Abstract
Let A_n be the class of functions f(z) which are analytic in the open unit disk U} with f(0)=0, f'(0)=1, f"(0)=f"'(0)=...=f^{(n)}=0 and f^{(n+1)}\neq0. Applying the results due to S. S. Miller (J. Math. Anal. Appl. 65(1978), 289-305), some interesting starlikeness problems concerned with subordinations are discussed. The results in the paper are extensions of results by M. Obradović (Hokkaido Math. J. 27(1998), 329-335).
Results & Lemmas (10)
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Lemma 1.
Lemma 1. Let the function w(z) defined by w(z) = anzn + an+1zn+1 + an+2zn+2 +... (n = 1, 2, 3,...) be analytic in U with w(0) = 0. If |w(z)|…
Lemma 1. Let the function w(z) defined by w(z) = anzn + an+1zn+1 + an+2zn+2 + . . . (n = 1, 2, 3, . . .) be analytic in U with w(0) = 0. If |w(z)| attains its maximum value on the circle |z| = r at a point z0 ∈U, then there exists a real number k ≧n such that z0w′(z0) w(z0) = k. 2. Main result Applying Lemma 1, we have the following lemma.
Lemma 2.
Lemma 2. Let p(z) ∈H[1, n] satisfy the condition p(z) −1 µzp′(z) ≺1 + λz (z ∈U) for some complex number µ (Re(µ) < n, µ ̸= 0) and some…
Lemma 2. Let p(z) ∈H[1, n] satisfy the condition p(z) −1 µzp′(z) ≺1 + λz (z ∈U) for some complex number µ (Re(µ) < n, µ ̸= 0) and some complex number λ (0 < |λ| ≦1), then p(z) ≺1 + λ1z (z ∈U), where λ1 is a complex number such that |λ1| = |λ| |µ| |n −µ|. (2.1)
Lemma 3.
Lemma 3. Let λ and λ1 be complex numbers such that 0 < |λ1| < |λ| < 1 and let Q(z) ∈H[1, n] such that Q(z) ≺1 + λ1z (z ∈U). (2.2) (1) If…
Lemma 3. Let λ and λ1 be complex numbers such that 0 < |λ1| < |λ| < 1 and let Q(z) ∈H[1, n] such that Q(z) ≺1 + λ1z (z ∈U). (2.2) (1) If p(z) ∈H[1, n] and Q(z)[α + (1 −α)p(z)] ≺1 + λz (z ∈U) (2.3) for some real α such that α ≦
Corollary 1.
Corollary 1. Let λ and λ1 be real with 0 < λ1 < λ < 1 and let Q(z) ∈H[1, n] satisfy Q(z) ≺1 + λ1z (z ∈U). (1) If p(z) ∈H[1, n] and Q(z)[α +…
Corollary 1. Let λ and λ1 be real with 0 < λ1 < λ < 1 and let Q(z) ∈H[1, n] satisfy Q(z) ≺1 + λ1z (z ∈U). (1) If p(z) ∈H[1, n] and Q(z)[α + (1 −α)p(z)] ≺1 + λz (z ∈U) for some real α such that α =
Theorem 1.
Theorem 1. If f(z) ∈An satisfies the condition f ′(z) z f(z) 1+µ ≺1 + λz (z ∈U) for some complex numbers µ (Re(µ) < n) and λ such that 0…
Theorem 1. If f(z) ∈An satisfies the condition f ′(z) z f(z) 1+µ ≺1 + λz (z ∈U) for some complex numbers µ (Re(µ) < n) and λ such that 0 < |λ| ≦ |n −µ| p |n −µ|2 + |µ|2 , then f(z) ∈S∗.
Corollary 2.
Corollary 2. If f(z) ∈A satisfies the condition f ′(z) z f(z) 1+µ −1 < λ (z ∈U) for some 0 < µ < 1 and 0 < λ < 1, then f(z) ∈S∗. Applying…
Corollary 2. If f(z) ∈A satisfies the condition f ′(z) z f(z) 1+µ −1 < λ (z ∈U) for some 0 < µ < 1 and 0 < λ < 1, then f(z) ∈S∗. Applying Lemma 3, we derive the following theorem.
Theorem 2.
Theorem 2. Let f(z) ∈An satisfy the condition f ′(z) z f(z) 1+µ ≺1 + λz (z ∈U) (2.10) for some complex number µ Re(µ) < n 2
Theorem 2. Let f(z) ∈An satisfy the condition f ′(z) z f(z) 1+µ ≺1 + λz (z ∈U) (2.10) for some complex number µ Re(µ) < n 2
Corollary 3.
Corollary 3. Let f(z) ∈A satisfy the condition f ′(z) z f(z) 1+µ −1 < λ (z ∈U) with some 0 < µ < 1 2. If the real number λ1 is given by…
Corollary 3. Let f(z) ∈A satisfy the condition f ′(z) z f(z) 1+µ −1 < λ (z ∈U) with some 0 < µ < 1 2. If the real number λ1 is given by λ1 = λ µ 1 −µ,
Theorem 3.
Theorem 3. If f(z) ∈An satisfies f ′(z) z f(z) 1+µ ≺1 + λz (z ∈U) and F(z) = z c −µ zc−µ Z z 0
Theorem 3. If f(z) ∈An satisfies f ′(z) z f(z) 1+µ ≺1 + λz (z ∈U) and F(z) = z c −µ zc−µ Z z 0
Corollary 4.
Corollary 4. If f(z) ∈A satisfies f ′(z) z f(z) 1+µ −1 < λ (z ∈U) and F(z) = z c −µ zc−µ Z z
Corollary 4. If f(z) ∈A satisfies f ′(z) z f(z) 1+µ −1 < λ (z ∈U) and F(z) = z c −µ zc−µ Z z
Function classes studied:
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