Exploring a Special Class of Bi-Univalent Functions: q-Bernoulli Polynomial, q-Convolution, and q-Exponential Perspective
Abstract
:1. Introduction and Definitions
- 1.
- The notation denotes the q-factorial, which can be expressed by:
- 2.
- A definition has been established for the q-derivative operator when .
- q-Numbers: Rationale and applications: q-numbers, an expansion of conventional numbers, emerge within the realm of quantum groups and quantum algebras. They find validation and practical utility in various domains. In particular, in the realm of quantum groups, they play a crucial role. This is significant in disciplines such as quantum field theory, condensed matter physics, and statistical mechanics. Q-numbers are instrumental in representing systems characterized by a non-commutative algebra, a fundamental element in the framework of quantum mechanics.
- Arik–Coon Oscillator: Rationale and applications: The Arik–Coon oscillator is a variant of the harmonic oscillator that incorporates a parameter called “q” for deformation. This alteration introduces a broader uncertainty principle, with potential significance in exploring non-commutative geometry, as well as in the realms of quantum optics and quantum information theory. This modification becomes crucial in cases where standard quantum mechanics needs to be expanded, and it can also serve as a tool for simulating systems in the presence of particular background fields.
- q-Deformed Oscillator: Rationale and applications: Similar to the Arik–Coon oscillator, q-deformed oscillators utilize the deformation parameter ‘q’ to modify the characteristics of regular oscillators. They find utility across diverse fields of physics, including quantum field theory, quantum optics, and nuclear physics, providing insights into systems operating within unconventional quantum mechanical contexts. These oscillators play a pivotal role in the exploration of quantum algebras and their corresponding representations.
- 1.
- Setting , , and , we obtain the operator defined by Hadi and Darus [39].
- 2.
- Setting , , , and , we obtain the operator defined by Hadi et al. [40].
- 3.
- Setting , , , and , we obtain the operator defined by Lasode and Opoola [41].
- 4.
- Setting , , , , and , the q-Al-Oboudi operator, originally introduced by Aouf et al. in their work cited as [42], is available to us.
- 5.
- Setting , , , , , and , the q-Salagean operator, introduced by Govindaraj and Sivasubramanian [43], is available to us.
- 6.
- Setting , , , and , we acquire the operator that is defined by Darus and Ibrahim in their work [44].
- 7.
- Setting , , , , and , we obtain the operator defined by Frasin [45].
- 8.
- Setting , , , , and , we obtain the operator defined by Opoola [46].
- 9.
- Setting , , , , , and , we have the Al-Oboudi operator that Al-Oboudi [47] presented.
- 10.
- Setting , , , , , , and , we have the Salagean operator that Salagean [48] presented.
- 1.
- 2.
- 3.
- When , , and , we obtain the subfamily
2. Bounds on the Initial Coefficients for Several Families Related to the -Bernoulli Polynomials
3. The Second Hankel Determinant and Fekete–Szegö Inequality for Several Families Related to the -Bernoulli Polynomials
- (Case 1: When (extreme point)). Assuming c equals 0, because , , and are all equal to zero, andThe function can be expressed in the following manner.The maximum value of the function can be observed at the edges of the enclosed square X.After applying differentiation techniques to the function with respect to , we obtain the following result.The function increases as increases and reaches its maximum when equals 1, as indicated by the condition . Therefore, we can conclude the following:By utilizing differentiation techniques on the function , we find that if is greater than zero, then is an ascending function and reaches its maximum value when is equal to 1. Consequently,Therefore, when c is equal to 0, we obtain the following.Since , we have
- (Case 2: when ). Now, when we set c equal to 2, we have andHence, we have
- (Case 3: When c is between 0 and 2). We are given a range for the variable c, which lies between 0 and 2. Our objective is to analyze the maximum value of the function while considering the sign of a certain expression denoted by . This expression is given by the equationWe have observed that the equation
- (a)
- In the given interval of c values, specifically between 0 and 2, it is required that remains less than or equal to zero. In this particular case, since both and are equal to and greater than or equal to zero, and is also greater than or equal to zero, the function cannot achieve its maximum value within the square X as per basic calculus principles.
- (b)
- Additionally, suppose there exists a value of c within the range of 0 to 2 where . In this particular case, if , it is not possible for the function to have a maximum within the square .
Due to these three occurrences, In the first case, we take the maximum, which is the extreme point, and then proceed to express it in writing.
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Shaba, T.G.; Araci, S.; Adebesin, B.O.; Esi, A. Exploring a Special Class of Bi-Univalent Functions: q-Bernoulli Polynomial, q-Convolution, and q-Exponential Perspective. Symmetry 2023, 15, 1928. https://doi.org/10.3390/sym15101928
Shaba TG, Araci S, Adebesin BO, Esi A. Exploring a Special Class of Bi-Univalent Functions: q-Bernoulli Polynomial, q-Convolution, and q-Exponential Perspective. Symmetry. 2023; 15(10):1928. https://doi.org/10.3390/sym15101928
Chicago/Turabian StyleShaba, Timilehin Gideon, Serkan Araci, Babatunde Olufemi Adebesin, and Ayhan Esi. 2023. "Exploring a Special Class of Bi-Univalent Functions: q-Bernoulli Polynomial, q-Convolution, and q-Exponential Perspective" Symmetry 15, no. 10: 1928. https://doi.org/10.3390/sym15101928
APA StyleShaba, T. G., Araci, S., Adebesin, B. O., & Esi, A. (2023). Exploring a Special Class of Bi-Univalent Functions: q-Bernoulli Polynomial, q-Convolution, and q-Exponential Perspective. Symmetry, 15(10), 1928. https://doi.org/10.3390/sym15101928