Some properties of a generalized multiplier transform on analytic $p$-valent functions

J. O.  Hamzat; R. M.  El-Ashwah

doi:10.37863/umzh.v74i9.6173

J. O. Hamzat Univ. Lagos, Nigeria https://orcid.org/0000-0002-9956-2673
R. M. El-Ashwah Faculty of Science Damietta Univ., New Damietta, Egypt

DOI: https://doi.org/10.37863/umzh.v74i9.6173

Keywords: Analytic, Univalent, Multiplier transform, Differential operator, Growth and Distortion Theorem.

Abstract

UDC 517.5

For a function $$f(z)=z^{p}+\sum^{\infty}_{k=1} a_{k+p}z^{k+p},$$ where $p\in\mathbb{N},$ the authors investigate some properties of a more general multiplier transform on analytic $p$-valent functions in an open unit disk. The applications of the obtained results to fractional calculus are pointed out, while several other corollaries follow as simple consequences.

References

Y. Avici, E. Zlotkiewicz, On harmonic univalent mappings, Ann. Univ. Marie Curie-Sklodowska Sect. A,~44, 1 – 7 (1990).

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

N. E. Cho, H. M. Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Modeling, 37, № 1-2, 39 – 49 (2003). DOI: https://doi.org/10.1016/S0895-7177(03)80004-3

J. Clunie, T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Fenn. Math., 9, 3 – 25 (1984). DOI: https://doi.org/10.5186/aasfm.1984.0905

I. V. Denega, Estimate of the inner radii of non-overlapping domains, J. Math. Sci., 242, № 6, 787 – 795 (2019). DOI: https://doi.org/10.1007/s10958-019-04516-2

J. O. Hamzat, M. O. Olayiwola, Application of fractional calculus on certain new subclasses of analytic function, Int. J. Sci. Tech., 3, Issue~10, 235 – 245 (2015).

Y. Komatu, On analytic prolongation family of integral operators, Mathematics (Cluj), 32(55), 141 – 145 (1990).

D. O. Makinde, J. O. Hamzat, A. M. Gbolagade, A generalized multiplier transform on a univalent integral operator, J. Contemp. Appl. Math., 9, № 1, 24 – 31 (2019).

H. M. Srivastava, S. Owa, Current topics in analytic function, World Sci., Singapore (1992). DOI: https://doi.org/10.1142/1628

S. R. Swamy, Inclusion properties of certain subclasses of analytic functions, Int. Math. Forum, 7, № 36, 1751 – 1760 (2012).

W. G. Atshan, Fractional calculus on a subclass of Spiralike functions defined by Komatu operator, Int. Math. Forum, 3, № 32, 1587 – 1594 (2008).