Iterated integral transforms of Caratheodory functions and their applications to analytic and univalent functions

Lemma 2.3. Let p ∈P(Ψγ). Then Re p(z) > γ, if 0 ≤γ < 1, < γ, if γ > 1.

Theorem 3.1. Let γ ̸= 1 be a nonnegative real number. Then for each n ∈N, Re pn−1(z) > γ ⇒Re pn(z) > γ for 0 ≤γ < 1 and Re pn−1(z) < γ ⇒Re pn(z) < γ for γ > 1

Corollary 3.2. Pn ⊂P, n ∈N.

Theorem 3.3. Pn+1 ⊂Pn

Corollary 3.4.([1]) Let p ∈P and γ + c > 0. Then q(z) = 1 + (γ + c) ∞ X k=1 pkzk (γ + c + k), z ∈E (3.5) is also in P. The above corollary and its extension in [6] follow easily by taking α = γ + c > 0, n = 0 in Theorem 3.3. The proofs in the two articles made use of a result of Miller and Mocanu [5, Theorem 10], which as observed in MR96j:30018, may not be applied directly except γ + c is an integer.

Theorem 3.5. The transformation (1.3) is starlikeness-preserving. In other words, if p ∈P is starlike in E, then its transform pn is also starlike in E.

Corollary 3.6. The transformation Ln(z) of the Moebius function is starlike and univalent in E.

Theorem 3.7. The transformation (1.3) is convexity-preserving. In other words, if p ∈P is convex in E, then its transform pn is also convex in E.

Corollary 3.8. The transformation Ln(z) of the Moebius function is convex in E.

Theorem 3.9. Let pn ∈Pn. Then |pn(z)| ≤1 + 2 ∞ X k=1 αn (α + k)n rk, |z| = r, (3.12) Re pn(z) ≥1 + 2 ∞ X k=1 αn (α + k)n (−r)k,

Corollary 3.10. pn ∈Pn if and only if pn(z) ≺Ln(z).

Lemma 4.2. Let f(z) be given by (1.2), and α, β and Dn as defined above. Then the following are equivalent: (i) f ∈T α n (β)

Theorem 4.3. Tn+1 α(β) ⊂Tn α(β).

Theorem 3.3 (f(z)α/zα −β)/(1 −β) ∈Pn. That is f ∈Tn α(β).

Corollary 4.4. For n ≥1, Tn α(β) ⊂S (the class of functions f(z) given by (1.2) which are univalent in E).

Theorem 4.5. Let f(z) given by (1.2) be in the class Tn α(β). Then the function F(z) defined by F(z)α = α + c zc Z z 0 tc−1f(t)αdt, α + c > 0 (4.4) is also in Tn α(β).

Lemma 4.2, F ∈Tn α(β).

Theorem 4.6. A function F(z) defined by F(z)α+υ = zυf(z)α (where f(z) is given by (1.2)) is in the class Tn α+υ(β) if and only if f(z) is in Tn α(β).

Theorem 4.7. Let f ∈Tn α(β) and define MT (n, α, β, r) = r n 1 + 2(1 −β)αn ∞ X k=1 rk (α + k)n o 1 α and mT (n, α, β, r) = r n

Theorem 4.8. Each function f(z) in the class Tn α(β) maps the unit disk onto a domain which covers the disk |ξ| < mT (n, α, β, 1). The result is sharp.

Theorem 4.9. Let f ∈Tn α(β) and define M ∗ T (n, α, β, r) = rα−1n 1 + 2(1 −β) ∞ X k=1 αn−1 (α + k)n−1 rko