansions of the Hurwitz–Lerch zeta function, J. Math. Anal. Appl. 298 (2004), no. 1, 210–224. Search in Google Scholar

[6] T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 746–765. Search in Google Scholar

[7] M. Garg, K. Jain and H. M. Srivastava, Some relationships between the generalized Apostol–Bernoulli polynomials and Hurwitz–Lerch zeta functions, Integral Transforms Spec. Funct. 17 (2006), no. 11, 803–815. Search in Google Scholar

[8] A. W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc. 8 (1957), 598–601. Search in Google Scholar

[9] A. W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl. 155 (1991), no. 2, 364–370. Search in Google Scholar

[10] I. B. Jung, Y. C. Kim and H. M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math. Anal. Appl. 176 (1993), no. 1, 138–147. Search in Google Scholar

[11] S.-D. Lin and H. M. Srivastava, Some families of the Hurwitz–Lerch zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154 (2004), no. 3, 725–733. Search in Google Scholar

[12] S.-D. Lin, H. M. Srivastava and P.-Y. Wang, Some expansion formulas for a class of generalized Hurwitz–Lerch zeta functions, Integral Transforms Spec. Funct. 17 (2006), no. 11, 817–827. Search in Google Scholar

[13] S. Owa, T. Sekine and R. Yamakawa, On Sakaguchi type functions, Appl. Math. Comput. 187 (2007), no. 1, 356–361. Search in Google Scholar

[14] J. K. Prajapat and S. P. Goyal, Applications of Srivastava-Attiya operator to the classes of strongly starlike and strongly convex functions, J. Math. Inequal. 3 (2009), no. 1, 129–137. Search in Google Scholar

[15] D. Răducanu and H. M. Srivastava, A new class of analytic functions defined by means of a convolution operator involving the Hurwitz–Lerch zeta function, Integral Transforms Spec. Funct. 18 (2007), no. 11–12, 933–943. Search in Google Scholar

[16] S. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81 (1981), no. 4, 521–527. Search in Google Scholar

[17] K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan 11 (1959), 72–75. Search in Google Scholar

[18] M. P. Santosh, R. N. Ingle, P. Thirupathi Reddy and B. Venkateswarlu, A new subclass of analytic functions defined by linear operator, Adv. in Math. Sci. J. 9 (2010), 205–217. Search in Google Scholar

[19] H. Silverman, Partial sums of starlike and convex functions, J. Math. Anal. Appl. 209 (1997), no. 1, 221–227. Search in Google Scholar

[20] E. M. Silvia, On partial sums of convex functions of order α, Houston J. Math. 11 (1985), no. 3, 397–404. Search in Google Scholar

[21] H. M. Srivastava and A. A. Attiya, An integral operator associated with the Hurwitz–Lerch zeta function and differential subordination, Integral Transforms Spec. Funct. 18 (2007), no. 3–4, 207–216. Search in Google Scholar

Received: 2020-07-02
Revised: 2020-10-12
Accepted: 2020-10-14
Published Online: 2021-09-25

© 2021 Walter de Gruyter GmbH, Berlin/Boston