Fekete-Szego results for certain bi-univalent functions involving $q$-analogues of logarithmic functions


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Authors

  • Ebrahim Amini
  • Shrideh Al-Omari Al-Balqa Applied University
  • Jafar Al-Omari

Keywords:

Salagean differential operator; geometric function theory; inequalities; bi-univalent functions.

Abstract

In this paper, we discuss a novel type of analytic bi-univalent functions by utilizing specialized q-Salagean differential operators. Then, we use the q-analogue of the logarithmic function to introduce definition and provide properties of a class of bi-univalent functions. Further, we use the subordination principle to estimate the initial Taylor and Maclaurin coefficients for these given
univalent functions. Additionally, we introduce new operators to demonstrate practical applications of the existing theory and establish Fakte-Szego results for each function in the defined sets. Further, we discuss certain coefficient inequalities in detail.

References

Gasper, G. Rahman, M. (2004). Basic Hypergeometric Series, 2nd edn. Cambridge university Press, Cambridge.

Annaby, M. H., & Mansour, Z. S. (2012). q-Fractional Calculus and Equations. Springer Publishing.

AlOmari, S. K. Q. (2020). On a qLaplacetype integral operator and certain class of series expansion. Mathematical

Methods in the Applied Sciences.

Al-Omari, S. (2021). Estimates and properties of certain q-Mellin transform on generalized q-calculus theory. Advances

in Difference Equations, 2021(1).

Osburn, R., & Schneider, C. (2009). Gaussian Hypergeometric series and supercongruences. Mathematics of

Computation, 78(265), 275.

Jackson, F. H. (1910). On q-definite integrals, Quart. J. Pure Appl. Math. 41, 193–203.

Srivastava, H. M. (1989). Univalent functions, fractional calculus, and associated generalized hypergeometric functions, In: H. M. Srivastava and S. Owa, Editors, Univalent Functions, Fractional Calculus, and Their Applications,

Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto,

Kanas, S., & Rducanu, D. (2014). Some class of analytic functions related to conic domains. Mathematica Slovaca,

(5), 11831196.

Srivastava, H. M., Khan, S., Ahmad, Q. Z., Khan, N., & Hussain, S. (2018). 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. Studia Universitatis Babes-Bolyai Matematica, 63(4), 419436.

Kanas, S., Altinkaya, ., & Yalin, S. (2019). Subclass of k-Uniformly Starlike Functions Defined by the Symmetric

q-Derivative Operator. Ukrainian Mathematical Journal, 70(11), 17271740.

Srivastava, H. M., Khan, N., Khan, S., Ahmad, Q. Z., & Khan, B. (2021). A Class of k-Symmetric Harmonic Functions

Involving a Certain q-Derivative Operator. Mathematics, 9(15), 1812.

Srivastava, H. M. (2020). Operators of Basic (or q-) Calculus and Fractional q-Calculus and Their Applications in

Geometric Function Theory of Complex Analysis. Iranian Journal of Science and Technology, Transactions A: Science,

(1), 327–344.

Arif, M., Srivastava, H. M., & Umar, S. (2018). Some applications of a q-analogue of the Ruscheweyh type operator

for multivalent functions. Revista de La Real Academia de Ciencias Exactas, Fsicas y Naturales. Serie A. Matemticas,

(2), 12111221.

Ismail, M. E. H., Merkes, E., & Styer, D. (1990). A generalization of starlike functions. Complex Variables, Theory and

Application: An International Journal, 14(14), 7784.

Khan, S., Hussain, S., Darus, M. (2021). Inclusion relations of q-Bessel functions associated with generalized conic

domain. AIMS Mathematics, 6(4), 36243640.

Zhang, X., Khan, S., Hussain, S., Tang, H., & Shareef, Z. (2020). New subclass of q-starlike functions associated with

generalized conic domain, AIMS Mathematics, 5(5), 4830–4848

Seoudy, T. M. (2022). Some subclasses of univalent functions associated with -Ruscheweyh derivative operator.

Ukrainskyi Matematychnyi Zhurnal, 74(1), 122136.

Matala-aho, T., Vnnen, K., & Zudilin, W. (2005). New irrationality measures for q-logarithms. Mathematics of

Computation, 75(254), 879889.

Yamano, T. (2002). Some properties of q-logarithm and q-exponential functions in Tsallis statistics. Physica A:

Statistical Mechanics and Its Applications, 305(34), 486496.

Graham, I, & Kohr, G. (2003). Geometric Function Theory in One and Higher Dimensions (1ste editie). Amsterdam

University Press.

Amini, E., Al-Omari, S., & Rahmatan, H. (2022b). On geometric properties of cer tain subclasses of univalent functions

defined by Noor integral operator. Analysis, 0(0). https://doi.org/10.1515/anly-2022-1043

Amini, E., Fardi, M., Al-Omari, S., & Nonlaopon, K. (2022). Results on Univalent Functions Defined by q-Analogues

of Salagean and Ruscheweh Operators. Symmetry, 14(8), 1725.

Miller, S. S., & Mocanu, P. T. (2000). Differential Subordinations. Taylor & Francis.

Duren, P. L. (1983). Univalent Functions, in: Grundlehren der Mathematischen Wissenschaften, Band 259, SpringerVerlag, New York, Berlin, Heidelberg and Tokyo.

Xu, Q. H., Xiao, H. G., & Srivastava, H. (2012). A certain general subclass of analytic and bi-univalent functions and

associated coefficient estimate problems. Applied Mathematics and Computation, 218(23), 1146111465.

Caglar, M., Orhan, H., & Yagmur, N. (2013). Coefficient bounds for new subclasses of bi-univalent functions. Filomat,

(7), 11651171.

Hussain, S., Khan, S., Zaighum, M. A., & Darus, M. (2018). On certain classes of bi-univalent functions related to

m-fold symmetry. Journal of Nonlinear Sciences and Applications, 11(04), 490499.

Salagean, G. S. (1983). subclass of univalent functios, Lecture Note in Math, Springer-Verlag, Berlin, 1013(1983),

Alb Lupa, A. (2011). On special differential superordinations using a generalized Slgean operator and Ruscheweyh

derivative. Computers & Mathematics with Applications, 61(4), 10481058.

Al-Oboudi, F. M. (2004). On univalent functions defined by a generalized Slgean operator. International Journal of

Mathematics and Mathematical Sciences, 2004(27), 14291436.

Bulut, S. (2008). A New Subclass of Analytic Functions Defined by Generalized Ruscheweyh Differential Operator.

Journal of Inequalities and Applications, 2008(1), 134932.

Oros, G. I., & Oros, G. (2008). On a class of univalent functions defined by a generalized Slgean operator. Complex

Variables and Elliptic Equations, 53(9), 869877.

Darus, M. & Ibrahim, W. (2011) On New Subclasses of Analytic Functions Involving Gener alized Differential and

Integral Operators, Eur. J. Pure Appl. Math. 1(4), 59–66.

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Published

2024-07-22

How to Cite

Amini, E., Al-Omari, S., & Al-Omari, J. (2024). Fekete-Szego results for certain bi-univalent functions involving $q$-analogues of logarithmic functions. Results in Nonlinear Analysis, 7(3), 65–79. Retrieved from https://nonlinear-analysis.com/index.php/pub/article/view/412