Certain subclasses of bi-univalent functions related to k-Fibonacci numbers

Lemma 5. ([11]) Let p ∈P with p(z) = 1 + c1z + c2z2 + · · · , then |cn| ≤2 for n ≥1. (4)

Theorem 9. Let f given by (1) be in the class SLMk α,Σ(epk(z)). Then |a2| ≤ k √ k|τ k| p (1 + α)2k −(1 + α)(2(1 + α) + αk2)τ k (17) and |a3| ≤k|τ k|  (1 + α)2k −  (k2 + 2)α2 + (5k2 + 4)α + 2(k2 + 1)

Corollary 10. Let f given by (1) be in the class SLk Σ(˜pk(z)). Then |a2| ≤k √ k|τ k| √k −2τ k and |a3| ≤k|τ k|  k −2(k2 + 1)τ k

Corollary 11. Let f given by (1) be in the class KSLk Σ(˜pk(z)). Then |a2| ≤ k √ k|τ k| p 4k −2(4 + k2)τ k and |a3| ≤k|τ k|  k −2(k2 + 1)τ k

Corollary 12. Let f given by (1) be in the class SLΣ(˜p(z)). Then |a2| ≤ |τ| √1 −2τ and |a3| ≤|τ|(1 −4τ) 2(1 −2τ) .

Corollary 13. Let f given by (1) be in the class KSLΣ(˜p(z)). Then |a2| ≤ |τ| √4 −10τ and |a3| ≤|τ|(1 −4τ) 3(2 −5τ) . 3. Bi-Univalent function class SLGk γ,Σ(epk(z)) In this section, we define a new class SLGk γ,Σ(epk(z)) of γ−bi-starlike functions associated with shell-like domain. Definition 14. Let 0 ≤γ ≤1, and k be any positive real number. A function f ∈Σ of the form (1) is said to be in the class SLGk γ,Σ(epk(z)) if the following

Theorem 17. Let f given by (1) be in the class SLGk γ,Σ(epk(z)). Then |a2| ≤ k √ 2k|τ k| p 2(2 −γ)2k −(4(2 −γ)2 + (γ2 −5γ + 4)k2)τ k

Theorem 19. Let f given by (1) be in the class SLMk α,Σ(epk(z)) and µ ∈R. Then we have |a3−µa2 2| ≤        k|τ k| 2(1+2α), |µ −1| ≤

Corollary 20. If f ∈SLMk α,Σ(epk(z)), then |a3 −a2 2| ≤ k|τ k| 2(1 + 2α). (4) The second theorem is the solution of the Fekete-Szegö problem in SLGk γ,Σ(epk(z)) and it looks like the following:

Theorem 21. Let f given by (1) be in the class SLGk γ,Σ(epk(z)) and µ ∈R. Then we have |a3−µa2 2| ≤      k|τ k| 2(3−2γ), |µ −1| ≤2(2−γ)2k−(4(2−γ)2+(γ2−5γ+4)k2)τ k 4(3−2γ)k2|τ k| ,

Corollary 22. If f ∈SLGk γ,Σ(epk(z)), then |a3 −a2 2| ≤ k|τ k| 2(3 −2γ). (5)

Corollary 23. Let f given by (1) be in the class SLk Σ(˜pk(z)) and µ ∈R. Then we have |a3 −µa2 2| ≤ ( k|τ k| 2 , |µ −1| ≤k−2τ k 2k2|τ k|, |1−µ|k3τ 2 k k−2τ k ,

Corollary 24. Let f given by (1) be in the class KSLk Σ(˜pk(z)) and µ ∈R. Then we have |a3 −µa2 2| ≤ ( k|τ k| 6 , |µ −1| ≤2k−(k2+4)τ k 3k2|τ k| , |1−µ|k3τ 2 k 2(2k−(k2+4)τ k),