Sharp Bounds of the Fekete–Szegö Problem and Second Hankel Determinant for Certain Bi-Univalent Functions Defined by a Novel q-Differential Operator Associated with q-Limaçon Domain
Abstract
:1. Introduction and Definitions
- 1.
- Let , , we have the Frasin differential operator [27].
- 2.
- Let , , we have the Opoola differential operator [28].
- 3.
- Let , and , we have the Al-Oboudi differential operator [26].
- 4.
- Let , and , we have the q-Salagean differential operator [29].
- 5.
- Let , , and , we have the Salagean differential operator [25].
2. Coefficients Bound Estimates
3. The Second Hankel Determinant and Fekete-Szegö Inequality
- 1.
- Let . Since andIt is obvious that the function reaches its maximum near the closed-square boundary . Now, applying some differentiation on the function with respect to , we haveThe function is an increasing function with regard to and reaches its maximum at since . Therefore,Taking the differentiation of the function , we obtainSince , the function is an increasing function and the maximum occurs at . Hence,Thus, in the instance of , we obtainWe know that , so we can have
- 2.
- Now, taking . Since andThus, we obtain
- 3.
- Let us say . In this instance, we must look into the maximum of the function while accounting for the sign of .The equation is clear to see. We will look into two instances of the sign .
- (a)
- Let for the same . In this case, since and , the function (having a minimum) cannot have a maximum on the square according to basic calculus.
- (b)
- Now, let for some . In this case, since , the function cannot have a maximum on the square .
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Aral, A.; Gupta, V. Generalized q-Baskakov operators. Math. Slovaca 2011, 61, 619–634. [Google Scholar] [CrossRef]
- Abu-Risha, M.H.; Annaby, M.H.; Ismail, M.E.H.; Mansour, Z.S. Linear q-difference equations. Anal. Anwend. 2007, 26, 481–494. [Google Scholar] [CrossRef] [Green Version]
- Kac, V.G.; Cheung, P. Quantum Calculus. Universitext; Springer: New York, NY, USA, 2002. [Google Scholar]
- Saliu, A.; Noor, K.I.; Hussain, S.; Darus, M. On Quantum Differential Subordination Related with Certain Family of Analytic Functions. J. Math. 2020, 2020, 6675732. [Google Scholar] [CrossRef]
- Ayasrah, M.; AL-Smadi, M.; Al-Omari, S.; Baleanu, D.; Momani, S. Structure of optical soliton solution for nonlinear resonant space-time Schrodinger equation in conformable sense with full nonlinearity term. Phys. Scr. 2020, 95, 105215. [Google Scholar]
- Al-Smadi, M.; Djeddi, N.; Momani, S.; Al-Omari, S.; Araci, S. An attractive numerical algorithm for solving nonlinear Caputo- Fabrizio fractional Abel differential equation in a Hilbert space. Adv. Differ. Equ. 2021, 2021, 271. [Google Scholar] [CrossRef]
- Momani, S.; Djeddi, N.; Al-Smadi, M.; Al-Omari, S. Numerical investigation for Caputo-Fabrizio fractional Riccati and Bernoulli equations using iterative reproducing kernel method. Appl. Numer. Math. 2021, 170, 418–434. [Google Scholar] [CrossRef]
- Mohammed, S.; Alabedalhad, M.; Al-Omari, S.; Al-Smadi, S. Abundant exact travelling wave solutions for a fractional massive Thirring model using extended Jacobi elliptic function method. Fractal Fract. 2022, 6, 252. [Google Scholar]
- Jackson, F.H. On q-definite integrals. Q. J. Pure Appl. Math. 1910, 41, 193–203. [Google Scholar]
- Gasper, G.; Rahman, M. Basic Hypergeometric Series, 2nd ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Srivastava, H.M. 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. 2020, 44, 327–344. [Google Scholar] [CrossRef]
- Arif, M.; Srivastava, H.M.; Umar, S. Some application of a q-analogue of the Ruscheweyh type operator for multivalent functions. Rev. Real Acad. Cienc. Exactas Fis. Natur. Ser. A Mat. 2019, 113, 1121–1221. [Google Scholar] [CrossRef]
- Ismail, M.E.H.; Merkes, E.; Styer, D. A generalization of starlike functions. Complex Var. Theory Appl. 1990, 14, 77–84. [Google Scholar] [CrossRef]
- Mahmood, S.; Sokol, J. New subclass of analytic functions in conical domain associated with Ruscheweyh q-differential operator. Results Math. 2017, 71, 1345–1357. [Google Scholar] [CrossRef]
- Kanas, S.; Raducanu, D. Some class of analytic functions related to conic domains. Math. Slovaca 2014, 64, 1183–1196. [Google Scholar] [CrossRef]
- Chandak, S.; Suthar, D.L.; Al-Omari, S.; Gulyaz-Ozyurt, S. Estimates of classes of generalized special functions and their application in the fractional (k, s)-calculus theory. J. Funct. Spaces 2021, 2021, 9582879. [Google Scholar] [CrossRef]
- Vijaya, K.; Murugusundaramoorthy, G.; Kasthuri, M. Starlike functions of complex order involving q-hypergeometric functions with fixed point. Annales Universitatis Paedagogicae Cracoviensis. Stud. Math. 2014, 13, 51–63. [Google Scholar]
- Murugusundaramoorthy, G.; Vijaya, K.; Janani, T. Subclasses of Starlike functions with a fixed Point involving q-hypergeometric function. J. Anal. Numb. Theory 2016, 4, 41–47. [Google Scholar] [CrossRef]
- Catas, A. On the Fekete–Szego Problem for Meromorphic Functions Associated with p, q-Wright Type Hypergeometric Function. Symmetry 2021, 13, 2143. [Google Scholar] [CrossRef]
- Amini, E.; Al-Omari, S.; Nonlaopon, K.; Baleanu, D. Estimates for coefficients of Bi-univalent functions associated with a fractional q-difference Operator. Symmetry 2022, 14, 879. [Google Scholar] [CrossRef]
- Adebesin, B.O.; Adeniyi, J.O.; Adimula, I.A.; Adebiyi, S.J.; Ikubanni, S.O.; Oladipo, O.A.; Olawepo, A.O. Pattern of ionization gradient, solar quiet magnetic element, and F2-layer bottomside thickness parameter at African equatorial location. Radio Sci. 2019, 54, 415–425. [Google Scholar] [CrossRef]
- Adebesin, B.O.; Pulkkinen, A.; Ngwira, C.M. The interplanetary and magnetospheric causes of extreme dB/dt at equatorial locations. Geophys. Res. Lett. 2016, 43, 11501–11509. [Google Scholar] [CrossRef]
- Jackson, F.H. q-difference equations. Am. J. Math. 1910, 32, 305–314. [Google Scholar] [CrossRef]
- Graham, I.; Kohr, G. Geometric Function Theory in One and Higher Dimensions, 1st ed.; Amsterdam University Press: Amsterdam, The Netherlands, 2003. [Google Scholar]
- Salagean, G.S. Subclass of univalent functions. Lect. Note Math. 1983, 1013, 362–372. [Google Scholar]
- Al-Oboudi, F.M. On univalent functions defined by a generalized Salagean operator. Int. J. Math. Math. Sci. 2009, 27, 1429–1436. [Google Scholar] [CrossRef] [Green Version]
- Frasin, B.A. A new differential operator of analytic functions involving binomial series. Bol. Soc. Paran. Mat. 2020, 38, 205–213. [Google Scholar] [CrossRef]
- Opoola, T.O. On a subclass of univalent functions defined by a generalised differential operator. Int. J. Math. Anal. 2017, 11, 869–876. [Google Scholar]
- Govindaraj, M.; Sivasubramanian, S. On a class of analytic functions related to conic domains involving q-calculus. Anal. Math. 2017, 43, 475–487. [Google Scholar] [CrossRef]
- Zhang, C.; Khan, B.; Shaba, T.G.; Ro, J.-S.; Araci, S.; Khan, M.G. Applications of q-Hermite polynomials to Subclasses of analytic and bi-Univalent Functions. Fractal Fract. 2022, 6, 420. [Google Scholar] [CrossRef]
- Hu, Q.; Shaba, T.G.; Younis, J.; Khan, B.; Mashwani, W.K.; Caglar, M. Applications of q-derivative operator to Subclasses of bi-Univalent Functions involving Gegenbauer polynomial. Appl. Math. Sci. Eng. 2022, 30, 501–520. [Google Scholar] [CrossRef]
- Duren, P.L. Univalent Functions. In Grundlehren der Mathematischen Wissenschaften. Band 259; Springer: New York, NY, USA; Berlin/Heidelberg, Germany; Tokyo, Japan, 1983. [Google Scholar]
- Srivastava, H.M.; Mishra, A.K.; Gochhayat, P. Certain sublcasses of analytic and bi-univalent functions. Appl. Math. Lett. 2010, 23, 1188–1192. [Google Scholar] [CrossRef] [Green Version]
- Brannan, D.A.; Taha, T.S. On some classes of bi-univalent functions. Stud. Univ. Babes-Bolyai Math. 1986, 31, 70–77. [Google Scholar]
- Brannan, D.A.; Clunie, J.; Kirwan, W.E. Coefficient estimates for a class of star-like functions. Canad. J. Math. 1970, 22, 476–485. [Google Scholar] [CrossRef]
- Frasin, B.A.; Aouf, M.K. New subclasses of bi-univalent functions. Appl. Math. Lett. 2011, 24, 1569–1573. [Google Scholar] [CrossRef] [Green Version]
- Lewin, M. On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 1967, 18, 63–68. [Google Scholar] [CrossRef]
- Li, X.-F.; Wang, A.-P. Two new subclasses of bi-univalent functions. Int. Math. Forum 2012, 7, 1495–1504. [Google Scholar]
- Netanyahu, E. The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z|<1. Arch. Ration. Mech. Anal. 1969, 32, 100–112. [Google Scholar]
- Deniz, E.; Çağlar, M.; Orhan, H. Second Hankel determinant for bi-stalike and bi-convex functions of order β. Appl. Math. Comput. 2015, 271, 301–307. [Google Scholar]
- Orhan, H.; Magesh, N.; Yamini, J. Bounds for the second Hankel determinant of certain bi-univalent functions. Turk. J. Math. 2016, 40, 678–687. [Google Scholar] [CrossRef]
- Mustafa, N.; Murugusundaramoorthy, G.; Janani, T. Second Hankel determinant for certain subclass of bi-univalent functions. Mediterr. J. Math. 2018, 15, 119–136. [Google Scholar] [CrossRef]
- Oyekan, E.A.; Olatunji, T.A.; Lasode, A.O. Applications of (p, q)-Gegenbauer Polynomials on a Family of Bi-univalent Functions. Earthline J. Math. Sci. 2023, 12, 271–284. [Google Scholar] [CrossRef]
- Ma, W.C.; Minda, D. A Unified Treatment of Some Special Classes of Univalent Functions. In Proceedings of the Conference on Complex Analysis, Tianjin, China, 19–23 June 1992; International Press Inc.: Boston, MA, USA, 1992; pp. 157–169. [Google Scholar]
- Masih, V.S.; Kanas, S. Subclasses of Starlike and Convex Functions Associated with the Limaçon Domain. Symmetry 2020, 12, 942. [Google Scholar] [CrossRef]
- Saliu, A.; Noor, K.I. On Janowski Close-to-Convex Functions Associated with Conic Regions. Int. J. Appl. Anal. 2020, 18, 614–623. [Google Scholar]
- Saliu, A.; Noor, K.I.; Hussain, S.; Darus, M. Some results for the family of univalent functions related with Limaçon domain. AIMS Math. 2021, 6, 3410–3431. [Google Scholar] [CrossRef]
- Saliu, A.; Jabeen, K.; Al-Shbeil, I.; Aloraini, N.; Malik, S.N. On q-Limaçon Functions. Symmetry 2022, 14, 2422. [Google Scholar] [CrossRef]
- Jabeen, K.; Saliu, A. A study of q-analogue of the analytic characterization of limacon function. Miskolc Math. Notes 2022, 24, 179–195. [Google Scholar] [CrossRef]
- Grenander, U.; Szego, G. Toeplitz form and Their Applications; California Monographs in Mathematical Sciences, University California Press: Berkeley, CA, USA, 1958. [Google Scholar]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Shaba, T.G.; Araci, S.; Adebesin, B.O.; Tchier, F.; Zainab, S.; Khan, B. Sharp Bounds of the Fekete–Szegö Problem and Second Hankel Determinant for Certain Bi-Univalent Functions Defined by a Novel q-Differential Operator Associated with q-Limaçon Domain. Fractal Fract. 2023, 7, 506. https://doi.org/10.3390/fractalfract7070506
Shaba TG, Araci S, Adebesin BO, Tchier F, Zainab S, Khan B. Sharp Bounds of the Fekete–Szegö Problem and Second Hankel Determinant for Certain Bi-Univalent Functions Defined by a Novel q-Differential Operator Associated with q-Limaçon Domain. Fractal and Fractional. 2023; 7(7):506. https://doi.org/10.3390/fractalfract7070506
Chicago/Turabian StyleShaba, Timilehin Gideon, Serkan Araci, Babatunde Olufemi Adebesin, Fairouz Tchier, Saira Zainab, and Bilal Khan. 2023. "Sharp Bounds of the Fekete–Szegö Problem and Second Hankel Determinant for Certain Bi-Univalent Functions Defined by a Novel q-Differential Operator Associated with q-Limaçon Domain" Fractal and Fractional 7, no. 7: 506. https://doi.org/10.3390/fractalfract7070506