Faber polynomial coefficients estimates for certain subclasses of $ q $-Mittag-Leffler-Type analytic and bi-univalent functions

Zeya Jia; Nazar Khan; Shahid Khan; Bilal Khan; Zeya Jia; Nazar Khan; Shahid Khan; Bilal Khan

doi:10.3934/math.2022141

AIMS Mathematics

2022, Volume 7, Issue 2: 2512-2528. doi: 10.3934/math.2022141

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Research article

Faber polynomial coefficients estimates for certain subclasses of $ q $-Mittag-Leffler-Type analytic and bi-univalent functions

1.
School of Mathematics and Statistics, Huanghuai University, Zhumadian 463000, Henan, China
2.
Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan
3.
Department of Mathematics and Statistics, Riphah International University Islamabad 44000, Pakistan
4.
School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China

Received: 08 August 2021 Accepted: 01 November 2021 Published: 16 November 2021
MSC : Primary 05A30, 30C45; Secondary 11B65, 47B38

In this paper, we introduce the $ q $-analogus of generalized differential operator involving $ q $-Mittag-Leffler function in open unit disk

$ \begin{equation*} E = \left \{ z:z\in \mathbb{C\ \ }\text{ and} \ \ \left \vert z\right \vert <1\right \} \end{equation*} $

and define new subclass of analytic and bi-univalent functions. By applying the Faber polynomial expansion method, we then determined general coefficient bounds $ |a_{n}| $, for $ n\geq 3 $. We also highlight some known consequences of our main results.
- analytic functions,
- univalent functions,
- analytic and bi-univalent function,
- $ q $-derivative,
- subordination,
- Faber polynomials,
- $ q $-Mittag-Leffler function
Citation: Zeya Jia, Nazar Khan, Shahid Khan, Bilal Khan. Faber polynomial coefficients estimates for certain subclasses of $ q $-Mittag-Leffler-Type analytic and bi-univalent functions[J]. AIMS Mathematics, 2022, 7(2): 2512-2528. doi: 10.3934/math.2022141

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Abstract

In this paper, we introduce the $ q $-analogus of generalized differential operator involving $ q $-Mittag-Leffler function in open unit disk

$ \begin{equation*} E = \left \{ z:z\in \mathbb{C\ \ }\text{ and} \ \ \left \vert z\right \vert <1\right \} \end{equation*} $

and define new subclass of analytic and bi-univalent functions. By applying the Faber polynomial expansion method, we then determined general coefficient bounds $ |a_{n}| $, for $ n\geq 3 $. We also highlight some known consequences of our main results.

References

[1]	Q. Z. Ahmad, N. Khan, M. Raza, M. Tahir, B. Khan, Certain $q$-difference operators and their applications to the subclass of meromorphic $q$-starlike functions, Filomat, 33 (2019), 3385–3397. Available from: https://doi.org/10.2298/FIL1911385A.
[2]	F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Int. J. Math. Math. Sci., 27 (2004), 1429–1436. Available from: https://doi.org/10.1155/S0161171204108090.
[3]	H. Airault, Remarks on Faber polynomials, Int. Math. Forum., 3 (2008), 449–456.
[4]	H. Airault, A. Bouali, Differential calculus on the Faber polynomials, Bull. Sci. Math., 130 (2006), 179–222. doi:10.1016/j.bulsci.2005.10.002. doi: 10.1016/j.bulsci.2005.10.002
[5]	H. Airault, J. Ren, An algebra of differential operators and generating functions on the set of univalent functions, Bull. Sci. Math., 126 (2002), 343–367. Available from: https://doi.org/10.1016/S0007-4497(02)01115-6.
[6]	H. Airault, Symmetric sums associated to the factorizations of Grunsky coefficients, In: Conference, Groups and Symmetries Montreal Canada, April 2007.
[7]	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. Available from: https://doi.org/10.1016/j.aml.2011.09.012.
[8]	S. Altınkaya, S. Yalçın, Coefficient estimates for two new subclasses of bi-univalent functions with respect to symmetric points, J. Funct. Spaces, (2015), Article ID: 145242. Available from: doi.org/10.1155/2015/145242.
[9]	S. Altınkaya, S. Yalçın, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Acta Univ. Apulensis, Mat. Inform., 40 (2014), 347–354.
[10]	S. Altınkaya, S. Yalçın, Initial coefficient bounds for a general class of bi-univalent functions, Int. J. Anal., (2014), Article ID: 867871.
[11]	S. Altinkaya, S. Yalcin, Faber polynomial coefficient bounds for a subclass of bi-univalent functions, C. R. Acad. Sci. Paris, Ser. I., 353 (2015), 1075–1080. Available from: https://doi.org/10.1016/j.crma.2015.09.003.
[12]	A. A. Attiya, Some applications of Mittag-Leffler function in the unit disk, Filomat, 30 (2016), 2075–2081.
[13]	A. Aral, On the generalized Picard and Gauss Weierstrass singular integrals, J. Comput. Anal. Appl., 8 (2006), 249–261.
[14]	A. Aral, V. Gupta, On $q$-Baskakov type operators, Demon-str. Math., 42 (2009), 109–122.
[15]	M. Arif, H. M. Srivastava, S. Uma, Some applications of a $q$ -analogue of the Ruscheweyh type operator for multivalent functions, RACSAM, 113 (2019), 1211–1221. Available from: https://doi.org/10.1007/s13398-018-0539-3.
[16]	D. A. Brannan, J. Clunie, Aspects of contemporary complex analysis, Proceedings of the NATO Advanced Study Instute Held at University of Durham, New York, Academic Press, 1979.
[17]	S. Bulut, Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions, C. R. Acad. Sci. Paris. Ser. I., 352 (2014), 479–484. Available from: https://doi.org/10.1016/j.crma.2014.04.004.
[18]	E. Deniz, J. M. Jahangiri, S. K. Kina, S. G. Hamidi, Faber polynomial coefficients for generalized bi-subordinate functions of complex order, J. Math. Ineq., 12 (2018), 645–653. Available from: dx.doi.org/10.7153/jmi-2018-12-49.
[19]	E. Deniz, H. T. Yolcu, Faber polynomial coefficients for meromorphic bi-subordinate functions of complex order, AIMS Math., 5 (2020), 640–649.
[20]	M. Çağlar, E. Deniz, Initial coefficients for a subclass of bi-univalent functions defined by Salagean differential operator, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 66 (2017), 85–91. Available from: https://doi.org/10.1501/Commua1_0000000777.
[21]	E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Classical Anal., 2 (2013), 49–60. Available from: dx.doi.org/10.7153/jca-02-05.
[22]	D. Raducanu, H. Orhan, Subclasses of analytic functions defined by a generalized differential operator, Int. J. Math. Anal., 4 (2010), 1–15.
[23]	E. Deniz, H. Orhan, The Fekete-Szegö Problem for A Generalized Subclass of Analytic Functions, Kyungpook Math. J., 50 (2010), 37–47.
[24]	P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer, New York, 1983.
[25]	G. Faber, Uber polynomische Entwickelungen, Math. Ann., 57 (1903), 389–408. Available from: https://doi.org/10.1007/BF01444293.
[26]	B. A. Frasin, M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., 24 (2011), 1569–1573. Available from: https://doi.org/10.1016/j.aml.2011.03.048.
[27]	M. Govindaraj, S. Sivasubramanian, On a class of analytic functions related to conic domains involving $q$-calculus, Anal. Math., 43 (2017), 475–487. Available from: https://doi.org/10.1007/s10476-017-0206-5.
[28]	H. Grunsky, Koffizientenbedingungen fur schlict abbildende meromorphe funktionen, Math. Zeit., 45 (1939), 29–61.
[29]	S. G. Hamidi, J. M. Jahangiri, Faber polynomial coefficient estimates for analytic bi-close-to-convex functions, C. R. Acad. Sci. Paris. Ser. I, 352 (2014), 17–20. Available from: https://doi.org/10.1016/j.crma.2013.11.005.
[30]	S. G. Hamidi, J. M. Jahangiri, Faber polynomial coefficients of bi-subordinate functions, C. R. Acad. Sci. Paris, Ser. I, 354 (2016), 365–370. Available from: https://doi.org/10.1016/j.crma.2016.01.013.
[31]	S. G. Hamidi, J. M. Jahangiri, Faber polynomial coefficient estimates for bi-univalent functions defined by subordinations, Bull. Iran. Math. Soc., 41 (2015), 1103–1119. Available from: https://doi.org/10.1007/s41980-018-0011-3.
[32]	S. Hussain, S. Khan, M. A. Zaighum, M. Darus, Z. Shareef, Coefficients bounds for certain subclass of bi-univalent functions associated with Ruscheweyh $q$-differential operator, J. Complex Anal., Article ID 2826514. Available from: https://doi.org/10.1155/2017/2826514.
[33]	S. Hussain, S. Khan, M. A. Zaighum, D. Darus, Certain subclass of analytic functions related with conic domains and associated with Salagean $q$-differential operator, AIMS Math., 2 (2017), 622–634. doi: 10.3934/Math.2017.4.622. doi: 10.3934/Math.2017.4.622
[34]	M. E. H. Ismail, E. Merkes, D. Styer, A generalization of starlike functions, Complex Variables Theory Appl., 14 (1990), 77–84. Available from: https://doi.org/10.1080/17476939008814407.
[35]	F. H. Jackson, On $q$-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193–203.
[36]	F. H. Jackson, On $q$-functions and a certain difference operator, T. Royal Soc. Edinburgh, 46 (1908), 253–281. doi: https://doi.org/10.1017/S0080456800002751. doi: 10.1017/S0080456800002751
[37]	J. M. Jahangiri, On the coefficients of powers of a class of Bazilevic functions, Indian J. Pure Appl. Math., 17 (1986), 1140–1144.
[38]	J. M. Jahangiri, S. G. Hamidi, Coefficient estimates for certain classes of bi-univalent functions, Int. J. Math. Math. Sci., (2013), Article ID 190560. Available from: https://doi.org/10.1155/2013/190560.
[39]	J. M. Jahangiri, S. G. Hamidi, S. Abd Halim, Coefficients of bi-univalent functions with positive real part derivatives, Bull. Malays. Math. Soc., 3 (2014), 633–640.
[40]	S. Kanas, D. Raducanu, Some class of analytic functions related to conic domains, Math. Slovaca, 64 (2014), 1183–1196. Available from: https://doi.org/10.2478/s12175-014-0268-9.
[41]	S. Khan, N. Khan, S. Hussain, Q. Z. Ahmad, M. A. Zaighum, Some subclasses of bi-univalent functions associated with Srivastva-Attiya operator, Bull. Math. Anal. Appl., 9 (2017), 37–44.
[42]	B. Khan, Z. G. Liu, H. M. Srivastava, N. Khan, M. Darus, M. Tahir, A study of some families of multivalent $q$-starlike functions involving higher-order $q$-Derivatives, Mathematics, 8 (2020), Article ID 1470, 1–12. Available from: https://doi.org/10.3390/math8091470.
[43]	B. Khan, Z. G. Liu, H. M. Srivastava, N. Khan, M. Tahir, Applications of higher-order derivatives to subclasses of multivalent $q$-starlike functions, Maejo Internat. J. Sci. Technol., 15 (2021), 61–72.
[44]	B. Khan, H. M. Srivastava, M. Tahir, M. Darus, Q. Z. Ahmad, N. Khan, Applications of a certain integral operator to the subclasses of analytic and bi-univalent functions, AIMS Math., 6 (2021), 1024–1039. doi:10.3934/math.2021061. doi: 10.3934/math.2021061
[45]	B. Khan, H. M. Srivastava, N. Khan, M. Darus, M. Tahir, Q. Z. Ahmad, Coefficient estimates for a subclass of analytic functions associated with a certain leaf-like domain, Mathematics, 8 (2020), Article ID 1334, 1–15. Available from: https://doi.org/10.3390/math8081334.
[46]	W. S. Chung, T. Kim, H. I. Kwon, On the $q$-analog of the Laplace transform, Russ. J. Math. Phys., 21 (2014), 156–168. Available from: https://doi.org/10.1134/S1061920814020034.
[47]	V. Gupta, T. Kim, On a $q$-analog of the Baskakov basis functions, Russ. J. Math. Phys., 20 (2013), 276–282. Available from: https://doi.org/10.1134/S1061920813030035.
[48]	T. Kim, D. S. Kim, W. S. Chung, H. I. Kwon, Some families of $q$-sums and $q$-products, Filomat, 31 (2017), 1611–1618. Available from: https://doi.org/10.2298/FIL1706611K.
[49]	T. Kim, Some identities on the $q$-integral representation of the product of several $q$-Bernstein-type polynomials, Abstr. Appl. Anal., 2011 (2011), Article ID 634675, 11.
[50]	M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), 63–68. Available from: https://doi.org/10.2307/2035225.
[51]	G. M. Mittag-Leffler, Sur la nouvelle fonction $E_{\alpha }(x)$, C R Acad. Sci. Paris, 137 (1903), 554–558.
[52]	G. M. Mittag-Leffler, Sur la representation analytique dune branche uniforme dune fonction monogene, Acta Math., 29 (1905), 101–181.
[53]	E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $ \left \vert z\right \vert < 1$, Arch. Ration. Mech. An., 32 (1969), 100–112. Available from: https://doi.org/10.1007/BF00247676.
[54]	H. Rehman, M. Darus, J. Salah, Coefficient properties involving the generalized k-Mittag-Leffler functions, Transyl. J. Math. Mech.(TJMM), 9 (2017), 155–164.
[55]	G. S. Salagean, Subclasses of univalent functions, Lecture Notes in Mathematics, 1013, Springer (Berlin, 1983), 362–372.
[56]	M. Schiffer, A method of variation within the family of simple functions, Proc. London Math. Soc., 44 (1938), 432–449. Available from: https://doi.org/10.1112/plms/s2-44.6.432.
[57]	A. C. Schaeffer, D. C. Spencer, The coefficients of schlict functions, Duke Math. J., 10 (1943), 611–635. doi: 10.1215/S0012-7094-43-01056-7. doi: 10.1215/S0012-7094-43-01056-7
[58]	S. K. Sharma, R. Jain, On some properties of generalized $q$ -Mittag Leffler function, Math. Aeterna, 4 (2014), 613–619.
[59]	L. Shi, M. Raza, K. Javed, S. Hussain, M. Arif, Class of analytic functions defined by $q$-integral operator in a symmetric region, Symmetry, 11 (2019), 1042. Available from: https://doi.org/10.3390/sym11081042.
[60]	H. M. Srivastava, Certain $q$-polynomial expansions fot functions of several variables. I and II, IMA J. Appl. Math., 30 (1983), 315-323. Available from: https://doi.org/10.1093/imamat/30.3.315.
[61]	H. M. Srivastava, P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, (1985).
[62]	H. M. Srivastava, Univalent functions, fractional calculus, and associated generalized hypergeometric functions, In: H. M. Srivastava, S. Owa, Univalent functions, fractional Calculus, and Their Applications, John Wiley & Sons, New York, etc. (1989).
[63]	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. Trans. A: Sci., 44 (2020), 327–344. Available from: https://doi.org/10.1007/s40995-019-00815-0.
[64]	H. M. Srivastava, S. S. Eker, R. M. Ali, Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29 (2015), 1839–1845. Available from: https://www.jstor.org/stable/24898346.
[65]	H. M. Srivastava, B. A. Frasin, V. Pescar, Univalence of integral operators involving Mittag-Leffler functions, Appl. Math. Inf. Sci., 11 (2017), 635–641.
[66]	H. M. Srivastava, S. Khan, Q. Z. Ahmad, N. Khan, S. Hussain, The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain $q$-integral operator, Stud. Univ. Babe s-Bolyai Math., 63 (2018), 419–436. doi: 10.24193/subbmath.2018.4.01. doi: 10.24193/subbmath.2018.4.01
[67]	H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23 (2010), 1188–1192. Available from: https://doi.org/10.1016/j.aml.2010.05.009.
[68]	H. M. Srivastava, Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198–210. Available from: https://doi.org/10.1016/j.amc.2009.01.055.
[69]	P. G. Todorov, On the Faber polynomials of the univalent functions of class, J. Math. Anal. Appl., 162 (1991), 268–276. Available from: https://doi.org/10.1016/0022-247X(91)90193-4.
[70]	A. Wiman, Uber den fundamentalsatz in der teorie der funktionen $ E(x)$, Acta Math., 29 (1905), 191–201. doi: 10.1007/BF02403202. doi: 10.1007/BF02403202

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