Carathéodory properties of Gaussian hypergeometric function associated with differential inequalities in the complex plane

Georgia Irina Oros; Georgia Irina Oros

doi:10.3934/math.2021759

AIMS Mathematics

2021, Volume 6, Issue 12: 13143-13156. doi: 10.3934/math.2021759

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Carathéodory properties of Gaussian hypergeometric function associated with differential inequalities in the complex plane

Georgia Irina Oros ^,

Department of Mathematics and Computer Science, University of Oradea, 1 Universitii str., 410087 Oradea, Romania

Received: 06 April 2021 Accepted: 25 August 2021 Published: 15 September 2021
MSC : 30C45, 30C80

The results presented in this paper highlight the property of the Gaussian hypergeometric function to be a Carathéodory function and refer to certain differential inequalities interpreted in form of inclusion relations for subsets of the complex plane using the means of the theory of differential superordination and the method of subordination chains also known as Löwner chains.
- differential superordination,
- analytic function,
- convex function,
- univalent function,
- subordinant,
- best subordinant,
- Gaussian hypergeometric function,
- Löwner chain
Citation: Georgia Irina Oros. Carathéodory properties of Gaussian hypergeometric function associated with differential inequalities in the complex plane[J]. AIMS Mathematics, 2021, 6(12): 13143-13156. doi: 10.3934/math.2021759

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Abstract

The results presented in this paper highlight the property of the Gaussian hypergeometric function to be a Carathéodory function and refer to certain differential inequalities interpreted in form of inclusion relations for subsets of the complex plane using the means of the theory of differential superordination and the method of subordination chains also known as Löwner chains.

References

[1]	H. M. Srivastava, Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. A, 44 (2020), 327–344.
[2]	L. de Branges, A proof of the Bieberbach conjecture, Acta Math., 154 (1985), 137–152. doi: 10.1007/BF02392821
[3]	E. P. Merkes, W. T. Scott, Starlike hypergeometric functions, P. Am. Math. Soc., 12 (1961), 885–888. doi: 10.1090/S0002-9939-1961-0143950-1
[4]	E. Kreyszig, J. Todd, The radius of univalence of the error function, Numer. Math., 1 (1959), 78–89. doi: 10.1007/BF01386375
[5]	E. Kreyszig, J. Todd, On the radius of univalence of the function $\exp z^{2} \int_{0}^{z}{\exp {(-t^{2})dt}}$, Pac. J. Math., 9 (1959), 123–127. doi: 10.2140/pjm.1959.9.123
[6]	S. Ruscheweyh, V. Singh, On the order of starlikeness of hypergeometric functions, J. Math. Anal. App., 113 (1986), 1–11. doi: 10.1016/0022-247X(86)90329-X
[7]	S. S. Miller, P. T. Mocanu, Univalence of Gaussian and confluent hypergeometric functions, P. Am. Math. Soc., 110 (1990), 333–342.
[8]	J. Dziok, H. M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput., 103 (1999), 1–13. doi: 10.1016/S0377-0427(98)00235-0
[9]	H. M. Srivastava, Some Fox-Wright generalized hypergeometric functions and associated families of convolution operators, Appl. Anal. Discr. Math., 1 (2007), 56–71. doi: 10.2298/AADM0701056S
[10]	J. Dziok, H. M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integr. Transf. Spec. F., 14 (2003), 7–18. doi: 10.1080/10652460304543
[11]	G. I. Oros, G. Oros, I. H. Kim, N. E. Cho, Differential subordinations associated with the Dziok-Srivastava operator, Math. Rep., 13 (2011), 57–64.
[12]	G. I. Oros, G. Oros, Second order differential subordinations and superordinations using the Dziok-Srivastava linear operator, Mathematica, 54 (2012), 155–164.
[13]	S. Devi, H. M. Srivastava, A. Swaminathan, Inclusion properties of a class of functions involving the Dziok-Srivastava operator, Korean J. Math., 24 (2016), 139–168. doi: 10.11568/kjm.2016.24.2.139
[14]	V. Kiryakova, Unified approach to univalency of the Dziok-Srivastava and the fractional calculus operators, Adv. Math., 1 (2012), 33–43.
[15]	M. A. Chaudhry, A. Qadir, H. M. Srivastava, R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159 (2004), 589–602.
[16]	H. M. Srivastava, P. Agarwal, S. Jain, Generating functions for the generalized Gauss hypergeometric functions, Appl. Math. Comput., 247 (2014), 348–352.
[17]	H. M. Srivastava, R. Jain, M. K. Bansal, A study of the s-generalized Gauss hypergeometric function and its associated integral transforms, Turkish J. Anal. Number Theory, 3 (2015), 116–119.
[18]	H. M. Srivastava, S. M. Khairnar, Meena More, Inclusion properties of a subclass of analytic functions defined by an integral operator involving the Gauss hypergeometric function, Appl. Math. Comput., 218 (2011), 3810–3821.
[19]	S. S. Miller, P. T. Mocanu, Subordinants of differential superordinations, Complex Var., 48 (2003), 815–826.
[20]	N. E. Cho, I. H. Kim, H. M. Srivastava, Sandwich-type theorems for multivalent functions associated with the Srivastava-Attiya operator, Appl. Math. Comput., 217 (2010), 918–928.
[21]	H. M. Srivastava, A. K. Wanas, Strong differential sandwich results of $\lambda$-pseudo-starlike functions with respect to symmetrical points, Mathematica Moravica, 23 (2019), 45–58. doi: 10.5937/MatMor1902045S
[22]	A. K. Wanas, H. M. Srivastava, Differential sandwich theorems for Bazilevič function defined by convolution structure, Turkish J. Ineq., 4 (2020), 10–21.
[23]	G. I. Oros, Applications of inequalities in the complex plane associated with confluent hypergeometric function, Symmetry, 13 (2021), 259. doi: 10.3390/sym13020259
[24]	S. S. Miller, P. T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl., 65 (1978), 289–305. doi: 10.1016/0022-247X(78)90181-6
[25]	S. S. Miller, P. T. Mocanu, Differential subordinations and univalent functions, Mich. Math. J., 28 (1981), 157–172.
[26]	J. A. Antonino, S. S. Miller, Systems of simultaneous differential inequalities, inclusions and subordinations in the complex plane, Anal. Math. Phys., 10 (2020), 32. doi: 10.1007/s13324-020-00372-5
[27]	S. S. Miller, P. T. Mocanu, M. O. Reade, All $\alpha$-convex functions are starlike, Revue Roumaine de Mathematiques Pures et Appliquees, 17 (1972), 1395–1397.
[28]	S. Miller, Differential inequalities and Carathéodory functions, B. Am. Math. Soc., 81 (1975), 79–81. doi: 10.1090/S0002-9904-1975-13643-3
[29]	P. Delsarte, Y. Genin, A simple proof of Livingston's inequality for Carathéodory functions, P. Am. Math. Soc., 107 (1989), 1017–1020.
[30]	M. Nunokawa, Differential inequalities and Carathéodory functions, P. JPN Acad. Ser. A Math., 65 (1989), 326–328.
[31]	M. Nunokawa, A. Ikeda, N. Koike, Y. Ota, H. Saitoh, Differential inequalities and Carathéodory functions, J. Math. Anal. Appl., 212 (1997), 324–332. doi: 10.1006/jmaa.1997.5519
[32]	M. Nunokawa, S. Owa, N. Takahashi, H. Saitoh, Sufficient conditions for Carathéodory functions, Indian J. Pure Ap. Math., 33 (2002), 1385–1390.
[33]	D. Yang, S. Owa, K. Ochiai, Sufficient conditions for Carath éodory functions, Comput. Math. Appl., 51 (2006), 467–474. doi: 10.1016/j.camwa.2005.10.008
[34]	H. M. Srivastava, S. Owa, Some new results associated with Carath éodory functions of order $\alpha $, J. Comput. Sci. Appl. Math., 2 (2016), 11–13. doi: 10.37418/jcsam.2.1.2
[35]	M. Nunokawa, O. S. Kwon, Y. J. Simb, N. E. Cho, Sufficient Conditions for Carathéodory functions, Filomat, 32 (2018), 1097–1106. doi: 10.2298/FIL1803097N
[36]	I. H. Kim, Y. J. Sim, N. E. Cho, New criteria for Carathéodory functions, J. Inequal. Appl., 13 (2019), 1–16.
[37]	O. S. Kwon, Y. J. Sim, Sufficient conditions for Carathéodory functions and applications to univalent functions, Math. Slovaca, 69 (2019), 1065–1076. doi: 10.1515/ms-2017-0290
[38]	N. E. Cho, H. M. Srivastava, E. A. Adegani, A. Motamednezhad, Criteria for a certain class of the Carathéodory functions and their applications, J. Inequal. Appl., 2020 (2020), 1–14. doi: 10.1186/s13660-019-2265-6
[39]	K. Löwner, Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I, Math. Ann., 89 (1923), 103–121. doi: 10.1007/BF01448091
[40]	C. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, 1975.
[41]	G. I. Oros, Univalence conditions for Gaussian hypergeometric function involving differential inequalities, Symmetry, 13 (2021), 904. doi: 10.3390/sym13050904
[42]	J. A. Antonino, S. S. Miller, Systems of simultaneous differential inclusions implying function containment, Mathematics, 9 (2021), 1252. doi: 10.3390/math9111252
[43]	J. A. Antonino, S. S. Miller, Systems of simultaneous differential containments and superordinations in the complex plane, Complex Var. Elliptic, 2021, DOI: 10.1080/17476933.2021.1953490.

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