Sufficient Conditions for Subordination of Multivalent Functions

Lemma 2.1 see 3, page 132. Let q be univalent in U and let θ and φ be analytic in a domain D containing qU, with φw / 0, when w ∈qU. Set Qz  zq′z · φqz, hz  θqz  Qz, 2.1 and suppose that either i Q is starlike, or ii h is convex. In addition, assume that iii Rezh′z/Qz  Reθ′qz/φqz  zQ′z/Qz > 0. If P is analytic in U, with P0  q0, PU ⊂D and θPz  zP ′z · φPz ≺θqz  zq′z·φqz  hz, 2.2 then P ≺q, and q is the

Lemma 2.2. Let q ∈Hp be univalent, qz / 0 and satisfies the following conditions: i zq′z qz is starlike; ii Re qz λ  1  zq′′z q′z −z′qz qz  > 0 2.3 for λ / 0 and for all z ∈U. For P ∈Hp with Pz / 0 in U if Pz  λzP ′z

Theorem 2.3. Let q ∈Hp be univalent, qz / 0 and satisfies the conditions 2.3 in Lemma 2.2. For f ∈Ap if Jλ, f; z ≺qz  λzq′z qz , 2.7 then zf′z fz ≺qz, 2.8 and q is the best dominant.

Lemma 2.4. Let q ∈H1 be univalent and satisfies the following conditions: i qz is convex; ii Re  1 μ  p   zq′′z q′z  > 0, p ∈N  {1, 2, 3, . . .} 2.11 for μ / 0 and for all z ∈U. For P ∈H1 in U if 1 −μ  μpPz  μzP ′z ≺1 −μ  μpqz  μzq′z,

Theorem 2.5. Let q ∈H1 be univalent and satisfies the conditions 2.11 in Lemma 2.4. For f ∈Ap if Ipμ, f; z ≺1 −μ  μpqz  μzq′z. 2.15 Then, fz zp ≺qz, 2.16 and q is the best dominant.

Corollary 2.6. Let q ∈H1 be univalent and satisfies the following conditions: i qz is convex; ii Re  1 μ  1   zq′′z q′z  > 0, p ∈N  {1, 2, 3, . . .} 2.19 for μ / 0 and for all z ∈U. For P ∈H1 in U if Pz  μzP ′z ≺qz  μzq′z, 2.20

Corollary 2.7. Suppose q ∈S satisfies the conditions 2.19 in Corollary 2.6. For f ∈A if Iμ, f; z ≺qz  μzq′z. 2.21 Then, fz z ≺qz, 2.22 and q is the best dominant.

Corollary 2.8. Let q ∈H1 be univalent; qz is convex for all z ∈U. For P ∈H1 in U if Pz  zP ′z ≺qz  zq′z, 2.23 then P ≺q, and q is the best dominant.

Corollary 2.9. Let q ∈S be convex. For f ∈A if f′z ≺qz  zq′z. 2.24 Then, fz z ≺qz, 2.25 and q is the best dominant.

Corollary 2.10. Let q ∈H1 be univalent, qz is convex for all z ∈U. For P ∈H1 in U if pPz  zP ′z ≺pqz  zq′z, 2.26 then P ≺q, and q is the best dominant.

Corollary 2.11. Let q ∈H1 be univalent, qz is convex, for all z ∈U. If f ∈Ap, and f′z zp−1 ≺pqz  zq′z, 2.27 then fz z ≺qz, 2.28 and q is the best dominant.

Corollary 2.12. Let q ∈S satisfies Ipμ, f; z ≺1 −μ  μp  2μ −α −αμpz −1 −2α1 −μ  μpz2 1 −z2 , 2.29 where f ∈Ap, then fz zp ∈CS∗α, 2.30 and q is the best dominant.

Corollary 2.13. Let q ∈S satisfies f′z zp−1 ≺p  21 −α −αpz −1 −2αpz2 1 −z2 , 2.32 where f ∈Ap, then fz zp ∈CS∗α, 2.33 and q is the best dominant.

Corollary 2.14. Let q ∈S satisfies f′z zp−1 ≺p  2z −pz2 1 −z2 , 2.34 where f ∈Ap, then f ∈CS∗, 2.35 and q is the best dominant.

Corollary 2.15. Let q ∈S satisfies f′z ≺1  2z −z2 1 −z2 , 2.36 where f ∈Ap, then f ∈CS∗, 2.37 and q is the best dominant.