Duality on a certain class of analytic functions related to parabolic domain

Mokhtari Heydari, Maryam; Shams, Saeid; Ebadian, Ali

doi:10.22103/jmmr.2026.27156.1966

Duality on a certain class of analytic functions related to parabolic domain

Articles in Press

Document Type : Research Paper

Authors

Maryam Mokhtari Heydari

Saeid Shams

Ali Ebadian

Department of Mathematics, Faculty of Sciences, Urmia University. Urmia, Iran

10.22103/jmmr.2026.27156.1966

Abstract

In the present paper a certain subclass of analytic functions related to parabolic domains is defined. We characterize duality of the class $ \small
M(\alpha) = \left\{ f \in \mathcal{A} : \operatorname{Re} \{ \alpha\frac{f(z)}{z} + (1 - \alpha) f'(z) \} >\left|\alpha\frac{f(z)}{z} + (1 - \alpha) f'(z) -1\right|,\ z \in \mathbb{D} \right\},$ which gives some interesting results such as neighborhood, necessary and sufficient condition for normalized analytic function $f$ in open unit disk $\mathbb{D}$ to be in $M(\alpha)$ in terms of convolution.

Keywords

Starlike convex functions

neighborhoods

stability

parabolic domain

convolution

Subjects

Analysis

[1] Annamalai, S., Sivasubramanian, S., & Ramachandran, C. (2017). Hankel determinant for a class of analytic functions involving conical domains de ned by subordination. Mathematica Slovaca, 67(4), 945{956. https://doi.org/10.1515/ms-2017-0023

[2] Azizi, S., Ebadian, A., & Yalcn, S. (2019). On T-neighborhoods of harmonic univalent functions. Iranian Journal of Science and Technology, Transactions A: Science, 43(5), 2269{2273. https://doi.org/10.1007/s40995-018-00673-2

[3] Bharati, R., Parvatham, R., & Swaminathan, A. (1997). On subclasses of uniformly convex functions and corresponding class of starlike functions. Tamkang Journal of Mathematics, 28(1), 17{32. https://doi.org/10.5556/j.tkjm.28.1997.4330

[4] Bracci, F., Contreras, M. D., Daz-Madrigal, S., Elin, M., & Shoikhet, D. (2018). Filtrations of in nitesimal generators. Func-
tiones et Approximatio Commentarii Mathematici, 59, 99{115. https://doi.org/10.7169/facm/1710

[5] Ebadian, A., & Kargar, R. (2018). Univalence of integral operators on neighborhoods of analytic functions. Iranian Journal of Science and Technology, Transactions A: Science, 42(2), 911{915. https://doi.org/10.1007/s40995-017-0223-z

[6] Gaderi, N., & Shams, S. (2020). Stability for comprehensive class of analytic functions. International Journal of Nonlinear Analysis and Applications, 11(2), 237{253. https://doi.org/10.22075/ijnaa.2018.16662.1887

[7] Goodman, A. W. (1991). On uniformly starlike functions. Journal of Mathematical Analysis and Applications, 155, 364{370. https://doi.org/10.1016/0022-247X(91)90006-L

[8] Kanas, S. (1998). Stability of convolution and dual sets for the class of k-uniformly convex and k-starlike functions. Folia Scientiarum Universitatis Technicae Resoviensis, 170, 51{64.

[9] Kanas, S., & Wisniowska, A. (1999). Conic regions and k-uniform convexity. Journal of Computational and Applied Mathematics, 105(1{2), 327{336. https://doi.org/10.1016/S0377-0427(99)00018-7

[10] Kanas, S., & Wisniowska, A. (2000). Conic domains and k-starlike functions. Revue Roumaine de Mathematiques Pures et Appliquees, 45, 647{657.

[11] Ma, W. C., & Minda, D. (1992). Uniformly convex functions. Annales Polonici Mathematici, 57(2), 165{175. https://doi.org/10.4064/ap-57-2-165-175

[12] Ma, W. C., & Minda, D. (1993). Uniformly convex functions II. Annales Polonici Mathematici, 58(3), 275{285. https://doi.org/10.4064/ap-58-3-275-285

[13] Nishiwaki, J., & Owa, S. (2007). Certain classes of analytic functions concerned with uniformly starlike and convex functions. Applied Mathematics and Computation, 187(1), 350{355. https://doi.org/10.1016/j.amc.2006.08.132

[14] Pascu, M. N., & Pascu, N. R. (2011). Neighborhoods of univalent functions. Bulletin of the Australian Mathematical Society, 83, 210{219. https://doi.org/10.1017/S0004972710000468

[15] Rogosinski, W. (1943). On the coecients of subordinate functions. Proceedings of the London Mathematical Society, 4, 48{82. https://doi.org/10.1112/PLMS/S2-48.1.48

[16] Rnning, F. (1993). Uniformly convex functions and corresponding class of starlike functions. Proceedings of the American Mathematical Society, 118, 189{196. https://doi.org/10.1090/S0002-9939-1993-1128729-7

[17] Ruscheweyh, S. (1975). New criteria for univalent functions. Proceedings of the American Mathematical Society, 49, 109{115. https://doi.org/10.1090/S0002-9939-1975-0367176-1

[18] Ruscheweyh, S. (1981). Neighborhoods of univalent functions. Proceedings of the American Mathematical Society, 81(4), 521{527. https://doi.org/10.1090/S0002-9939-1981-0601721-6

[19] Ruscheweyh, S. (1982). Convolutions in Geometric Function Theory. Presses de l'Universite de Montreal, Montreal.

[20] Shams, S., Kulkarni, S. R., & Jahangiri, J. M. (2004). Classes of uniformly starlike and convex functions. International Journal of Mathematics and Mathematical Sciences, 2959{2961. https://doi.org/10.1155/S0161171204402014

[21] Shams, S., Ebadian, A., Sayadiazar, M., & Soko l, J. (2014). T-neighborhoods in various classes of analytic functions. Bulletin of the Korean Mathematical Society, 51(3), 659{666. https://doi.org/10.4134/BKMS.2014.51.3.659

[22] Silverman, H. (1995). Neighborhoods of a class of analytic functions. Far East Journal of Mathematical Sciences, 3(2), 175{183.

[23] Sim, Y. J., Kwon, O. S., Cho, N. E., & Srivastava, H. M. (2012). Some classes of analytic functions associated with
conic regions. Taiwanese Journal of Mathematics, 16(1), 387{408. https://doi.org/10.11650/twjm/1500406547