Applications of the Second Kind Chebyshev Polynomials of Bi-Starlike and Bi-Convex λ-Pseudo Functions Associated with Sakaguchi Type Functions

  • Abbas Kareem Wanas Department of Mathematics, College of Science, University of Al-Qadisiyah, Diwaniyah, Iraq
  • S. R. Swamy Department of Information and Engineering, Acharya Institute of Technology, Bengaluru-560 107, Karnataka, India
Keywords: bi-univalent function, λ-pseudo functions, coefficient estimates, Chebyshev polynomials, subordination


The purpose of this paper is to use the second kind Chebyshev polynomials to introduce a new class of analytic and bi-univalent functions associating bi-starlike and biconvex $\lambda$-pseudo functions with Sakaguchi type functions defined in the open unit disk. We determinate upper bounds for the initial Taylor-Maclaurin coefficients $\left|a_{2}\right|$ and $\left|a_{3}\right|$ for functions in this class.


R. M. Ali, S. K. Lee, V. Ravichandran and S. Supramaniam, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett. 25 (2012), 344-351.

I. Al-Shbeil, A. K. Wanas, A. Saliu and A. Catas, Applications of beta negative binomial distribution and Laguerre polynomials on Ozaki bi-close-to-convex functions, Axioms 11 (2022), Art. ID 451, 1-7.

K. O. Babalola, On λ-pseudo-starlike functions, J. Class. Anal. 3(2) (2013), 137-147.

L.-I. Cotirla and A. K. Wanas, Coefficient-related studies and Fekete-Szego inequalities for new classes of bi-starlike and bi-convex functions, Symmetry 14 (2022), Art. ID 2263, 1-9.

E. H. Doha, The first and second kind Chebyshev coefficients of the moments of the general-order derivative of an infinitely differentiable function, Int. J. Comput. Math. 51 (1994), 21-35.

P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Band 259, Springer Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.

J. Dziok, R. K. Raina and J. Sokol, Application of Chebyshev polynomials to classes of analytic functions, C. R. Math. Acad. Sci. Paris, Ser I 353 (2015), 433-438.

B. A. Frasin, Coefficient inequalities for certain classes of Sakaguchi type functions, Int. J. Nonlinear Sci. 10 (2010), 206-211.

J. O. Hamzat, M. O. Oluwayemi, A. A. Lupas and A. K. Wanas, Bi-univalent problems involving generalized multiplier transform with respect to symmetric and conjugate points, Fractal Fract. 6 (2022), Art. ID 483, 1-11.

A. R. S. Juma, A. Al-Fayadh, S. P. Vijayalakshmi and T. V. Sudharsan, Upper bound on the third hankel determinant of the class of univalent functions using an operator, Afrika Matematika 33 (2022), 1-10.

J. C. Mason, Chebyshev polynomials approximations for the L-membrane eigenvalue problem, SIAM J. Appl. Math. 15 (1967), 172-186.

S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York and Basel, 2000.

K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan 11(1) (1959), 72-75.

H. M. Srivastava, S. Gaboury and F. Ghanim, Coefficient estimates for some general subclasses of analytic and bi-univalent functions, Africa Math. 28 (2017), 693-706.

H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010), 1188-1192.

H. M. Srivastava and A. K. Wanas, Applications of the Horadam polynomials involving λ-pseudo-starlike bi-univalent functions associated with a certain convolution operator, Filomat 35 (2021), 4645-4655.

S. R. Swamy, P. K. Mamatha, N. Magesh and J. Yamini, Certain subclasses of biunivalent functions defined by Salagean operator associated with the (p, q)-Lucas polynomials, Advances in Mathematics Scientific Journal 9(8) (2020), 6017-6025.

S. R. Swamy and A. K. Wanas, A comprehensive family of bi-univalent functions defined by (m, n)-Lucas polynomials, Bol. Soc. Mat. Mex. 28 (2022), 1-10.

A. K. Wanas, Coefficient estimates of bi-starlike and bi-convex functions with respect to symmetrical points associated with the second kind Chebyshev polynomials, Earthline Journal of Mathematical Sciences 3(2) (2020), 191-198.

A. K. Wanas and A. H. Majeed, Chebyshev polynomial bounded for analytic and biunivalent functions with respect to symmetric conjugate points, Applied Mathematics E-Notes 19 (2019), 14-21.

G. Wang, C. Y. Gao and S. M. Yuan, On certain subclasses of close-to-convex and quasi-convex functions with respect to k-symmetric points, J. Math. Anal. Appl. 322(1) (2006), 97-106.

T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge: Cambridge Univ. Press, 1996.

How to Cite
Wanas, A. K., & Swamy, S. R. (2023). Applications of the Second Kind Chebyshev Polynomials of Bi-Starlike and Bi-Convex λ-Pseudo Functions Associated with Sakaguchi Type Functions. Earthline Journal of Mathematical Sciences, 13(2), 497-507.