Subordination for Higher-Order Derivatives of Multivalent Functions

Lemma 1.1 see 12, page 135, Corollary 3.4h.1. Let Q be univalent in U, and ϕ be analytic in a domain D containing QU. If zQ′z · ϕQz is starlike, and P is analytic in U with P0  Q0 and PU ⊂D, then zP ′z · ϕPz ≺zQ′z · ϕQz ⇒P ≺Q, 1.7 and Q is the best dominant.

Lemma 1.2 see 12, page 135, Corollary 3.4h.2. Let Q be convex univalent in U, and let θ be analytic in a domain D containing QU. Assume that R

Theorem 2.1. Let Qz be univalent and nonzero in U, Q0  1, and let zQ′z/Qz be starlike in U. If a function f ∈Ap satisfies the subordination zfq1z fqz ≺zQ′z Qz  p −q, 2.2 then fqz λp; qzp−q ≺Qz, 2.3 and Q is the best dominant.

Theorem 2.1 is the correct formulation of their result in a more general setting.

Corollary 2.3. Let −1 ≤B < A ≤1. If f ∈Ap satisfies zfq1z fqz ≺ zA −B 1  Az1  Bz  p −q, 2.10

Corollary 2.5 see 13. If f ∈S∗ pφ, then fz zp ≺hφ,p zp . 2.19 Similarly, choosing q  1 and Qz  k′ φ,p/pzp−1, Theorem 2.1 yields the following corollary.

Corollary 2.6 see 13. If f ∈C∗ pφ, then f′z zp−1 ≺ k′ φ,p zp−1 . 2.20

Theorem 2.7. Let Qz be convex univalent in U and Q0  1. If f ∈Ap satisfies fqz λp; qzp−q · zfq1z fqz −p  q  ≺zQ′z, 2.21 then fqz λp; qzp−q ≺Qz, 2.22 and Q is the best dominant.

Theorem 2.7 is reduced to the following result in 5, page 4, Theorem 2.4. For f ∈Ap, fqz · zfq1z fqz −p  q  ≤|z|p−q ⇒ fqz −λp; qzp−q ≤|z|p−q. 2.26 In the special case q  1, this result gives a sufficient condition for the multivalent function fz to be close-to-convex.

Theorem 2.9. Let Qz be convex univalent in U and Q0  1. If f ∈Ap satisfies zfq1z λp; qzp−q ≺zQ′z  p −qQz, 2.27 then fqz λp; qzp−q ≺Qz, 2.28 and Q is the best dominant.

Corollary 2.10 see 17, Corollary 2.11. Let Qz be convex univalent in U, and Q0  1. If f ∈Ap satisfies f′z zp−1 ≺zQ′z  pQz, 2.31

Corollary 2.11 see 17, Corollary 2.9. Let Qz be convex univalent in U, and Q0  1. If f ∈A satisfies f′z ≺zQ′z  Qz, 2.33 then fz z ≺Qz. 2.34

Theorem 2.12. Let Qz be univalent and nonzero in U, Q0  1, and zQ′z/Q2z be starlike. If f ∈Ap satisfies λp; qzp−q fqz · zfq1z fqz −p  q  ≺zQ′z Q2z , 2.35 then fqz λp; qzp−q ≺Qz,

Theorem 2.13. Let Qz be univalent and nonzero in U, Q0  1, Qz / q −p  1, and zQ′z/QzQz  p −q −1 be starlike in U. If f ∈Ap satisfies 1  zfq2z/fq1z −p  q  1 zfq1z/fqz −p  q  1 ≺1  zQ′z QzQz  p −q −1, 2.40 then zfq1z fqz −p  q  1 ≺Qz, 2.41 and Q is the best dominant.

Lemma 1.1.

Theorem 2.13 contains a result in 18, page 122, Corollary 4 as a special case. In particular, we note that Theorem 2.13 with p  1, q  0, and Qz  1  Az/1  Bz for −1 ≤B < A ≤1 yields the following corollary.

Corollary 2.14 see 18, page 123, Corollary 6. Let −1 ≤B < A ≤1. If f ∈A satisfies 1  zf′′z/f′z zf′z/fz ≺1  A −Bz 1  Az2 , 2.48 then f ∈S∗A, B. For A  0, B  b and A  1, B  −1, Corollary 2.14 gives the results of Obradoviˇc and Tuneski 19.

Theorem 2.15. Let Qz be univalent and nonzero in U, Q0  1, Qz / q −p  1, and let zQ′z/Qz  p −q −1 be starlike in U. If f ∈Ap satisfies 1  zfq2z fq1z −zfq1z fqz ≺ zQ′z Qz  p −q −1, 2.49 then zfq1z fqz −p  q  1 ≺Qz, 2.50 and Q is the best dominant.

Theorem 2.16. Let Qz be a convex function in U, and Q0  1. If f ∈Ap satisfies zfq1z fqz

Theorem 2.17. Let Qz be a convex function in U, with Q0  1. If f ∈Ap satisfies zfq1z fqz