Some Properties for Convolution Structures of Holomorphic Functions
Abstract
In the paper submitted, by employing the linear operator \(I_{\alpha,\beta}^{j}\) defined using the convolution of univalent functions \(k\) and \(h\) belonging to the class \(\mathbb{A}\), which consists of holomorphic functions in the disk \(\mathcal{U =}\left\{ s:s \in \mathbb{C}:|s| < 1 \right\}\) that satisfy the condition \(k(0) = k^{'}(0) - 1 = 0\), we proved that the convolution under this operator belongs to the new class
\(\mathfrak{S}_{(\upsilon,\beta,\alpha)(h)}(\Phi)\), according to the concept of the convex hull method of differential subordination, for any convex univalent function mapping the unit disk into the right half-plane, \(\Phi\), in \(\mathcal{U}\) with \(\Phi(0) = 1\).
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References
Goodman, A. W. (1983a). Univalent functions (Vols. 1–2). Polygonal Publishing House.
Akbarally, A., Soh, S. C., & Ismail, M. (2013). Coefficient bounds for a class multivalent function defined by Salagean operator. International Journal of Pure and Applied Mathematics, 83, 417–423. https://doi.org/10.12732/ijpam.v83i3.4
Ravichandran, V., Polatoglu, Y., Bolcal, M., & Sen, A. (2005). Certain subclasses of starlike and convex functions of complex order. Hacettepe Journal of Mathematics and Statistics, 34, 9–15.
Suchithra, K., Stephen, B. A., & Sivasubramanian, S. (2006). A coefficient inequality for certain classes of analytic functions of complex order. Journal of Inequalities in Pure and Applied Mathematics, 7, 1–6.
Nasr, M., & Aouf, M. (1985). Starlike function of complex order. Journal of Natural Sciences and Mathematics, 25, 1–12.
Wiatrowski, P. (1970). On the coefficients of some family of holomorphic functions. Scientific Papers of the University of Lodz, Mathematical and Natural Sciences, 2, 75–85.
Swamy, S. (2012). Inclusion properties of certain subclasses of analytic functions. International Mathematical Forum, 7, 1751–1760.
Ruscheweyh, S., & Sheil-Small, T. (1973). Hadamard products of Schlicht functions and the Pólya–Schoenberg conjecture. Commentarii Mathematici Helvetici, 48, 119–135.
Shanmugam, T. (1989). Convolution and differential subordination. International Journal of Mathematics and Mathematical Sciences, 12, 333–340. https://doi.org/10.1155/S0161171289000384
Padmanabhan, K., & Parvatham, R. (1985). Some applications of differential subordination. Bulletin of the Australian Mathematical Society, 32, 321–330. https://doi.org/10.1017/S0004972700002410
Goodman, A. W. (1983b). Univalent functions (Vol. 2). Mariner Publishing Company.
Oyekan, E., & Opoola, T. (2014). On a subordination result for analytic functions defined by convolution. Annales Universitatis Mariae Curie-Skłodowska, Sectio A, Mathematica, 68, 59–66. https://doi.org/10.17951/a.2014.68.1.59
Babalola, K. O. (2010). Quasi-convolution of analytic functions with applications. arXiv. https://doi.org/10.48550/arXiv.1004.2658
Cho, N. E., & Kim, J. A. (2006). Inclusion properties of certain subclasses of analytic functions defined by a multiplier transformation. Computers & Mathematics with Applications, 52, 323–330. https://doi.org/10.1016/j.camwa.2006.08.022
Xu, N., & Raina, R. K. (2019). Some properties of a class of analytic functions. Boletim da Sociedade Paranaense de Matemática, 37, 173–186. https://doi.org/10.5269/bspm.v37i4.34901
El-Ashwah, R., Drbuk, M., & Aouf, M. (2016). A new class defined by subordination for
γ-spirallike functions. Journal of the Egyptian Mathematical Society, 24, 355–360. https://doi.org/10.1016/j.joems.2015.08.005
Barnard, R. W., & Kellogg, C. (1980). Applications of convolution operators to problems in univalent function theory. Michigan Mathematical Journal, 27, 81–94. https://doi.org/10.1307/mmj/1029002312
Choi, J. H., Kim, Y. C., & Sugawa, T. (2000). Coefficient bounds and convolution properties for certain classes of close-to-convex functions. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 76, 95–98. https://doi.org/10.3792/pjaa.76.95
Odeh, Z., & Jassim, K. (2019). Subordination for certain analytic multivalent functions defined by using differential operator. Journal of Physics: Conference Series, 1234, Article 012110. https://doi.org/10.1088/1742-6596/1234/1/012110
Hussain, A., Aouf, M., & Mostafa, A. (2023). Subordination results of analytic functions defined by convolution. Sirte University Scientific Journal, 13, 55–59. https://doi.org/10.37375/susj.v13i1.1373
Ruscheweyh, S. (1982). Convolutions in geometric function theory. University of Montreal Press.
Miller, S. S., & Mocanu, P. T. (2000). Differential subordinations: Theory and applications. CRC Press.
Miller, S. S., & Mocanu, P. T. (1981). Differential subordinations and univalent functions. Michigan Mathematical Journal, 28, 157–171.
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