On certain subclasses of analytic functions associated with Poisson distribution series

B. A. Frasin

doi:10.2478/ausm-2019-0007

Abstract

In this paper, we find the necessary and sufficient conditions, inclusion relations for Poisson distribution series $𝒦 (m, z) = z + \sum_{n = 2}^{\infty} \frac{m^{n - 1}}{(n - 1)!} e^{- m} z^{n}$ $\mathcal{K}\left( {{\rm{m, z}}} \right) = {\rm{z + }}\sum\limits_{{\rm{n}} = 2}^\infty {{{{{\rm{m}}^{{\rm{n}} - 1}}} \over {\left( {n - 1} \right)!}}{{\rm{e}}^{ - {\rm{m}}}}{{\rm{z}}^{\rm{n}}}} $ to be in the subclasses 𝒮(k, λ) and 𝒞(k, λ) of analytic functions with negative coefficients. Further, we obtain necessary and sufficient conditions for the integral operator $𝒢 (m, z) = \int_{0}^{z} \frac{ℱ (m, t)}{t} dt$ ${\rm{\mathcal{G}}}\left( {{\rm{m}},{\rm{z}}} \right) = \int_0^{\rm{z}} {{{{\rm{\mathcal{F}}}\left( {{\rm{m}},{\rm{t}}} \right)} \over {\rm{t}}}} {\rm{dt}}$ to be in the above classes.

References

[1] N. E. Cho, S. Y. Woo, S. Owa, Uniform convexity properties for hyper-geometric functions, Fract. Cal. Appl. Anal., 5 (3) (2002), 303–313.
Search in Google Scholar Back to article
[2] K. K. Dixit, S. K. Pal, On a class of univalent functions related to complex order, Indian J. Pure Appl. Math., 26 (9) (1995), 889–896.
Search in Google Scholar Back to article
[3] B. A. Frasin, T. Al-Hawary, F. Yousef, Necessary and sufficient conditions for hypergeometric functions to be in a subclass of analytic functions, Afr. Mat., 30 (1–2) (2019), 223–230.10.1007/s13370-018-0638-5
Search in Google Scholar Back to article
[4] E. Merkes, B. T. Scott, Starlike hypergeometric functions, Proc. Amer. Math. Soc., 12 (1961), 885–888.10.1090/S0002-9939-1961-0143950-1
Search in Google Scholar Back to article
[5] M. L. Mogra, On a class of starlike functions in the unit disc I, J. Indian Math. Soc. 40 (1976), 159–161.
Search in Google Scholar Back to article
[6] G. Murugusundaramoorthy, Subclasses of starlike and convex functions involving Poisson distribution series, Afr. Mat. (2017) 28:1357-1366.10.1007/s13370-017-0520-x
Search in Google Scholar Back to article
[7] G. Murugusundaramoorthy, K. Vijaya, S. Porwal, Some inclusion results of certain subclass of analytic functions associated with Poisson distribution series, Hacettepe J. Math. Stat., 45 (4) (2016), 1101–1107.10.15672/HJMS.20164513110
Search in Google Scholar Back to article
[8] S. Owa, On certain classes of univalent functions in the unit disc, Kyungpook Math. J., 24 (2) (1984), 127–136.
Search in Google Scholar Back to article
[9] K. S. Padmanabhan, On certain classes of starlike functions in the unit disc, J. Indian Math. Soc., 32 (1968), 89–103.
Search in Google Scholar Back to article
[10] S. Porwal, An application of a Poisson distribution series on certain analytic functions, J. Complex Anal., (2014), Art. ID 984135, 1–3.10.1155/2014/984135
Search in Google Scholar Back to article
[11] S. Porwal, Mapping properties of generalized Bessel functions on some subclasses of univalent functions, Anal. Univ. Oradea Fasc. Matematica, 20 (2) (2013), 51–60.
Search in Google Scholar Back to article
[12] S. Porwal, M. Kumar, A unified study on starlike and convex functions associated with Poisson distribution series, Afr. Mat., 27 (5) (2016), 1021–1027.10.1007/s13370-016-0398-z
Search in Google Scholar Back to article
[13] H. Silverman, Starlike and convexity properties for hypergeometric functions, J. Math. Anal. Appl., 172 (1993), 574–581.10.1006/jmaa.1993.1044
Search in Google Scholar Back to article
[14] H. M. Srivastava, G. Murugusundaramoorthy, S. Sivasubramanian, Hypergeometric functions in the parabolic starlike and uniformly convex domains, Integr. Transf. Spec. Func., 18 (2007), 511–520.101-1107.10.1080/10652460701391324
Search in Google Scholar Back to article

On certain subclasses of analytic functions associated with Poisson distribution series

Abstract

Paradigm

My account