Multivalent harmonic functions involving multiplier transformation

In the present investigation we study a subclass of multivalent harmonic functions involving multiplier transformation. An equivalent convolution class condition and a sufficient coefficient condition for this class is acquired. We also show that this coefficient condition is necessary for functions belonging to its subclass. As an application of coefficient condition, a necessary and sufficient hypergeometric inequality is also given. Further, results on bounds, inclusion relation, extreme points, a convolution property and a result based on the integral operator are obtained.

___

  • Ahuja, O. P., Jahangiri, J. M., Multivalent harmonic starlike functions, Ann. Univ. Mariac Curie-Sklodowska Section A, 55(1) (2001), 1–13.
  • Ahuja, O. P., Jahangiri, J. M., Errata to Multivalent harmonic starlike function, Ann. Univ. Mariac Curie-Sklodowska Section A 56(1) (2002), 105.
  • Ahuja, O. P., Aghalary, R., Joshi, S. B., Harmonic univalent functions associated with k-uniformly starlike functions, Math. Sci. Res. J., 9(1) (2005), 9–17.
  • Ahuja, O. P., Güney, H. Ö., Sakar, F. M., Certain classes of harmonic multivalent functions based on Hadamard product, J. Inequal. Appl., 2010 (2009), Art. ID 759251, 12pp. https://doi.org/10.1155/2009/759251
  • Güney, H. Ö., Ahuja, O. P., Inequalities involving multipliers for multivalent harmonic functions, J. Inequal. Pure Appl. Math., 7(5) Art. 190 (2006), 1–9.
  • Clunie, J., Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Aci. Fenn. Ser. A Math., 9 (1984), 3–25.
  • Carlson, B. C., Shaffer, D. B., Starlike and prestarlike hypergeometric functions, SIAM, J. Math. Anal., 15 (1984), 737–745.
  • Duren, P., Hengartner, W., Laugesen, R. S., The argument principle for harmonic functions, Amer. Math. Monthly, 103 (1996), 411–415.
  • Dziok, J., Srivastava, H. M., Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput., 103 (1999), 1–13. https://doi.org/10.1016/S0096-3003(98)10042-5
  • Ebadian, A., Tehranchi, A., On certain classes of harmonic p- valent functions by applying the Ruscheweyh derivatives, Filomat, 23(1) (2009), 91–101.
  • Murugusundaramoorthy, G., Raina, R. K., On a subclass of harmonic functions associated with Wright’s generalized hypergeometric functions, Hacettepe J. Math. Stat., 38(2) (2009), 129–136.
  • Murugusundaramoorthy, G., Vijaya, K., Starlike harmonic functions in parabolic region associated with a convolution structure, Acta Univ. Sapientiae Math., 2(2) (2010), 168–183.
  • Murugusundaramoorthy, G., Vijaya, K., Raina, R. K., A subclass of harmonic univalent functions with varying arguments defined by Dziok-Srivastava operator, Archivun Mathematicum (BRNO) Tomus, 45 (2009), 37-46.
  • Murugusundaramoorthy, G., Harmonic starlike functions of complex order involving hypergeometric functions, Math. Vesnik, 64(4) (2012), 316–325.
  • Murugusundaramoorthy, G., Uma, K., Harmonic univalent functions associated with generalized hypergeometric functions, Bull. Math. Anal. Appl., 2(2) (2010), 69–76.
  • Omar, R., Halim, S. A., Multivalent Harmonic Functions defined by Dziok-Srivastava operator, Bull. Malays. Math. Sci. Soc., 35(3)(2) (2012), 601–610.
  • Owa, S., On the distortion theorems - I, Kyungpook. Math. J., 18 (1978) 53–59.
  • Porwal, S., Some properties of a subclass of harmonic univalent functions defined by the multiplier transformations, Indian J. Pure Appl. Math., 46(3) (2015), 309–335. https://doi.org/10.1007/s13226-015-0132-9
  • Porwal, S., On a new subclass of harmonic univalent functions defined by multiplier transformation, Mathematica Moravica, 19(2) (2015), 75–87. http://dx.doi.org/10.5937/MatMor1502075P
  • Srivastava, H. M., Li, Shu-Hai, Tang, H., Certain classes of k-uniformly close-to-convex functions and other related functions defined by using the Dziok-Srivastava operator, Bull. Math. Anal. Appl., 1(3) (2009), 49–63.
  • Ruscheweyh, S., New criteria for univalent functions, Proc. Amer. Math. Soc., 49 (1975), 109–115. https://doi.org/10.1090/S0002-9939-1975-0367176-1
  • Atshan, W. G., Kulkarni, S. R., Raina, R. K., A class of multivalent harmonic functions involving a generalized Ruscheweyh type operator, Math. Vesnik, 60 (2008), 207–213.