Subordination and Superordination on Schwarzian Derivatives

Lemma 1.3 see 2, Theorem 2.3b, page 28 . Let ψ ∈ΨnΩ, q with q0  a. If the analytic function pz  a  anzn  an1zn1  · · · satisfies ψpz, zp′z, z2p′′z; z ∈Ω, 1.20 then pz ≺qz.

Lemma 1.4 see 3, Theorem 1, page 818 . Let ψ ∈Ψ′ nΩ, q with q0  a. If p ∈Qa and ψpz, zp′z, z2p′′z; z is univalent in U, then Ω ⊂ ψpz, zp′z, z2p′′z; z : z ∈U 1.21 implies qz ≺pz.

Theorem 2.2. Let f ∈A with fzf′z/z / 0. If φ ∈ΦSΩ, q and  φ zf′z fz , 1  zf′′z f′z , z2{f, z}; z  : z ∈U  ⊂Ω, 2.3 then zf′z fz ≺qz. 2.4

Theorem 2.3. Let φ ∈ΦSh, q . If f ∈A with fzf′z/z / 0 satisfies φ zf′z fz , 1  zf′′z f′z , z2{f, z}; z  ≺hz, 2.14 then zf′z fz ≺qz. 2.15

Theorem 2.4. Let h and q be univalent in U with q0  1, and set qρz  qρz and hρz  hρz. Let φ : C3 × U →C satisfy one of the following conditions: i φ ∈ΦSh, qρ for some ρ ∈0, 1, or ii there exists ρ0 ∈0, 1 such that φ ∈ΦShρ, qρ for all ρ ∈ρ0, 1. If f ∈A with fzf′z/z / 0 satisfies 2.14, then zf′z fz ≺qz. 2.16 The next theorem yields the best dominant of the differential subordination 2.14.

Theorem 2.5. Let h be univalent in U, and φ : C3 × U →C. Suppose that the differential equation φ  qz, qz  zq′z qz , zq′z  z2q′′z qz −3 2 zq′z qz 2  1 −q2z 2 ; z 

Theorem 2.6. Let Ω be a set in C, and φ : C3 × U →C satisfy the admissibility condition φ  1  Meiθ, 1  Meiθ  kMeiθ 1  Meiθ , L; z  /∈Ω 2.19

Theorem 2.6 yields 1 −αzf′z fz  α  1  zf′′z f′z  −1  < M ⇒  zf′z fz −1  < M α ≥2M −1 ≥0, 

Theorem 2.8. Let Ω be a set in C and let the function φ : C3 × U →C satisfy the admissibility condition φiρ, iτ, ξ  iη; z /∈Ω 2.26 for all z ∈U and for all real ρ, τ, ξ and η with ρτ ≥1 21  3ρ2, ρη ≥0. 2.27 Let f ∈A with f′zfz/z / 0. If φ zf′z fz , 1  zf′′z f′z , z2{f, z}; z 

Corollary 2.9 see 2, Theorem 4.6a, page 244 . Let φ ∈ΦSΔ . If f ∈A with fzf′z/z / 0 satisfies R  φ zf′z fz , 1  zf′′z f′z , z2{f, z}; z  > 0, 2.33 then f ∈S∗. 3. Superordination and starlikeness Now we will give the dual result of Theorem 2.2 for differential superordination. Definition 3.1. Let Ω be a set in C, q ∈H with zq′z / 0. The class of admissible functions

Theorem 3.2. Let φ ∈Φ′ SΩ, q , and f ∈A with f′zfz/z / 0. If zf′z/fz ∈Q1 and φzf′z/fz, 1  zf′′z/f′z, z2{f, z}; z is univalent in U, then Ω ⊂  φ zf′z fz , 1  zf′′z f′z , z2{f, z}; z  : z ∈U  3.3 implies qz ≺zf′z

Theorem 3.3. Let q ∈H, h be analytic in U and φ ∈Φ′ Sh, q . If f ∈A, f′zfz/z / 0, zf′z/fz ∈Q1 and φzf′z/fz, 1  zf′′z/f′z, z2{f, z}; z is univalent in U, then hz ≺φ zf′z fz , 1  zf′′z f′z , z2{f, z}; z  3.9 implies qz ≺zf′z fz . 3.10 Theorems 3.2 and 3.3 can only be used to obtain subordinants of differential superordinations of the form 3.3 or 3.9. The following theorem proves the existence of

Theorem 3.4. Let h be analytic in U and φ : C3 × U →C. Suppose that the differential equation φ  qz, qz  zq′z qz , zq′z  z2q′′z qz −3 2 zq′z qz 2  1 −q2z 2 ; z 

Corollary 3.5. Let h1 and q1 be analytic functions in U, let h1 be an analytic univalent function in U, q2 ∈Q1 with q10  q20  1 and φ ∈ΦSh2, q2 ∩Φ′ Sh1, q1 . Let f ∈A with f′zfz/z / 0. If zf′z/fz ∈H∩Q1 and φzf′z/fz, 1zf′′z/f′z, z2{f, z}; z is univalent in U, then h1z ≺φ zf′z fz , 1  zf′′z f′z , z2{f, z}; z  ≺h2z 3.15 implies q1z ≺zf′z fz ≺q2z. 3.16

Theorem 4.2. Let φ ∈ΦScΩ, q , and f ∈A with f′z / 0. If  φ  1  zf′′z f′z , z2{f, z}; z  : z ∈U  ⊂Ω, 4.2 then 1  zf′′z f′z ≺qz. 4.3

Theorem 4.3. Let φ ∈ΦSch, q . If f ∈A with f′z / 0 satisfies φ  1  zf′′z f′z , z2{f, z}; z  ≺hz, 4.11

Theorem 4.4. Let Ω ⊂C and let q be univalent in U with q0  1. Let φ ∈ΦSch, qρ for some ρ ∈0, 1 where qρz  qρz. If f ∈A with f′z / 0 satisfies 4.2, then 4.12 holds.

Theorem 4.5. Let Ω be a set in C, qz  1  Mz, M > 0, and φ : C2 × U →C satisfy φ  1  Meiθ, 2k −1 −Meiθ 2 Meiθ; z  /∈Ω 4.13 whenever z ∈U, θ ∈R and k ≥1. Let f ∈A with f′z / 0. If φ  1  zf′′z f′z , z2{f, z}; z 

Theorem 4.7. Let Ω be a set in C. Let φ : C2 × U →C satisfy the admissibility condition φiρ, η; z /∈Ω 4.21 for all z ∈U, and for all real ρ and η with η ≤0. Let f ∈A with f′z / 0. If φ  1  zf′′z f′z , z2{f, z}; z  ∈Ω, 4.22 then f ∈K. Let hz  1  z/1 −z. Clearly, hU  Δ. Writing the class of admissible functions ΦSchU, Δ as ΦScΔ , the following result is a restatement of 1.6, which is an immediate consequence of Theorem 4.7.

Corollary 4.8 see 2, Theorem 4.6b, page 246 . Let φ ∈ΦScΔ . If f ∈A with f′z / 0 satisfies R  φ  1  zf′′z f′z , z2{f, z}; z  > 0, 4.23 then f ∈K. Definition 4.9. Let Ω be a set in C and q ∈H. The class of admissible functions Φ′ ScΩ, q consists of those functions φ : C2 × U →C that satisfy the admissibility condition

Theorem 4.10. Let φ ∈Φ′ ScΩ, q , and f ∈A with f′z / 0. If 1  zf′′z/f′z ∈Q1 and φ1  zf′′z/f′z, z2{f, z}; z is univalent in U, then Ω ⊂  φ  1  zf′′z f′z , z2{f, z}; z  : z ∈U  4.25

Lemma 1.4, qz ≺pz or qz ≺1  zf′′z f′z . 4.29 Proceeding similarly as in the previous section, the following result is an immediate consequence of Theorem 4.10.

Theorem 4.11. Let q ∈H, let h be analytic in U and φ ∈Φ′ Sch, q . Let f ∈A with f′z / 0. If 1  zf′′z/f′z ∈Q1 and φ1  zf′′z/f′z, z2{f, z}; z is univalent in U, then hz ≺φ  1  zf′′z f′z , z2{f, z}; z  4.30 implies qz ≺1  zf′′z f′z . 4.31 Combining Theorems 4.3 and 4.11, we obtain the following sandwich-type theorem.

Corollary 4.12. Let h1 and q1 be analytic functions in U, let h1 be analytic univalent in U, q2 ∈ Q1 with q10  q20  1 and φ ∈ΦSch2, q2 ∩Φ′ Sch1, q1 . Let f ∈A with f′z / 0. If 1  zf′′z/f′z ∈H ∩Q1 and φ 1  zf′′z/f′z, z2{f, z}; z is univalent in U, then h1z ≺φ  1  zf′′z f′z , z2{f, z}; z  ≺h2z 4.32