New criteria for Carathéodory functions

Lemma 2.1 Suppose that function w is analytic for |z| ≤r, w(0) = 0 and |w(z0)| = max|z|=r |w(z)|. Then z0w′(z0) = kw(z0), where k is a real number with k ≥1.

Lemma 2.2 Let q ∈Q, with q(0) = a, and let p(z) = a + anzn + ··· be analytic in U with p(z) ̸≡a and n ≥1. If p is not subordinate to q, then there exist points z0 = r0eiθ0 ∈U and ζ0 ∈T \ E(q), and an m ≥n ≥1 for which p(Ur0) ⊂q(U), (i) p(z0) = q(ζ0), (ii) z0p′(z0) = mζ0q′(ζ0) and (iii) Re{1 + z0p′′(z0) p′(z0) } ≥mRe{1 + ζ0q′′(ζ0) q′(ζ0) }. By using Lemma 2.1, we now derive the following theorem.

Theorem 2.3 Let p be analytic in U with p(0) = 1. If Re  p(z) + βzp′(z)  > α – β 2(1 – α)  1 – 2α +

Corollary 2.4 Let p be analytic in U with p(0) = 1. If Re  p(z) + zp′(z)  > –1 + |p(z)|2 2 , then p ∈P.

Theorem 2.6 Let p be analytic in U with p(0) = 1. If Re  p(z) + βzp′(z)  > α – β(1 – α) 2 (0 ≤α < 1,β ≥0), then p ∈P(α). Letting β = 1 in Theorem 2.6, we have the following corollary.

Corollary 2.7 Let p be analytic in U with p(0) = 1. If Re  p(z) + zp′(z)  > 3α – 1 2 (0 ≤α < 1), then p ∈P(α).

Theorem 2.10 Let p be analytic in U with p(0) = 1. If Re  p(z) + zp′(z) βp(z) + γ  > δ  α,β,γ ,

Corollary 2.12 Let p be analytic in U with p(0) = 1 and 0 ≤α < 1. If Re  p(z) + zp′(z) p(z)  > α(1 – 2α)(|p(z)|2 – 1) 2(1 – α)|p(z)|2 , then p ∈P(α). Applying Theorem 2.10 leads us to get the following theorem which doesn’t depend on |p(z)|.

Theorem 2.13 Let p be analytic in U with p(0) = 1 and 0 ≤α < 1. If p satisfies one of the following conditions: (i) Re{p(z) + zp′(z) βp(z)+γ } > α – αβ+γ 2β2(1–α) (–αβ < γ < β(1 – 2α) for β > 0 or –αβ < γ < –β for β < 0), (ii) Re{p(z) + zp′(z) βp(z)+γ } > α – 1–α 2(αβ+γ ) (γ ≥β(1 – 2α) for β > 0 or γ ≥–β for β < 0), then p ∈P(α).

Corollary 2.14 Let p be analytic in U with p(0) = 1. If function p satisfies the following condition, then p ∈P(α): Re  p(z) + zp′(z) p(z)  > ⎧ ⎨ ⎩ α–2α2 2(1–α), when 0 ≤α < 1/2, (1+α)(2α–1)

Theorem 2.16 Let α and β be real numbers such that 0 ≤α < 1 and β ≥(3α – 1)/2. Let p be analytic in U with p(0) = 1. If

Corollary 2.17 Let 0 ≤α < 1 and β ≥(3α – 1)/2. And let p be an analytic function in U with p(0) = 1. If p satisfies

Corollary 2.18 Let p be analytic in U with p(0) = 1. If

Theorem 2.19 Let α and A be real numbers with 0 ≤α < 1 and A ≥0. And let B and C be functions defined in U such that Re{B(z)} > A for all z ∈U. If p is analytic in U with p(0) = 1 and Re  Az2p′′(z) + B(z)zp′(z) + C(z)p(z)  > δ  α,A,B(z),C(z)

Corollary 2.20 Let p be analytic in U with p(0) = 1. Then Re  p(z) + zp′(z)  > –1 2 ⇒ Re  p(z)  > 0.

Corollary 2.7. Taking A = 0, B(z) = C(z) ≡1 and α = 1/2 in Theorem 2.19, we have the following result.

Corollary 2.22 Let p be analytic in U with p(0) = 1. Then Re  p(z) + zp′(z)  > 1 4 ⇒ Re  p(z)  > 1 2. Letting p(z) = (Iγ ,β[f ](z)/z)β (f ∈A), where Iγ ,β : A →A is the integral operator defined

Corollary 2.23 Let f ∈A and let β and γ be complex numbers. If Re  (γ + β) f (z) z β > 1 2  Im{γ + β} 2 + 2α Re{γ + β} – 1  , then

Theorem 2.24 Let α and A be real numbers with 0 ≤α < 1 and A ≥0. And let B and C be functions defined in U such that Re{B(z)} = A and Im{C(z)} = 0 for all z ∈U. If p is analytic in U with p(0) = 1 and Re  Az2p′′(z) + B(z)zp′(z) + C(z)p(z)  > α Re  C(z)  , then p ∈P(α). Taking A = 1, B(z) = C(z) ≡1 in Theorem 2.19, then we have the following result.

Corollary 2.25 Let p be analytic in U with p(0) = 1. Then Re  z2p′′(z) + zp′(z) + p(z)  > α ⇒ Re  p(z)  > α. Next, we derive another conditions for Carathéodory functions of order α in Theo- rems 2.26 and 2.27 below.

Theorem 2.26 Let p be analytic in U with p(0) = 1 and 0 ≤α < 1. If p satisfies zp′(z) p(z) – α ̸= iΛ (27) for all Λ ∈R with |Λ| ≥1, then p ∈P(α).

Theorem 2.27 Let 0 ≤α < 1 and let p be analytic in U with p(0) = 1. If p satisfies zp′(z) p(z) – α (p(z) – α)2 – (1 – α)2 (p(z) – α)2 + (1 – α)2 ̸= i (29) for all ∈R with | | ≥2, then p ∈P(α).

Theorem 2.29 Let 0 ≤α < 1 and 0 < β ≤1. If p is analytic in U with p(0) = 1 and Re  p(z) – α β

Corollary 2.31 Let f ∈A and 0 < β ≤1. If Re  f ′(z) f (z) z β–1 > h  δ(0,β),0,β