Sufficient and necessary conditions for the generalized distribution series to be in subclasses of univalent functions

W. Y.  Kota; R. M.  El-Ashwah

doi:10.3842/umzh.v75i10.7267

W. Y. Kota Department of Mathematics, Faculty of Science, Damietta University New Damietta, Egypt
R. M. El-Ashwah Department of Mathematics, Faculty of Science, Damietta University New Damietta, Egypt

DOI: https://doi.org/10.3842/umzh.v75i10.7267

Keywords: Generalized distribution; univalent functions; analytic functions; Hadamard product.

Abstract

UDC 517.5

We establish a relationship between the subclasses of univalent functions and generalized distribution series. The main aim of our investigation is to obtain necessary and sufficient conditions for the generalized distribution series to belong to the classes ${\mathcal{TF}}(\rho,\vartheta),$ $\mathcal{{TH}}(\rho,\vartheta), \,\mathcal{{TJ}}(\rho,\vartheta)$, and $\mathcal{{TX}}(\rho,\vartheta)$. In addition, we obtain some particular cases of our main results.

References

S. Altinkaya, S. Yalcn, Poisson distribution series for analytic univalent functions, Complex Anal. and Oper. Theory, 12, № 5, 1315–1319 (2018). DOI: https://doi.org/10.1007/s11785-018-0764-y

A. K. Bakhtin, I. V. Denega, Weakened problem on extremal decomposition of the complex plane, Mat. Stud., 51, № 1, 35–40 (2019). DOI: https://doi.org/10.15330/ms.51.1.35-40

A. K. Bakhtin, I. V. Denega, Extremal decomposition of the complex plane with free poles II, J. Math. Sci., 246, № 1, 1–17 (2020). DOI: https://doi.org/10.1007/s10958-020-04718-z

A. Baricz, Generalized Bessel functions of the first kind, Lecture Notes in Math., 1994, Springer-Verlag, Berlin (2010). DOI: https://doi.org/10.1007/978-3-642-12230-9

V. B. L. Chaurasia, H. S. Parihar, Certain sufficiency conditions on Fox–Wright functions, Demonstr. Math., 41, № 4, 813–822 (2008). DOI: https://doi.org/10.1515/dema-2008-0409

P. Chichra, New subclasses of the class of close-to-convex functions, Proc. Amer. Math. Soc., 62, № 1, 37–43 (1977). DOI: https://doi.org/10.1090/S0002-9939-1977-0425097-1

S. S. Ding, Y. Ling, G. J. Bao, Some properties of a class of analytic functions, J. Math. Anal. and Appl., 195, № 1, 71–81 (1995). DOI: https://doi.org/10.1006/jmaa.1995.1342

R. M. El-Ashwah, Subclasses of bi-univalent functions defined by convolution, J. Egyptian Math. Soc., 22, 348–351 (2014). DOI: https://doi.org/10.1016/j.joems.2013.06.017

R. M. El-Ashwah, A. H. El-Qadeem, Certain geometric properties of some Bessel functions}; arXiv:1712.01687v1 [math.CV] 2 Dec 2017.

R. M. El-Ashwah, W. Y. Kota, Some condition on a poisson distribution series to be in subclasses of univalent functions, Acta Univ. Apulensis, 51, 89–103 (2017). DOI: https://doi.org/10.17114/j.aua.2017.51.08

R. M. El-Ashwah, W. Y. Kota, Some application of a Poisson distribution series on subclasses of univalent functions, J. Fract. Calc. and Appl., 9, № 1, 167–179 (2018).

B. A. Frasin, Poisson distribution series on a general class of analytic functions, Acta Comment. Univ. Tartu. Math., 24, № 2, 241–251 (2020). DOI: https://doi.org/10.12697/ACUTM.2020.24.16

A. Gangadharan, T. N. Shanmugam, H. M. Srivastava, Generalized hypergeometric functions associated with $k$-uniformly convex functions, Comput. Math. Appl., 44, 1515–1526 (2002). DOI: https://doi.org/10.1016/S0898-1221(02)00275-4

A. Y. Lashin, On a certain subclass of starlike functions with negative coefficients, J. Inequal. Pure and Appl. Math., 10, № 2, 1–8 (2009).

A. Y. Lashin, A. O. Badghaish, A. Z. Bajamal, The sufficient and necessary conditions for the Poisson distribution series to be in some subclasses of analytic functions, J. Funct. Spaces, 2022, Article ID 2419196 (2022). DOI: https://doi.org/10.1155/2022/2419196

A. Y. Lashin, A. O. Badghaish, A. Z. Bajamal, Certain subclasses of univalent functions involving Pascal distribution series, Bol. Soc. Mat. Mex., 28, № 1, 1–11 (2022). DOI: https://doi.org/10.1007/s40590-021-00403-6

R. J. Libera, Some radius of convexity problems, Duke Math. J., 31, 143–158 (1964). DOI: https://doi.org/10.1215/S0012-7094-64-03114-X

G. Murugusundaramoorthy, K. Vijaya, S. Porwal, Some inclusion results of certain subclasses of analytic functions associated with Poisson distribution series, Hacet. J. Math. Stat., 45, № 4, 1101–1107 (2016). DOI: https://doi.org/10.15672/HJMS.20164513110

M. Obradovic, S. B. Joshi, On certain classes of strongly starlike functions, Taiwanese J. Math., 2, № 3, 297–302 (1998). DOI: https://doi.org/10.11650/twjm/1500406969

H. Orhan, On a subclass of analytic functions with negative coefficients, Appl. Math. and Comput., 142, № 2-3, 243–253 (2003). DOI: https://doi.org/10.1016/S0096-3003(02)00299-0

S. Ponnusamy, F. Ronning, Starlikeness properties for convolution involving hypergeometric series, Ann. Univ. Mariae Curie-Sklodowska, 1, № 16, 141–155 (1998).

S. Porwal, Generalized distribution and its geometric properties associated with univalent functions, J. Complex Anal., 2018, Article ID 8654506 (2018). DOI: https://doi.org/10.1155/2018/8654506

S. Porwal, An application of a Poisson distribution series on certain analytic functions, J. Complex Anal., Article ID~984135 (2014). DOI: https://doi.org/10.1155/2014/984135

S. Porwal, M. Ahmad, Some sufficient condition for generalized Bessel functions associated with conic regions, Vietnam J. Math., 43, 163–172 (2015). DOI: https://doi.org/10.1007/s10013-014-0089-8

R. K. Raina, On univalent and starlike Wright's hypergeometric functions, Rend. Semin. Mat. Univ. Padova, 95, 11–22 (1996).

M. S. Robertson, On the theory of univalent functions, Ann. Math. J., 37, № 2, 374–408 (1936). DOI: https://doi.org/10.2307/1968451

A. Swaminathan, Certain sufficiency conditions on Gaussian hypergeometric functions, J. Inequal. Pure and Appl. Math., 5, № 4, Article ID 83 (2004).