Asia Pac. J. Math. 2023 10:21

Lemma 1.3. [17,18] Suppose that P is the set of all analytic functions a of the form a(z) = 1 + ∞ X λ=1 aλzλ (1.3) satisfying ℜ(a(z)) > 0, z ∈⋋and a(0) = 1. Then, |aλ| ≤2, λ = 1, 2, 3, · · · . For any value of λ = 1, 2, 3, · · · , this inequality is sharp. For example, the function a(z) = 1 + z 1 −z is equal for all λ.

Lemma 1.4. [17,18] Suppose that P is the set of all analytic functions a of the form a(z) = 1 + ∞ X λ=1 aλzλ (1.4) satisfying ℜ(a(z)) > 0, z ∈⋋and a(0) = 1. Then, 2a2 = a2 1 + (4 −a2 1)n 4a3 = a3 1 + 2(4 −a2 1)a1n −(4 −a2 1)a1n2 + 2(4 −a2

Lemma 1.5. [18] If and only if the Toeplitz determinants Hj =

Proposition 1.7. [19] If the function a(z) = 1 + X j≥1 ajzj, v(z) = 1 + X j≥1 vjzj which are in the class of P and a1 = −v1, then a2 −v2 = (4 −a2 1)(n −y) 2 (1.6) a2 + v2 = h2 1 + (4 −a2

Theorem 2.1. Let f ∈FDΣ. Then: |n2| ≤5 12, |n3| ≤max  5 18 , 25 144  , |n4| ≤max  5 24 , 25 216

Theorem 3.1. Let f(z) ∈FDΣ. Then: |n2n4 −n2 3| ≤max  25 324 , 1625 20736  . The result obtained here are sharp for f2(z) = Z z 0  1 + 5

Theorem 3.1 has now been successfully proved. □ We now provide the subsequent theorem on the Fekete-Szegö inequality.

Theorem 3.2. Let f(z) ∈FDΣ, χ ∈C. Then: n3 −χn2 2 ≤          5 18

Theorem 3.2 is presented in the following manner for the case χ ∈R.

Theorem 3.3. Let f(z) ∈FDΣ, χ ∈R. Then: |n3 −χn2 2| ≤            

Corollary 3.4. Let f(z) ∈FDΣ. Then: |n3 −n2 2| ≤5 18. 4. Conclusion Recently, well-known mathematicians have been drawn to special domains and polynomials in Geometric Functions Theory due to their usefulness in various fields of mathematics and other sciences. In our paper, we addressed the Fekete-Szegö problem and provided solutions for functions belonging to the class f ∈FDΣ, which consists of analytic and bi-univalent functions that involve the four leaf domain. We also derived estimates for