Bi-Univalent Function Classes Defined by Using an Einstein Function and a New Generalised Operator
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Keywords:
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Einstein Function, Subordination, Distortion Theorem, Covering Theorem, Radius of Convexity, Bi-Univalent Functions
Abstract
Let A be the class of all analytic and univalent functions f (z) = z+Σ∞k=2 akzk in the open unit disc D = {z:|z|<1 }. S then represents the classes of every function in A that is univalent in D. For every f ∈ S, there is an inverse f−1. A function f ∈ A in D is categorised as bi-univalent if f and its inverse g = f−1 are both univalent. Motivated by the generalised operator, subordination principle, and the first Einstein function, we present a new family of bi-univalent analytic functions on the open unit disc of the complex plane. The functions contained in the subclasses are used to account for the initial coefficient estimate of |a2|. In this study, we derive the results for the covering theorem, distortion theorem, rotation theorem, growth theorem, and the convexity radius for functions of the class Ns,m,kλ,α (Σ, E) of bi-univalent functions related to an Einstein function and a generalised differential operator Ds,m,kλ,α f (z). We use the elementary transformations that preserve the class Ns,m,kλ,α (Σ, E) in order to attain the intended results. The required properties
are then obtained.
References
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Srivastava, H. M., A. K. Mishra, and P. Gochhayat (2010). Certain Subclasses of Analytic and Bi Univalent Functions. Applied Mathematics Letters, 23(10); 1188–1192
Srivastava, H. M., M. F. Sakar, and G. H. Özlem (2018). Some General Coefficient Estimates for a New Class of Analytic and Bi-Univalent Functions Defined by a Linear Combination. Filomat, 32(4); 1313–1322
Wanas, A. K. (2019). New Differential Operator for Holomorphic Functions. Earthline Journal of Mathematical Sciences, 2(2); 527–537
Wei, D. K. (2017). A Study on Univalent Functions and their Geometrical Properties. Ph.D. thesis, Universiti Tunku Abdul Rahman
Weisstein, E. W. (2022). Einstein Functions. MathWorld-A Wolfram Web Resource
Xu, Q. H., Y. C. Gui, and H. M. Srivastava (2012). Coefficient Estimates for a Certain Subclass of Analytic and Bi-Univalent Functions. Applied Mathematics Letters, 25(6); 990–994
Yunus, Y., A. Akbarally, and S. A. Halim (2017). Strongly Starlike Functions Associated with a New Operator. In AIP Conference Proceedings, 1870; 040001
Zhang, C., S. Khan, A. Hussain, N. Khan, S. Hussain, and N. Khan (2021). Applications of q-Difference Symmetric Operator in Harmonic Univalent Functions. AIMS Mathematics, 7; 667–680
Bureau of Standards, USA. 10th printing
Al-Ameedee, S. A., W. G. Atshan, and F. A. Al-Maamori (2020). Coefficients Estimates of Bi-Univalent Functions Defined by New Subclass Function. In Journal of Physics: Conference Series, 1530; 012105
Ali, R. M., S. K. Lee, V. Ravichandran, and S. Supramaniam (2012). Coefficient Estimates for Bi Univalent Ma-Minda Starlike and Convex Functions. Applied Mathematics Letters, 25(3); 344–351
Arfken, G. B. and H. J. Weber (1999). Mathematical Methods for Physicists. Academic Press
Awasthi, J. (2017). A Generalized Sub Class of Univalent Starlike Functions with a Linear Operator. Journal of Applied Research and Technology, 2(2); 113–122
Brannan, D. A. and J. Clunie (1980). Aspects of Contemporary Complex Analysis. Academic Press
Brannan, D. A. and T. Taha (1988). On Some Classes of Bi-Univalent Functions. In Mathematical Analysis and Its Applications. Elsevier; 53–60
Cho, N., V. Kumar, S. S. Kumar, and V. Ravichandran (2019). Radius Problems for Starlike Functions Associated with the Sine Function. Bulletin of the Iranian Mathematical Society, 45; 213–232
De Branges, L. (1985). A Proof of the Bieberbach Conjecture.
Acta Mathematica, 154(1-2); 137–152
Duren, P. L. (2001). Univalent Functions, Volume 259. Springer Science & Business Media
El-Qadeem, A. H., M. A. Mamon, and I. S. Elshazly (2022a). Application of Einstein Function on Bi Univalent Functions Defined on the Unit Disc. Symmetry, 14(4); 758
El-Qadeem, A. H., S. A. Saleh, and M. A. Mamon (2022b). Bi-Univalent Function Classes Defined by Using a Second Einstein Function. Journal of Function Spaces, 2022; 1–10
Elhaddad, S. and M. Darus (2021). Differential Subordination and Superordination for a New Differential Operator Containing Mittag-Leffler Function. Kragujevac Journal of Mathematics, 45(5); 699–708
Frasin, B. (2020). A New Differential Operator of Analytic Functions Involving Binomial Series. Boletim da Sociedade Paranaense de Matemática, 38(5); 205–213
Gradshteyn, I. S. and I. M. Ryzhik (2014). Table of Integrals, Series, and Products. Academic Press
Janowski, W. (1970). Extremal Problems for a Family of Functions with Positive Real Part and for Some Related Families. In Annales Polonici Mathematici, 2; 159–177
Kumar, S. and S. K. Sahoo (2021). Radius of Convexity for Integral Operators Involving Hornich Operations. Journal of Mathematical Analysis and Applications, 502(2); 125265
Lewin, M. (1967). On a Coefficient Problem for Bi-Univalent Functions. Proceedings of the American Mathematical Society, 18(1); 63–68
Ma, W. (1992). A Unified Treatment of Some Special Classes of Univalent Functions. In Proceedings of the Conference on Complex Analysis, 1992. International Press Inc., page 157–169
Mendiratta, R., S. Nagpal, and V. Ravichandran (2015). On a Subclass of Strongly Starlike Functions Associated with Exponential Function. Bulletin of the Malaysian Mathematical Sciences Society, 38; 365–386
Miller, S. S. and P. T. Mocanu (2000). Differential Subordinations: Theory and Applications. CRC Press
Natalini, P. and A. Bernardini (2003). A Generalization of the Bernoulli Polynomials. Journal of Applied Mathematics, 2003(3); 155–163
Netanyahu, E. (1969). The Minimal Distance of the Image Boundary from the Origin and the Second Coefficient of a Univalent Function in |z|<1. Archive for Rational Mechanics and Analysis, 32(2); 100–112
Omar, R. and S. Abdul Halim (2012). Subclasses of Analytic Functions Associated with Generalised Multiplier Transformations. International Scholarly Research Notices, 2012; 1–9
Orloff, J. (2018). Complex Variables with Applications. MIT Open Course Ware
Oros, G. I. and L. I. Cotîrlă (2022). Coefficient Estimates and the Fekete–Szegö Problem for New Classes of m-Fold Symmetric Bi-Univalent Functions. Mathematics, 10(1); 129
Rossdy, M., R. Omar, and S. C. Soh (2021). Coefficient Problems of Bi-Univalent Functions with Respect to Symmetric and Symmetric Conjugate Points Defined by the Al-Oboudi Operator. In Journal of Physics: Conference Series, 1988; 012076
Rossdy, M., R. Omar, and S. C. Soh (2022). New Differential Operator for Analytic Univalent Functions associated with Binomial Series. Proceedings of the 29th National Symposium on Mathematical Sciences
Ruscheweyh, S. (1975). New Criteria for Univalent Functions. Proceedings of the American Mathematical Society, 49(1); 109–115
Saheb, A. H. and A. K. Al-Khafaji (2021). On the Class of Analytic and Univalent Functions Defined by Differential Operator. In Journal of Physics: Conference Series, 1818; 012184
Salagean, G. (1983). Subclasses of Univalent Functions. Lecture Notes in Math., 1013; 362–372
Sivasubramanian, S., R. Sivakumar, T. Bulboacă, and T. N. Shanmugam (2014). On the Class of Bi Univalent Functions. Comptes Rendus Mathematique, 352(11); 895–900
Soni, A., K. K. Mishra, and S. Issar (2018). Coefficient Estimates for Certain Subclass of Bi Univalent Functions Associated with the Class Pm (b, β ). In AIP Conference Proceedings, 1953; 140093
Srivastava, H. M., A. K. Mishra, and P. Gochhayat (2010). Certain Subclasses of Analytic and Bi Univalent Functions. Applied Mathematics Letters, 23(10); 1188–1192
Srivastava, H. M., M. F. Sakar, and G. H. Özlem (2018). Some General Coefficient Estimates for a New Class of Analytic and Bi-Univalent Functions Defined by a Linear Combination. Filomat, 32(4); 1313–1322
Wanas, A. K. (2019). New Differential Operator for Holomorphic Functions. Earthline Journal of Mathematical Sciences, 2(2); 527–537
Wei, D. K. (2017). A Study on Univalent Functions and their Geometrical Properties. Ph.D. thesis, Universiti Tunku Abdul Rahman
Weisstein, E. W. (2022). Einstein Functions. MathWorld-A Wolfram Web Resource
Xu, Q. H., Y. C. Gui, and H. M. Srivastava (2012). Coefficient Estimates for a Certain Subclass of Analytic and Bi-Univalent Functions. Applied Mathematics Letters, 25(6); 990–994
Yunus, Y., A. Akbarally, and S. A. Halim (2017). Strongly Starlike Functions Associated with a New Operator. In AIP Conference Proceedings, 1870; 040001
Zhang, C., S. Khan, A. Hussain, N. Khan, S. Hussain, and N. Khan (2021). Applications of q-Difference Symmetric Operator in Harmonic Univalent Functions. AIMS Mathematics, 7; 667–680
Authors
Rossdy, M. ., Omar, R. ., & Soh, S. C. . (2023). Bi-Univalent Function Classes Defined by Using an Einstein Function and a New Generalised Operator. Science and Technology Indonesia, 8(2), 195–204. https://doi.org/10.26554/sti.2023.8.2.195-204
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