Abstract: In this paper, we make use of the concept of fractional q-calculus to introduce two new classes of bi-

Lemma 2.3. if the function Ωin the Caratheodory class, then it can be written as Ω(z) = 1 + c1z + c2z2 + c3z3 + · · ·, for z ∈D. Moreover, |cn| ≤2 for each n ∈N. The lemma presented in the following discussion is extensively referenced in existing literature (refer to, for example, [27]) and is regarded as a foundational principle that significantly influences the research we are conducting.

Lemma 2.4. Let u and v be real numbers. Let p and q be complex numbers. If |p| < r and |q| < r, |(u + v)p + (u −v)q| ≤ ( 2r|u|, if |u| ≥|v| 2r|v|, if |u| ≤|v|. By selecting particular values of β in Definition 2.1, it is possible to obtain the following subclasses. Example 1. Let f be a bi-univalent function that is given by Equation (1). If f belongs to the subclass B0(Rα q , Gδ(x, z)), then the following subordinations hold z

Theorem 3.1. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class Bβ(Rα q , Gδ(x, z)), then the following inequalities hold: |a2| ≤ 4δ3/2|x|3/2 v u u u t 4x2δ2 (2(β + 2)ψ3+ (β −1)(β + 2)ψ2 2 −2(β + 1)2ψ2

Theorem 3.1 is now complete. The subsequent theorem establishes the estimates concerning the modulus of the initial coefficients of functions that are part of the class Bβ(Rα q , tanh). The methods employed in its proof are akin to those uti- lized in the proof of Theorem 3.1.

Theorem 3.2. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class Bβ(Rα q , tanh), then the following inequalities hold |a2| ≤ √ 2q p qK + 2(β + 1)2ψ2 2 , (35) and |a3| ≤

Theorem 3.1, contingent upon the conditions specified in the earlier examples. The techniques employed in deriving these corollaries closely mirror those applied in the proof of aforementioned theorem, which is the rationale behind our decision to exclude the detailed proofs.

Corollary 3.3. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class B0(Rα q , Gδ(x, z)), then the following inequalities hold: |a2| ≤ 2 (δ|x|)3/2 p |8x2δ2ψ3 + ψ2 2δ(2x2(1 −δ)) −1| , and |a3| ≤δ|x| ψ3 + 4δ2x2

Corollary 3.4. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class B1(Rα q , Gδ(x, z)), then the following inequalities hold: |a2| ≤ √ 2 (δ|x|)3/2 p |3x2δ2ψ3 −ψ2 2gδ 2(x)| , and |a3| ≤2δ|x|

Corollary 3.5. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class B∗(Gδ(x, z)), then the following inequalities hold: |a2| ≤ 2 (δ|x|)3/2 p |2x2δ(3δ + 1) −δ| , and |a3| ≤δ|x| + 4δ2x2.

Corollary 3.6. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class Bβ(Rα q , Tn(x, z)), then the following inequalities hold: |a2| ≤ 4|x|3/2 v u u t

Theorem 3.2, contingent upon the conditions specified in the earlier examples. The techniques employed in deriving these corollaries closely mirror those applied WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2025.24.15 Waleed Al-Rawashdeh E-ISSN: 2224-2880 151 Volume 24, 2025

Corollary 3.7. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class B0(Rα q , tanh), then the following inequalities hold |a2| ≤ q p 2qψ3 + (1 −q)ψ2 2 , and |a3| ≤ q 2ψ3

Corollary 3.8. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class B1(Rα q , tanh), then the following inequalities hold |a2| ≤ q p 3qψ3 + 4ψ2 2 , and |a3| ≤ q 3ψ3

Corollary 3.9. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class S∗(tanh), then the following inequalities hold |a2| ≤ 1 √ 2, and |a3| ≤1. 4 Main Results and Corollaries on Fekete-Szeg¨o problem In this section, we will explore how to establish the Fekete-Szeg¨o inequalities for functions that fall within the specified class. Bβ(Rα q , Gδ(x, z)) and to the class

Theorem 4.1. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class Bβ(Rα q , Gδ(x, z)), then for a real number ζ and δ ̸= 0 the following inequality holds |a3 −ζa2 2| ≤ ( 2δ|x| (β+2)ψ3 , if |1 −ζ| ≤|Y| 8δ3|x3||1−ζ| |2δ2x2K−B2gδ 2(x)|, if |1 −ζ| ≥|Y|,

Corollary 4.2. Let f be a bi-univalent function that is given by Equation (1). If the function f is obeying the Subordination conditions (5) and (6), then for a real number ζ and δ ̸= 0 the following holds |a3 −ζa2 2| ≤ ( δ|x| 2ψ3 , if |1 −ζ| ≤|Y1| 8δ3|x3||1−ζ||gδ 1(x)|3 |A| , if |1 −ζ| ≥|Y1|, where

Corollary 4.3. Let f be a bi-univalent function that is given by Equation (1). If the function f is obeying WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2025.24.15 Waleed Al-Rawashdeh E-ISSN: 2224-2880 152 Volume 24, 2025

Corollary 4.4. Let f be a bi-univalent function that is given by Equation (1). If the function f is obeying the Subordination conditions (9) and (10), then for a real number ζ and δ ̸= 0 the following holds |a3−ζa2 2| ≤ ( δ|x|, if |1 −ζ| ≤| 1−2x2(1−δ) 8xδ | 8δ2|x3||1−ζ| |δ2x2−2x2+1|, if |1 −ζ| ≥| 1−2x2(1−δ) 8xδ

Theorem 3.1.

Theorem 4.5. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class Bβ(Rα q , tanh), then for a real number ζ and δ ̸= 0 the following inequality holds |a3 −ζa2 2| ≤ ( q 2(β+2)ψ3 , if |1 −ζ| ≤B 2q2|1−ζ| qK+2(β+1)2ψ2 2 , if |1 −ζ| ≥B,

Theorem 4.5, provided that the conditions outlined in the preceding examples are met. The methods utilized in the derivation of these corollaries closely resemble those used in the proof of the aforementioned theo- rem, which is the reason for our choice to omit the comprehensive proofs.

Corollary 4.6. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class B0(Rα q , tanh), then the following inequalities hold |a3 −ζa2 2| ≤ ( q 4ψ3 , if ζ ∈[ζ1, ζ2] q2|1−ζ| 2qψ3+(1−q)ψ2 2 , if ζ /∈[ζ1, ζ2],

Corollary 4.7. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class B1(Rα q , tanh), then the following inequalities hold |a3 −ζa2 2| ≤ ( q 6ψ3 , if ζ ∈[ζ3, ζ4] q2|1−ζ| 3qψ3+4ψ2 2 , if ζ /∈[ζ3, ζ4],

Corollary 4.8. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class S∗(tanh), then the following inequalities hold |a3 −ζa2 2| ≤ ( 1 4, if ζ ∈[0, 2] |1−ζ| 2 , if ζ /∈[0, 2]. 5 Conclusion

Definition 2.1. A bi-univalent function f that is given by Equtaion (1) is said to be in the class Bβ(Rα q , Gδ(x, z)) if the following subordinations hold: z1−β Rα q f(z)
′ Rαq f(z)
1−β ≺Gδ(x, z),

Definition 2.2. A bi-univalent function f that is given by Equtaion (1) is said to be in the class Bβ(Rα q , tanh) if the following subordinations hold : z1−β Rα q f(z)
′ Rαq f(z)
1−β ≺1 + tanh(qz),