Subclasses of analytic and bi-univalent functions involving a generalized Mittag-Leffler function based on quasi-subordination
Volume 26, Issue 4, pp 379--394
http://dx.doi.org/10.22436/jmcs.026.04.06
Publication Date: January 06, 2022
Submission Date: September 29, 2021
Revision Date: November 07, 2021
Accteptance Date: December 02, 2021
-
1081
Downloads
-
2809
Views
Authors
P. Long
- School of Mathematics and Statistics, Ningxia University, Yinchuan, Ningxia 750021, People's Republic of China.
G. Murugusundaramoorthy
- Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, India.
H. Tang
- School of Mathematics and Computer Sciences, Chifeng University, Chifeng, Inner Mongolia 024000, People's Republic of China.
W. Wang
- School of Mathematics and Statistics, Ningxia University, Yinchuan, Ningxia 750021, People's Republic of China.
Abstract
Two quasi-subordination subclasses \(\mathcal{Q}\Sigma^{\gamma,k}_{
\alpha,\beta}(\vartheta,\rho;\phi)\) and
\(\mathcal{M}\Sigma^{\gamma,k}_{
\alpha,\beta}(\tau,\vartheta,\rho;\phi)\) of the class \(\Sigma\) of
analytic and bi-univalent functions associated with the
convolution operator involving Mittag-Leffler function are
introduced and investigated. Then, the corresponding bound
estimates of the coefficients \(a_2\) and \(a_3\) are provided.
Meanwhile, Fekete-Szegö functional inequalities for these
classes are proved. Besides, some consequences and connections to
all the theorems would be interpreted, which generalize and
improve earlier known results.
Share and Cite
ISRP Style
P. Long, G. Murugusundaramoorthy, H. Tang, W. Wang, Subclasses of analytic and bi-univalent functions involving a generalized Mittag-Leffler function based on quasi-subordination, Journal of Mathematics and Computer Science, 26 (2022), no. 4, 379--394
AMA Style
Long P., Murugusundaramoorthy G., Tang H., Wang W., Subclasses of analytic and bi-univalent functions involving a generalized Mittag-Leffler function based on quasi-subordination. J Math Comput SCI-JM. (2022); 26(4):379--394
Chicago/Turabian Style
Long, P., Murugusundaramoorthy, G., Tang, H., Wang, W.. "Subclasses of analytic and bi-univalent functions involving a generalized Mittag-Leffler function based on quasi-subordination." Journal of Mathematics and Computer Science, 26, no. 4 (2022): 379--394
Keywords
- Fekete-Szegö inequality
- bi-univalent function
- Mittag-Leffler function
- quasi-subordination
MSC
References
-
[1]
H. R. Abdel-Gawad, On the Fekete-Szegö problem for alpha-quasi-convex functions, Tamkang J. Math., 31 (2000), 251--255
-
[2]
R. P. Agarwal, A propos d'une note de M. Pierre Humbert, CR Acad. Sci. Paris, 236 (1953), 2031--2032
-
[3]
P. Agarwal, R. P. Agarwal, M. Ruzhansky, Special Functions and Analysis of Differential Equations, Chapman and Hall/CRC, New York (2020)
-
[4]
P. Agarwal, S. S. Dragomir, M. Jleli, B. Samet, Advances in Mathematical Inequalities and Applications, Birkhäuser/Springer, Singapore (2018)
-
[5]
P. Agarwal, J. J. Nieto, Some fractional integral formulas for the Mittag-Leffler type function with four parameters, Open math., 13 (2015), 537--546
-
[6]
R. M. Ali, S. K. Lee, V. Ravichandran, S. Supramaniam, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett., 25 (2012), 344--351
-
[7]
M. K. Aouf, T. M. Seoudy, Some family of meromorphic $p$-valent functions involving a new operator defined by generalized Mittag-Leffler function, J. Egypt. Math. Soc., 26 (2018), 406--411
-
[8]
A. A. Attiya, Some applications of Mittag-Leffler function in the unit disk, Filomat, 30 (2016), 2075--2081
-
[9]
A. A. Attiya, A. M. Lashin, E. E. Ali, P. Agarwal, Coefficient bounds for certain classes of analytic functions associated with Faber polynomial, Symmetry, 2021 (2021), 13 pages
-
[10]
D. Bansal, J. K. Prajapat, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic Equ., 61 (2016), 338--350
-
[11]
D. A. Brannan, J. G. Clunie, Aspects of contemporary complex analysis, Academic Press, New York--London (1980)
-
[12]
D. A. Brannan, T. S. Taha, On some classes of bi-univalent functions, Studia Univ. Babes-Bolyai Math., 31 (1986), 70--77
-
[13]
T. Bulboacă, Differential subordinations and superordinations: Recent Results, House of Scientific Book Publ., Cluj-Napoca (2005)
-
[14]
M. Caglar, H. Orhan, N. Yagmur, Coefficient bounds for new subclasses of bi-univalent functions, Filomat, 27 (2013), 1165--1171
-
[15]
N. E. Cho, G. Murugusundarmoorthy, K. Vijaya, Bi-univalent functions of complex order based on quasi-subordinate conditions involving Wright hypergeometric functions, J. Comput. Anal. Appl., 24 (2018), 58--70
-
[16]
E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Class. Anal., 2 (2013), 49--60
-
[17]
P. L. Duren, Univalent functions, Springer-Verlag, New York (1983)
-
[18]
M. Fekete, G. Szegö, Eine bemerkung Über ungerade schlichte funktionen, J. London Math. Soc., 8 (1933), 85--89
-
[19]
B. A. Frasin, M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., 24 (2011), 1569--1573
-
[20]
R. Garra, R. Garrappa, The Prabhakar or three parameter Mittag-Leffler function: Theory and application, Commun. Nonlinear Sci. Numer. Simulat., 56 (2018), 314--329
-
[21]
A. W. Goodman, Univalent Functions, Vol: I & II, Mariner Publishing Co., Tampa (1983)
-
[22]
S. P. Goyal, O. Singh, R. Mukherjee, Certain results on a subclass of analytic and bi-univalent functions associated with coefficient estimates and quasi-subordination, Palest. J. Math., 5 (2015), 79--85
-
[23]
M. Grag, P. Manohar, S. L. Kalla, A Mittag-Leffler-type function of two variables, Integral Transforms Spec. Funct., 24 (2013), 934--944
-
[24]
T. Hayami, S. Owa, Coefficient bounds for bi-univalent functions, PanAmer. Math. J., 22 (2012), 15--26
-
[25]
P. Humbert, Quelques resultants retifs a la fonction de Mittag-Leffler, Comptes Rendus de L'Academie Des Sciences, 236 (1953), 1467--1468
-
[26]
P. Humbert, R. P. Agarwal, Sur la fonction de Mittag-Leffler et quelques unes de ses generalizations, Bull. Sci. Math. Ser. II, 77 (1953), 180--185
-
[27]
S. Jain, R. P. Agarwal, P. Agarwal, P. Singh, Certain Unified Integrals Involving a Multivariate Mittag-Leffler Function, Axioms, 2021 (2021), 10 pages
-
[28]
M. Kamarujjama, N. U. Khan, O. Khan, Fractional calculus of generalized $p$-$k$-Mittag-Leffler function using Marichev-Saigo-Maeda operators, Arab J. Math. Sci., 25 (2019), 156--168
-
[29]
B. S. Keerth, B. Raja, Coefficient inequality for certain new subclasses of analytic bi-univalent functions, Theor. Math. Appl., 3 (2013), 1--10
-
[30]
V. S. Kiryakova, The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus, Comput. Math. Appl., 59 (2010), 1885--1895
-
[31]
W. Koepf, On the Fekete-Szegö Problem for Close-to-Convex Functions, Proc. Amer. Math. Soc., 101 (1987), 89--95
-
[32]
S. S. Kumar, V. Kumar, V. Ravichandran, Estimates for initial coefficients of bi-univalent functions, Tamsui Oxf. J. Inf. Math. Sci., 29 (2013), 487--504
-
[33]
A. Y. Lashin, Coefficient estimates for two subclasses of analytic and bi-univalent functions, Ukrainian Math. J., 70 (2019), 1484--1492
-
[34]
M. Lewin, On a coefficients problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), 63--68
-
[35]
N. Magesh, J. Yamini, Coefficient estimates for a certain general subclass of analytic and bi-univalent functions, Appl. Math. Soc., 5 (2014), 1047--1052
-
[36]
F. Y. Mansour, A. A. Attiya, A. Praveen, Subordination and superordination properties for certain family of analytic functions associated with Mittag Leffler function, Symmetry, 2020 (2020), 12 pages
-
[37]
S. S. Miller, P. T. Mocanu, Differential Subordinations: Theory and Applications, Marcel Dekker, New York (2000)
-
[38]
A. K. Mishra, S. Barik, Estimation for initial coefficients of bi-univalent $\vartheta$-convex analytic functions in the unit disc, J. Class. Anal., 7 (2015), 73--81
-
[39]
A. K. Mishra, P. Gochhayat, The Fekete-Szegö problem for $k$-uniformly convex functions and for a class defined by the Owa-Srivastava operator, J. Math. Anal. Appl., 347 (2008), 563--572
-
[40]
G. Mittag-Leffler, Sur la Nouvelle Fonction $E_{\alpha}(x)$, Comptes Rendus de l'Academie des Sciences Paris, 137 (1903), 554--558
-
[41]
G. M. Mittag-Leffler, Sur la représentation analytique d’une branche uniforme d’une fonction monogène: cinquième note, Acta Math., 29 (1905), 101--182
-
[42]
V. F. Morales-Delgado, J. F. Gomez-Aguilar, K. M. Saad, M. A. Khan, P. Agarwal, Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: a fractional calculus approach, Phys. A, 523 (2019), 48--65
-
[43]
E. Netanyahau, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $\mid z\mid<1$, Arch. Rational Mech. Anal., 32 (1969), 100--112
-
[44]
S. Noreen, M. Raza, S. N. Malik, Certain geometric properties of Mittag-Leffler functions, J. Inequal. Appl., 2019 (2019), 15 pages
-
[45]
H. Orhan, N. Magesh, V. K. Balaji, Fekete-Szegö problem for certain class of Ma-Minda bi-univalent functions, Afr. Mat., 27 (2016), 889--897
-
[46]
H. Orhan, D. Răducanu, The Fekete-Szegö problems for strongly starlike functions associted with generalized hypergeometric functions, Math. Comput. Model., 50 (2009), 430--438
-
[47]
T. Panigrahi, G. Murugusundaramoorthy, Coefficient bounds for bi-univalent functions analytic functions associated with Hohlov operator, Proc. Jangjeon Math. Soc., 16 (2013), 91--100
-
[48]
T. Panigrahi, R. K. Raina, Fekete–Szegö coefficient functional for quasi-subordination class, Afr. Mat., 28 (2017), 707--716
-
[49]
A. B. Patil, U. H. Naik, Estimates on initial coefficients of certain subclasses of bi-univalent functions associated with the Hohlov operator, Palest. J. Math., 7 (2018), 487--497
-
[50]
M. S. Robertson, Quasi-subordination and coefficient conjecture, Bull. Amer. Math. Soc., 76 (1970), 1--9
-
[51]
G. Rudolf, M. Francesco, R. Sergei, Mittag-Leffer function: properties and applications, in: Handbook of Fractional Calculus with Applications, Vol. 1, 2019 (2019), 269--296
-
[52]
M. Ruzhansky, Y. J. Cho, P. Agarwal, I. Area, Advances in Real and Complex Analysis with Applications, Birkhäuser/Springer, Singapore, Singapore (2017)
-
[53]
A. Serkan, A. Mehmet, G. Aynur, Analytic continuation of weighted $q$-Genocchi numbers and polynomials, Commun. Korean Math. Soc., 28 (2013), 457--462
-
[54]
H. M. Srivastava, D. Bansal, Coefficient estimates for a subclasses of analytic and bi-univalent functions, J. Egypt. Math. Soc., 23 (2015), 242--246
-
[55]
H. M. Srivastava, S. Bulut, M. Çağlar, N. Yağmur, Coefficient estimates for a general subclasses of analytic and bi-univalent functions, Filomat, 27 (2013), 831--842
-
[56]
H. M. Srivastava, S. Gaboury, F. Ghanim, Coefficient estimates for a general subclass of analytic and bi-univalent functions of the Ma-Minda type, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 112 (2018), 1157--1168
-
[57]
H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23 (2010), 1188--1192
-
[58]
H. M. Srivastava, G. Murugusundaramoorthy, N. Magesh, Certain subclasses of bi-univalent functions associated with the Hohlov operator, Global J. Math. Anal., 1 (2013), 67--73
-
[59]
H. M. Srivastava, G. Murugusundaramoorthy, K. Vijaya, Coefficient estimates for some families of bi-Bazilevic functions of the Ma-Minda type involving the Hohlov operator, J. Class. Anal., 2013 (2013), 167--181
-
[60]
H. M. Srivastava, Ž.. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198--210
-
[61]
D. Styer, D. J. Wright, Result on bi-univalent functions, Proc. Amer. Math. Soc., 82 (1981), 243--248
-
[62]
T. S. Taha, Topic in univalent function theory, Ph.D. thesis, University of London, London (1981)
-
[63]
D. L. Tan, Coefficient estimates for bi-univalent functions, Chinese Ann. Math. Ser. A, 5 (1984), 559--568
-
[64]
H. Tang, G.-T. Deng, S.-H. Li, Coefficient estimates for new subclasses of Ma-Minda bi-univalent functions, J. Inequal. Appl., 2013 (2013), 10 pages
-
[65]
H. Tang, H. M. Srivastava, S. Sivasubramanian, P. Gurusamy, The Fekete-Szegö functional problems for some subclasses of m-fold symmetric bi-univalent functions, J. Math. Inequal., 10 (2016), 1063--1092
-
[66]
A. Wiman, Über die nullstellen der funktionen $E_{\alpha}x$, Acta Math., 29 (1905), 217--234
-
[67]
A. Wiman, Über den fundamental satz in der theorie der funktionen $E_{\alpha}(x)$, Acta Mathematica, 29 (1995), 191--201
-
[68]
Q.-H. Xu, Y.-C. Gui, H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett., 25 (2012), 990--994
-
[69]
Q.-H. Xu, H.-G. Xiao, H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimates problems, Appl. Math. Comput., 218 (2012), 11461--11465
-
[70]
C.-M. Yan, J.-L. Liu, A family of meromorphic functions involving generalized Mittag-Leffler function, J. Math. Inequal., 12 (2018), 943--951