Аннотация: В настоящей работе мы рассматриваем класс гармонических функций типа Яновского, введенный и изученный Дзиоком, члены которого задаются формулой \(h(z) = z + \sum\nolimits_{n=2}^{\infty} h_n z^n\), \(g(z) = \sum\nolimits_{n =1}^{\infty} g_n z^n\) такой, что \( \mathcal{ST}_{H}(F,G)=\left\{ f = h + \overline g \in {H}:\frac{\mathfrak{D}_H f(z)}{f(z )}\prec\frac{1+Fz}{1+G z};\quad (-G \leq F < G \leq 1, \ g_1=0)\right\},\) где \(\mathfrak{D}_H f(z) = zh'(z)-\overline{zg'(z)},\) \(z\in \mathbb{U}=\{z: \; z\in \mathbb{C} \; \hbox{и} \; |z| < 1 \} .\) Мы изучаем связь между этими подклассами гармонических однолистных функций, применяя определенный оператор свертки, касающийся обобщенных гипергеометрических функций Райта, и в качестве следствия приводятся несколько частных случаев. Кроме того, мы указали на определенные связи между классом гармонических функций типа Яновского, включающими обобщенные функции Миттаг-Леффлера. Кратко указаны соответствующие связи представленных результатов с различными известными результатами.
Ключевые слова: гармонические функции, однолистные функции, обобщенные гипергеометрические функции Райта
Образец цитирования: Murugusundaramoorthy G. and Porwal S. On Janowski Type Harmonic Functions Associated with the Wright Hypergeometric Functions // Владикавк. матем. журн. 2023. Т. 25, № 4. C. 91-102.
DOI 10.46698/b2503-7977-9793-e
1. Clunie, J. and Sheil-Small, T. Harmonic Univalent Functions,
Annales Academiae Scientiarum Fennicae, Series A I, Mathematica, 1984, vol. 9, pp. 3-25.
2. Duren, P. Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, 156,
Cambridge, Cambridge University Press, 2004.
DOI: 10.1017/CBO9780511546600.
3. Ahuja, Om. P. Planar Harmonic Univalent and Related Mappings,
Journal of Inequalities in Pure and Applied Mathematics,
2005, vol. 6, no. 4, Article 122, 18 p.
4. Silverman, H. Harmonic Univalent Functions with Negative Coefficients,
Journal of Mathematical Analysis and Applications,
1998, vol. 220, pp. 283-289. DOI: 10.1006/jmaa.1997.5882.
5. Wright, E. M. The Asymptotic Expansion of the Generalized Hypergeometric Function,
Proceedings of the London Mathematical Society, 1940, vol. s2-46, no. 1, pp. 389-408.
DOI: 10.1112/plms/s2-46.1.389.
6. Srivastava, H. M. Some Fox-Wright Generalized Hypergeometric Functions
and Associated Families of Convolution Operators,
Applicable Analysis and Discrete Mathematics,
2007, vol. 1, no. 1, pp. 56-71.
7. Srivastava, H. M. and Manocha, H. L. A Treatise on Generating Functions,
Ellis Horwood Series: Mathematics and its Applications,
Chichester, Ellis Horwood Ltd., 1984.
8. Dziok, J. and Srivastava, H. M. Certain Subclasses of Analytic Functions Associated
with the Generalized Hypergeometric Function, Integral Transforms and Special Functions,
2003, vol. 14, no. 1, pp. 7-18.
DOI: 10.1080/10652460304543.
9. Themangani R., Porwal, S. and Magesh, N. Inclusion Relation Between Various
Subclasses of Harmonic Univalent Functions Associated with the Wright's Generalized Hypergeometric Functions,
Abstract and Applied Analysis, 2020, vol. 2020, Article ID 8811810, pp. 1-6.
DOI: 10.1155/2020/8811810.
10. Dziok, J. On Janowski Harmonic Functions, Journal of Applied Analysis,
2015, vol. 21, no. 2, pp. 99-107. DOI: 10.1515/jaa-2015-0010.
11. Janowski, W. Some Extremal Problems For Certain Families Of Analytic Functions I,
Annales Polonici Mathematici, 1973, vol. 28, pp. 297-326.
12. Jahangiri, J. M. Coefficient Bounds and Univalence Criteria for
Harmonic Functions with Negative Coefficients,
Annales Universitatis Mariae Curie-Sklodowska, sectio A -- Mathematica,
1998, vol. 52, no. 2, pp. 57-66.
13. Jahangiri, J. M. Harmonic Functions Starlike in the Unit Disk,
Journal of Mathematical Analysis and Applications,
1999, vol. 235, pp. 470-477. DOI: 10.1006/jmaa.1999.6377.
14. Ahuja, Om. P. Connections Between Various Subclasses of Planar Harmonic
Mappings Involving Hypergeometric Functions, Applied Mathematics and Computation,
2008, vol. 198, no. 1, pp. 305-316. DOI: 10.1016/j.amc.2007.08.035.
15. Ponnusamy, S. and Ronning, F. Starlikeness Properties for Convolutions Involving Hypergeometric Series,
Annales Universitatis Mariae Curie-Sklodowska, sectio A -- Mathematica,
1998, vol. 52, no. 1, pp. 141-155.
16. Porwal, S. and Dixit, K. K. An Application of Hypergeometric Functions on Harmonic Univalent Functions,
Bulletin of Mathematical Analysis and Applications, 2010, vol. 2, no. 4, pp. 97-105.
17. Miller, S. S. and Mocanu, P. T. Univalence of Gaussian and Confluent Hypergeometric Functions,
Proceedings of the American Mathematical Society, 1990, vol. 110, no. 2, pp. 333-342.
18. Owa, S. and Srivastava, H. M. Univalent and Starlike Generalized Hypergeometric Functions,
Canadian Journal of Mathematics, 1987, vol. 39, no. 5, pp. 1057-1077.
19. Aouf, M. K. and Dziok, J. Distortion and Convolution Theorems for Operators
of Generalized Fractional Calculus Involving Wright Function,
Journal of Applied Analysis, 2008, vol. 14, no. 2, pp. 183-192.
DOI: 10.1515/JAA.2008.183.
20. Murugusundaramoorthy, G. and Raina, R. K. On a Subclass of Harmonic Functions Associated
with Wright's Generalized Hypergeometric Functions, Synthesis Lectures on Mathematics and Statistics,
2009, vol. 38, no. 2, pp. 129-136.
21. Raina, R. K. and Sharma, P. Harmonic Univalent Functions Associated with Wright's
Generalized Hypergeometric Functions, Integral Transforms and Special Functions,
2011, vol. 22, no. 8, pp. 561-572. DOI: 10.1080/10652469.2010.535797.
22. Raina, R. K. and Nahar, T. S. On Characterization of Certain Wright's Generalized Hypergeometric
Functions Involving Certain Subclasses of Analytic Functions,
Informatica, Vilnius, Institute of Mathematics and Informatics,
1999, vol. 10, no. 2, pp. 219-230.
23. Chaurasia, V. B. L. and Parihar, H. S. Certain Sufficiency Conditions on Fox-Wright Functions,
Demonstratio Mathematica, 2008, vol. 41, no. 4, pp. 813-822.
DOI: 10.1515/dema-2008-0409.
24. Attiya, A. A. Some Applications of Mittag-Leffler Function in the Unit Disk,
Filomat, 2016, vol. 30, no. 7, pp. 2075-2081. DOI: 10.2298/FIL1607075A.
25. Porwal, S. Connections Between Various Subclasses of Planar Harmonic
Mappings Involving Generalized Bessel Functions,
Thai Journal of Mathematics, 2015, vol. 13, no. 1, pp. 33-42.
26. Maharana, S. and Sahoo, S. K. Inclusion Properties of Planar Harmonic Mappings
Associated with the Wright Function, Complex Variables and Elliptic Equations,
2020, vol. 66, no. 10, pp. 1619-1641.
DOI: 10.1080/17476933.2020.1772765.
27. Vijaya, K., Dutta, H. and Murugusundaramoorthy, G. Inclusion Relation Between Subclasses of Harmonic Functions
Associated with Mittag-LeffEr Functions, Mathematics in Engineering, Science and Aerospace, 2020, vol. 11, no. 4, pp. 959-968.
