Abstract
In the present paper, we study a new subclass $\mathcal{M}_p(α,β)$ of $p$--valent functions and obtain some inequalities concerning the coefficients for the desired class. Also, by use of the Hadamard product, we define a general operator and find a condition such that it belongs to the class $\mathcal{M}_p(α,β)$.
Results & Lemmas (8)
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Theorem 2.1.
Theorem 2.1. Let the function f be in the class Mp(α, β) and let f be of the form (1.3) for some bk such that bk ≥0 for k ∈ p, p + 1, p +…
Theorem 2.1. Let the function f be in the class Mp(α, β) and let f be of the form (1.3) for some bk such that bk ≥0 for k ∈{p, p + 1, p + 2, . . .} and ∞ X k=p bk < 1. Then (2.1) ∞ X k=p
Theorem 2.2.
Theorem 2.2. Let f ∈Ap be of the form (1.3) with µ > 0. If (2.4) ∞ X k=p [pµ(α −1) + k(1 −β)]|bk| < pµ(α −1), then f is in the class Mp(α,…
Theorem 2.2. Let f ∈Ap be of the form (1.3) with µ > 0. If (2.4) ∞ X k=p [pµ(α −1) + k(1 −β)]|bk| < pµ(α −1), then f is in the class Mp(α, β), where β ≤0 and α > 1.
Corollary 2.1.
Corollary 2.1. Assume that f ∈A and (z/f(z))µ = 1 −P∞ k=1 bkzk with µ > 0. If the function f satisfies the condition ∞ X k=1 [µ(α −1) + k(1…
Corollary 2.1. Assume that f ∈A and (z/f(z))µ = 1 −P∞ k=1 bkzk with µ > 0. If the function f satisfies the condition ∞ X k=1 [µ(α −1) + k(1 −β)]|bk| < µ(α −1), then f is in the class MD(α, β), where β ≤0 and α > 1.
Theorem 2.3.
Theorem 2.3. A function f of the form f(z) = zp + P∞ k=p+1 akzk is in the class Mp(α, β), if (2.8) ∞ X k=p+1 [pα + β + k(1 −β)]|ak| < p(α…
Theorem 2.3. A function f of the form f(z) = zp + P∞ k=p+1 akzk is in the class Mp(α, β), if (2.8) ∞ X k=p+1 [pα + β + k(1 −β)]|ak| < p(α −1).
Corollary 2.2.
Corollary 2.2. If f ∈A satisfies ∞ X k=2 [α + β + k(1 −β)]|ak| < α −1, for some α > 1 and β ≤0, then f ∈MD(α, β). At the end of this…
Corollary 2.2. If f ∈A satisfies ∞ X k=2 [α + β + k(1 −β)]|ak| < α −1, for some α > 1 and β ≤0, then f ∈MD(α, β). At the end of this section, by Theorem 2.3 we consider an example for the class Mp(α, β). Example 2.1. Define the function f ∈Ap as follows f(z) = zp + ∞ X k=p+1 p(p −1)(α −1)eiθ [pα + β + k(1 −β)]k(k −1)zk,
Theorem 3.1.
Theorem 3.1. Let a, b ∈C 0 and e, d ∈C. Also, assume that δ, d + e+1 2 > 0 and c be a real number such that c > |a| + |b| + 1. Then Ia,b…
Theorem 3.1. Let a, b ∈C\{0} and e, d ∈C. Also, assume that δ, d + e+1 2 > 0 and c be a real number such that c > |a| + |b| + 1. Then Ia,b c,d(p, e, δ)(z) ∈Mp(α, β) if (3.3) ∞ X k=p+1 [pα + β + k(1 −β)] |(a)k−p(b)k−p|(δ/4)k−p (c)k−p d + e+1 2
Corollary 3.1.
Corollary 3.1. If a, b ∈C 0, e, d ∈C, δ, d + e+1 2 > 0 and c be a real number such that c > |a| + |b| + 1, then a sufficient condition for…
Corollary 3.1. If a, b ∈C\{0}, e, d ∈C, δ, d + e+1 2 > 0 and c be a real number such that c > |a| + |b| + 1, then a sufficient condition for Ia,b c,d(e, δ)(z) ∈MD(α, β) is that ∞ X k=2 [α + β + k(1 −β)] |(a)k−1(b)k−1|(δ/4)k−1 (c)k−1 d + e+1 2
Corollary 3.2.
Corollary 3.2. If a, b ∈C 0, e, d ∈C, δ, d + e+1 2 > 0 and c be a real number such that c > |a| + |b| + 1, then a sufficient condition for…
Corollary 3.2. If a, b ∈C\{0}, e, d ∈C, δ, d + e+1 2 > 0 and c be a real number such that c > |a| + |b| + 1, then a sufficient condition for Ia,b c,d(e, δ)(z) ∈M(α) is that ∞ X k=2 [α + k] |(a)k−1(b)k−1|(δ/4)k−1 (c)k−1 d + e+1 2
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