A Study of Meromorphically Univalent Functions Define by a Linear Operator Associated with the λ-Generalized Hurwitz-Lerch Zeta Function

Firas Ghanim, H.M. Srivastava, S´ebastien Gaboury

Abstract


By using a linear operator associated with the λ-generalized Hurwitz-Lerch zeta function, which is defined here by means of the Hadamard product (orconvolution), the authors introduce and investigate certain sufficient conditions for this meromorphic functions to satisfy a subordination. In fact, these results extend known results of starlikeness, convexity, and close to convexity.

Keywords


Analytic functions; Univalent functions; Meromorphic functions; $\lambda$-generalized Hurwitz-Lerch zeta function; Srivastava-Attiya operator; Dziok-Srivastava and Srivastava-Wright operators; Hadamard product (or convolution).

References


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

B. C. Carlson and D. B. Shaer, Starlike and prestarlike hypergeometric functions, SIAMJ. Math. Anal. 15(1984), 737–745.

N. E. Cho, I. H. Kim and H. M. Srivastava, Sandwich-type theorems for multivalent functions associated with the Srivastava-Attiya operator, Appl. Math. Comput. 217 (2010), 918–928.

N. E. Cho and H. M. Srivastava, Argument estimation of certain analytic functions defied by a class of multiplier transformation, Math. Comput. Modelling. 37(2003),39–49.

J. H. Choi, M. Saigo and H. M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl. 276 (2002), 432–445.

J. Dziok and H. M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999),1–13.

J. Dziok and H. M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms Spec. Funct. 14 (2003),7–18.

F. Ghanim, A study of acertain subclass of Hurwitz-Lerch zeta function related to a linear operator, Abstr. Appl. Anal. 2013 Article ID 763756 (2013),1–7.

F. Ghanim and M. Darus, New result of analytic functions related to Hurwitz-Zeta function, The Scientific World Journal, vol. 2013, Article ID 475643, 5 pages, 2013. doi:10. 1155/2013/ 475643.

D. I. Hallenbeck and S. Ruscheweyh, Subordination by convex functions, Proc.Amer.Math.Soc.52(1975),191–195.

Yu.E.Hohlov, Operators and operations in the class of univalent functions, Izv. Vyss. Ucebn. Zaved. Matematika, 10(1978),83–89.

V. Kiryakova, Criteria for univalence of the Dziok-Srivastava and the Srivastava-Owa operators in the class A, Appl. Math. Comput. 218 (2011), 883–892.

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), 138–147.

A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Function: Theory and Applications, Springer, NewYork, Dordrecht, Heidelberg and London, 2010.

K. I. Noor and S. Z. H.Bukhari, Some subclasses of analytic and spiral-like functions of complex order involving the Srivastava-Attiya integral operator, Integral Transforms Spec. Funct. 21 (2010), 907–916.

S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 (1987), 1057–1077.

S. S. Miller and P.T. Mocanu, Dierential subordinations: theory and applications, Seriesin Pure and Applied Mathematics, 225, Marcel Dekker, New York,(2000).

S. Owa, M. Nunokawa, H. Saitoh and H. M. Srivastava, Close-to-convexity, starlikeness, and convexity of certain analytic functions, Appl. Math. Lett., 15(1) (2002),63-69.

J. K. Prajapat and T. Bulboac˘a, Double subordination preserving properties for a new generalized Srivastava-Attiya operator, Chin. Ann. Math. 33(2012), 569–582.

D. Raducanu and H. M. Srivastava, A newclassofanalyticfunctionsdefindbymeansofaconvolutionoperatorinvolvingtheHurwitz-Lerch zeta function, Integral Transforms Spec.Funct. 18(2007), 933–943.

S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975),109–115.

H. M. Srivastava, Some Fox-Wright generalized hypergeometric functions and associated families of convolution operators, Appl. Anal. Discrete Math.1(2007),56–71.

H. M. Srivastava, Some formulas for the Bernoulli and Euler polynomial satrational arguments, Math.Proc. Cambridge Philos. Soc. 129(2000), 77–84.

H.M.Srivastava, Some generalizations and basic extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inform. Sci. 5(2011), 390–444.

H. M. Srivastava ,Generating relations and other results associated with some families of the extended Hurwitz-Lerch Zeta functions, Springer Plus2 (2013), Article ID 2: 67,1–14.

H. M. Srivastava, A new family of the generalized Hurwitz-Lerch zeta functions with applications, Appl. Math. Inform. Sci. 8 (2014), 1485–1500.

H. M. Srivastava and A. A. Attiya, An integral operator associated with the Hurwitz-Lerch zeta function and dierential subordination, Integral Transforms Spec. Funct. 18(2007),207–216.

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

H. M. Srivastava and J. Choi, Zeta and q Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York,2012.

H. M. Srivastava and S. Gaboury, New expansion formulas for a family of the generalized Hurwitz Lerch zeta functions, Internat. J. Math. Math. Sci. 2014 (2014), Article ID 131067,1–13.

H. M. Srivastava and S. Gaboury, A new class of analytic functions define by means of a generalization of the Srivastava-Attiya operator,J. Inequal. Appl. 25(2015),Article ID39,1–15.

H. M. Srivastava, S. Gaboury and A. Bayad, Expansion formulas for an extended Hurwitz-Lerch zeta function obtained via fractional calculus, Adv.Dierence Equations 2014(2014), Article ID 169,1–17.

H. M. Srivastava, S. Gaboury and B. J. Fug`ere, Further results involving a class of generalized Hurwitz-Lerch zeta functions, Russian J. Math. Phys. 21(2014),521-537.

H. M. Srivastava, S. Gaboury and F. Ghanim, Certain subclasses of meromorphically univalent functions defned by a linear operator associated with the-generalized Hurwitz-Lerch zeta function, Integral Transforms Spec. Funct. 26 (4)(2015),258–272.

H. M. Srivastava, S. Gaboury and F.Ghanim, Some Further Properties of a Linear Operator Associated with the λ-Generalized Hurwitz-Lerch Zeta Function Related to the Class of Meromorphically Univalent Functions, Applied Mathematics and Computation, 259 (2015), 1019-1029.

H. M. Srivastava, S. Gaboury and R. Tremblay, New relations involving an extended multi-parameter Hurwitz-Lerch zeta function with applications, Int. J. Anal. 2014 (2014), Article ID 680850,1–21.

H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi,New Delhi and Madras,1982.

H. M. Srivastava, D. Jankov, T. K. Pog´any and R. K. Saxena, Two-sided inequalities for the extended Hurwitz-Lerch Zeta function, Comput. Math. Appl. 62 (2011), 516–522.

H. M. Srivastava and H. L. Manocha, A Treatiseon Generating Functions, Halsted Press (Ellis Horwoord Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto,1984.

H. M. Srivastava,D.R˘aducanu and G. S. S˘al˘agean, A new class of generalized close-to-starlike functions defned by the Srivastava-Attiya operator, Acta Math. Sinica (English Ser.) 29 (2013), 833–840.

H. M. Srivastava, R. K. Saxena, T. K. Pog´any and R. Saxena, Integral and computational representations of the extended Hurwitz-Lerch zeta function, Integral Transforms Spec. Funct. 22 (2011), 487–506.

Z. -G. Wang, H. M. Srivastava and S.M. Yuan, Some basic properties of certain subclasses of meromorphically starlike functions, J. Inequal. Appl. 2014(2014), Article ID 2014:29,1–12.

S.-M.Yuan,Z.M.Liu and H. M. Srivastava, Some inclusion relation ships and integral preserving properties of certain subclasses of meromorphic functions associated with a family of integral operators, J. Math. Anal. Appl. 337(2008),505-515.


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