Abstract
For functions $f(z)=z^p+a_{n+1}z^{p+1}+...$ defined on the open unit disk, the condition $\Re (f'(z)/z^{p-1})>0$ is sufficient for close-to-convexity of $f$. By making use of this result, several sufficient conditions for close-to-convexity are investigated and relevant connections with previously known results are indicated.
Results & Lemmas (7)
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THEOREM 1.1.
THEOREM 1.1. [11, Theorems 1-3] Let 0 ≤α < 1 and β,γ ≥0. If f ∈A, then Re 1+ z f ′′(z) f ′(z) > 1+3α 2(1+α) =⇒Re f ′(z) > 1+α 2, Re
THEOREM 1.1. [11, Theorems 1-3] Let 0 ≤α < 1 and β,γ ≥0. If f ∈A , then Re 1+ z f ′′(z) f ′(z) > 1+3α 2(1+α) =⇒Re f ′(z) > 1+α 2 , Re
THEOREM 1.2.
THEOREM 1.2. [11, Theorem 4] Let 1 < λ < 3. If f ∈A, then Re 1+ z f ′′(z) f ′(z) < 5λ−1 2(λ+1), 1 < λ ≤2; λ+1 2(λ−1),
THEOREM 1.2. [11, Theorem 4] Let 1 < λ < 3. If f ∈A , then Re 1+ z f ′′(z) f ′(z) < 5λ−1 2(λ+1), 1 < λ ≤2; λ+1 2(λ−1),
THEOREM 2.1.
THEOREM 2.1. If the function f ∈Ap,n satisfies the inequality (2.1) Re 1+ z f ′′(z) f ′(z) > (2p−n)+α(2p+n) 2(α +1), then Re f ′(z)…
THEOREM 2.1. If the function f ∈Ap,n satisfies the inequality (2.1) Re 1+ z f ′′(z) f ′(z) > (2p−n)+α(2p+n) 2(α +1) , then Re f ′(z) pzp−1
LEMMA 2.1.
LEMMA 2.1. [6, Lemma 2.2a] Let z0 ∈D and r0 = |z0|. Let f(z) = anzn +an+1zn+1 +··· be continuous on Dr0 and analytic on Dr0 ∪ z0 with f(z)…
LEMMA 2.1. [6, Lemma 2.2a] Let z0 ∈D and r0 = |z0|. Let f(z) = anzn +an+1zn+1 +··· be continuous on Dr0 and analytic on Dr0 ∪{z0} with f(z) ̸≡0 and n ≥1. If | f(z0)| = max{| f(z)| : z ∈Dr0}, then there exists an m ≥n such that (1) z0 f ′(z0) f(z0) = m, and (2) Re z0 f ′′(z0) f ′(z0) +1 ≥m. PROOF OF THEOREM 2.1. Let the function w be defined by (2.2) f ′(z) pzp−1 = 1+αw(z) 1+w(z) . Then clearly w is analytic in D with w(0) = 0. From (2.2), some computation yields
THEOREM 2.2.
THEOREM 2.2. If the function f ∈Ap,n satisfies the inequality (2.5) Re 1+ z f ′′(z) f ′(z) < (p+n)α +(2p+n) (α +2), then
THEOREM 2.2. If the function f ∈Ap,n satisfies the inequality (2.5) Re 1+ z f ′′(z) f ′(z) < (p+n)α +(2p+n) (α +2) , then
THEOREM 2.3.
THEOREM 2.3. If f ∈Ap,n, then (2.9)
THEOREM 2.3. If f ∈Ap,n, then (2.9)
THEOREM 2.4.
THEOREM 2.4. Let λ1 and λ2 be given by λ1 = 2n+4(2p−1) 4+n−2p+ p 16n+n2 +32p−12np−28p2, λ2 = 2n+4(2p−1) −n+2p+ p 16−8n+n2 −48p+4np+36p2,…
THEOREM 2.4. Let λ1 and λ2 be given by λ1 = 2n+4(2p−1) 4+n−2p+ p 16n+n2 +32p−12np−28p2, λ2 = 2n+4(2p−1) −n+2p+ p 16−8n+n2 −48p+4np+36p2, and λ1 < λ < λ2. If the function f ∈Ap,n satisfies the inequality (2.17) Re
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