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Abstract
In the present paper we introduce and investigate an interesting subclass K_{s}^{(k)}(γ,p) of analytic and p-valently close-to-convex functions in the open unit disk U. For functions belonging to this class, we derive several properties as the inclusion relationships and distortion theorems. The various results presented here would generalize many known recent results.
Results & Lemmas (11)
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Lemma 1.
Lemma 1. If g(z) = zp + ∞ X n=1 bn+pzn+p ∈S∗ p (k −1) p k , (2.1) then Gk(z) = gk(z) z(k−1)p = zp +
Lemma 1. If g(z) = zp + ∞ X n=1 bn+pzn+p ∈S∗ p (k −1) p k , (2.1) then Gk(z) = gk(z) z(k−1)p = zp +
Lemma 2.
Lemma 2. Let the function H (z) = p + h1z + h2z2 + · · · (z ∈U) be analytic in the unit disk U. Then, the function H satisfies the condition
Lemma 2. Let the function H (z) = p + h1z + h2z2 + · · · (z ∈U) be analytic in the unit disk U. Then, the function H satisfies the condition
Lemma 3.
Lemma 3. [4] A function p ∈P satisfies the following condition: ℜ(p(z)) > 0 (z ∈U) if and only if p(z) ̸= ζ −1 ζ + 1 (z ∈U; ζ ∈C; |ζ| = 1).
Lemma 3. [4] A function p ∈P satisfies the following condition: ℜ(p(z)) > 0 (z ∈U) if and only if p(z) ̸= ζ −1 ζ + 1 (z ∈U; ζ ∈C; |ζ| = 1) .
Lemma 4.
Lemma 4. A function f ∈Ap given by (1.1) is in the class K(k) s (γ, p) if and only if 1 + ∞ X n=1 An+pzn ̸= 0 (z ∈U), where An+p = (ζ + 1)…
Lemma 4. A function f ∈Ap given by (1.1) is in the class K(k) s (γ, p) if and only if 1 + ∞ X n=1 An+pzn ̸= 0 (z ∈U) , where An+p = (ζ + 1) (n + p) an+p + (p −2γ −pζ) Bn+p 2 (p −γ) (ζ ∈C; |ζ| = 1) .
Lemma 5.
Lemma 5. [6] Let −1 ≤B2 ≤B1 < A1 ≤A2 ≤1. Then 1 + A1z 1 + B1z ≺1 + A2z 1 + B2z.
Lemma 5. [6] Let −1 ≤B2 ≤B1 < A1 ≤A2 ≤1. Then 1 + A1z 1 + B1z ≺1 + A2z 1 + B2z.
Theorem 1.
Theorem 1. Let f be an analytic function in U given by (1.1). Then f ∈K(k) s (γ, p) if and only if there exists a function g ∈S∗ p (k−1)p…
Theorem 1. Let f be an analytic function in U given by (1.1). Then f ∈K(k) s (γ, p) if and only if there exists a function g ∈S∗ p (k−1)p k such that z(k−1)p+1f ′(z) gk(z) ≺p + (p −2γ) z 1 −z (z ∈U) ,
Theorem 2.
Theorem 2. We have K(k) s (γ, p) ⊂Kp.
Theorem 2. We have K(k) s (γ, p) ⊂Kp.
Theorem 3.
Theorem 3. Suppose that g ∈S∗ p (k−1)p k , where gk is given by (1.3). If f is an analytic function in U of the form (1.1), such that 2…
Theorem 3. Suppose that g ∈S∗ p (k−1)p k , where gk is given by (1.3). If f is an analytic function in U of the form (1.1), such that 2 ∞ X n=1 (n + p) |an+p| + (|p −2γ| + p) ∞ X
Theorem 4.
Theorem 4. Suppose that an analytic function f given by (1.1) and g ∈S∗ p (k−1)p k given by (2.1) are such that the condition (1.2)…
Theorem 4. Suppose that an analytic function f given by (1.1) and g ∈S∗ p (k−1)p k given by (2.1) are such that the condition (1.2) holds. Then, for n ≥1, we have |(n + p) an+p −pBn+p|2−4 (p −γ)2 ≤2 (p −γ) n+p−1 X m=p+1 2m |amBm| + (|p −2γ| + p) |Bm|2 ,
Theorem 5.
Theorem 5. If f ∈K(k) s (γ, p), then for |z| = r (0 ≤r < 1), we have (i) [p −(p −2γ) r] rp−1 (1 + r)2p+1 ≤|f ′(z)| ≤[p + (p −2γ) r] rp−1 (1…
Theorem 5. If f ∈K(k) s (γ, p), then for |z| = r (0 ≤r < 1), we have (i) [p −(p −2γ) r] rp−1 (1 + r)2p+1 ≤|f ′(z)| ≤[p + (p −2γ) r] rp−1 (1 −r)2p+1 , (3.7) (ii) Z r 0 [p −(p −2γ) τ] τ p−1 (1 + τ)2p+1
Theorem 6.
Theorem 6. Let 0 ≤γ2 ≤γ1 < p. Then we have K(k) s (γ1, p) ⊂K(k) s (γ2, p).
Theorem 6. Let 0 ≤γ2 ≤γ1 < p. Then we have K(k) s (γ1, p) ⊂K(k) s (γ2, p) .
Function classes studied:
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