Some Special Families of Holomorphic and Sălăgean Type Bi-univalent Functions Associated with \((m,n)\)-Lucas Polynomials

Authors

  • S. R. Swamy Department of Computer Science and Engineering, R.V. College of Engineering, Bengaluru 560059, Karnataka
  • Abbas Kareem Wanas Department of Mathematics, College of Science, University of Al-Qadisiyah
  • Y. Sailaja Department of Mathematics, R.V. College of Engineering, Bengaluru 560059, Karnataka

DOI:

https://doi.org/10.26713/cma.v11i4.1411

Keywords:

Holomorphic function, Bi-univalent function, Fekete-Szegö inequality, Lucas polynomial, Sălăgean operator

Abstract

The aim of the present paper is to introduce some special families of holomorphic and Sălăgean type bi-univalent functions associated with \((m,n)\)-Lucas polynomials in the open unit disc \(\mathfrak{D}\). We investigate the upper bounds on initial coefficients for functions in these newly introduced special families and also discuss the Fekete-Szegö problem.\ Some interesting consequences of the results established here are indicated.

Downloads

Download data is not yet available.

References

A. Akgül, (P,Q)-Lucas polynomial coefficient inequalities of the bi-univalent function class, Turkish Journal of Mathematics 43 (2019), 2170 – 2176, URL: https://journals.tubitak.gov.tr/math/issues/mat-19-43-5/mat-43-5-8-1903-38.pdf.

S. Altınkaya and S. Yalçın, On the Chebyshev polynomial coefficient problem of some subclasses of bi-univalent functions, Gulf Journal of Mathematics 5(3) (2017), 34 – 40, URL: https://gjom.org/index.php/gjom/article/view/105/95.

S. Altınkaya and S. Yalçın, On the (p, q)-Lucas polynomial coefficient bounds of the biunivalent function class, Boletí­n de la Sociedad Matemática Mexicana 25 (2019), 567 – 575, DOI: 10.1007/s40590-018-0212-z.

M. í‡aglar, E. Deniz and H. M. Srivastava, Second Hankel determinant for certain subclasses of bi-univalent functions, Turkish Journal of Mathematics 41 (2017), 694 – 706, URL: https://journals.tubitak.gov.tr/math/issues/mat-17-41-3/mat-41-3-19-1602-25.pdf.

P. L. Duren, Univalent Functions, Grundlehren der mathematischen Wissenschaften 259, Springer-Verlag, New York (1983), URL: https://books.google.co.in/books?hl=en&lr=&id=8mVIaZmsy0gC&oi=fnd&pg=PA1&dq=related:E6E-mxhj4gwJ:scholar.google.com/&ots=mlchb9Kgws&sig=SPVFHP4-C1BuHiVLEvceilfpk0g#v=onepage&q&f=false.

M. Fekete and G. Szegö, Eine Bemerkung íœber Ungerade Schlichte Funktionen, Journal of London Mathematical Society 89 (1933), 85 – 89, DOI: 10.1112/jlms/s1-8.2.85.

P. Filipponi and A. F. Horadam, Second derivative sequence of Fibonacci and Lucas polynomials, The Fibonacci Quarterly 31 (1993), 194 – 204, URL: https://www.fq.math.ca/31-3.html.

B. A. Frasin and M. K. Aouf, New subclass of bi-univalent functions, Applied Mathematics Letters 24 (2011), 1569 – 1573, DOI: 10.1016/j.aml.2011.03.048.

A. F. Horadam and J. M. Mahon, Pell and Pell-Lucas polynomials, The Fibonacci Quarterly 23 (1985), 7 – 20, URL: https://www.fq.math.ca/23-1.html.

G. Lee and M. Asci, Some properties of the (p, q)-Fibonacci and (p, q)-Lucas polynomials, Journal of Applied Mathematics 2012 (2012), Article ID 264842, 1 – 18, DOI: 10.1155/2012/264842.

M. Lewin, On a coefficient problem for bi-univalent functions, Proceedings of the American Mathematical Society 18 (1967), 63 – 68, URL: https://www.ams.org/journals/proc/1967-018-01/S0002-9939-1967-0206255-1/S0002-9939-1967-0206255-1.pdf.

A. Lupas, A guide of Fibonacci and Lucas polynomials, Octogon Mathematical Magazine 7 (1999), 2 – 12.

G. S. Salagean, Subclasses of univalent functions, Lecture Notes in Mathematics 1013 (1983), 362 – 372, Springer, Berlin, DOI: 10.1007/BFb0066543.

H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Applied Mathematics Letters 23 (2010), 1188 – 1192, DOI: 10.1016/j.aml.2010.05.009.

S. R. Swamy, Ruscheweyh derivative and a new generalized multiplier differential operator, Annals of Pure and Applied Mathematics 10(2) (2015), 229 – 238, URL: http://www.researchmathsci.org/apamart/APAM-v10n2-13.pdf.

T.-T. Wang and W.-P. Zhang, Some identities involving Fibonacci, Lucas polynomials and their applications, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie (New Series) 55(103) (2012), 95 – 103, URL: https://www.jstor.org/stable/43679243?seq=1.

Downloads

Published

31-12-2020
CITATION

How to Cite

Swamy, S. R., Wanas, A. K., & Sailaja, Y. (2020). Some Special Families of Holomorphic and Sălăgean Type Bi-univalent Functions Associated with \((m,n)\)-Lucas Polynomials. Communications in Mathematics and Applications, 11(4), 563–574. https://doi.org/10.26713/cma.v11i4.1411

Issue

Section

Research Article