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FullText JMA 34_08.pdf - Journal of Mathematics and Applications

FullText JMA 34_08.pdf - Journal of Mathematics and Applications

88 Inclusion

88 Inclusion relationship and Fekete-Szegö like inequalities . . . The class of all meromorphic convex functions will be denoted by Σ c . Let f ∈ Σ be of the form (1) and let α, β be real numbers with α ≥ β ≥ 0. We define the analogue of the differential operator given in [13] as follows D 0 α,βf(z) = f(z) D 1 α,βf(z) = D α,β f(z) = = αβ(z 2 f(z)) ′′ + (α − β) (z2 f(z)) ′ + (1 − α + β)f(z) z (4) ( ) Dα,βf(z) m = D α,β D m−1 α,β f(z) , z ∈ U ∗ , m ∈ N = {1, 2, . . .} . (5) If f ∈ Σ is given by (1), then from (4) and (5) we get where Dα,βf(z) m = 1 ∞ z + ∑ A(α, β, n) m a n z n , z ∈ U ∗ (6) n=0 A(α, β, n) = [(n + 2)αβ + α − β](n + 1) + 1. (7) Note that for α = 1 and β = 0 we obtain the differential operator defined in [1]. Making use of the operator Dα,β m f(z) we introduce the following subclasses of meromorphic functions. Definition 1.1 Let Σ ∗ m(α, β) be the class of functions f ∈ Σ for which D m α,β f(z) ∈ Σ ∗ , that is Note that Σ ∗ 0(α, β) = Σ ∗ . R z(Dm α,β f(z))′ D m α,β f(z) < 0 , z ∈ U ∗ . Definition 1.2 Let γ be a complex number. We say that a function f ∈ Σ belongs to the class HΣ ∗ m(α, β, γ) if the function F defined by 1 F (z) = 1 − γ D m α,β f(z) − is a meromorphic starlike function. γ z(D m α,β f(z))′ , z ∈ U∗ (8) By specializing parameters γ and m we obtain the following subclasses: 1. HΣ ∗ m(α, β, 0) = Σ ∗ m(α, β). 2. HΣ ∗ 0(α, β, 0) = Σ ∗ . 3. HΣ ∗ 0(α, β, 1) = Σ c .

Dorina Răducanu and Halit Orhan and Erhan Deniz 89 Also, if we consider m = 0 in Definition 1.2, we obtain another subclass of Σ consisting of functions f for which the function F given by 1 F (z) = 1 − γ f(z) − γ zf ′ (z) is in the class Σ ∗ . We denote this class of functions by HΣ ∗ (γ). In this paper we find the relationship between the classes HΣ ∗ m(α, β, γ) and Σ ∗ m(α, β). Sharp upper bounds for the Fekete-Szegö like functional |a 1 − µa 2 0| are also obtained. 2 Relationship property In order to prove the relationship between the classes HΣ ∗ m(α, β, γ) and Σ ∗ m(α, β) we need the following lemma. Lemma 2.1 ([7]) Let p(z) be an analytic function in the open unit disk U = {z ∈ C : |z| < 1} with p(0) = 1 and p(z) ≠ 1. If 0 < |z 0 | < 1 and then Rp(z 0 ) = min Rp(z) |z|≤|z 0| z 0 p ′ (z 0 ) ≤ − |1 − p(z 0)| 2 2[1 − Rp(z 0 )] . Theorem 2.1 Let γ be a complex number such that ∣ γ − 1 2∣ ≤ 1 2 . Then HΣ ∗ m(α, β, γ) ⊂ Σ ∗ m(α, β). Proof. Assume that f belongs to the class HΣ ∗ m(α, β, γ). Elementary calculations show [ that if f ∈ HΣ ∗ m(α, β, γ), then R 1 + z(Dm α,β f(z))′ Dα,β m f(z) + z(Dm α,β f(z))′′ (Dα,β m f(z))′ − (1 − ] 2γ)z(Dm α,β f(z))′ + (1 − γ)z 2 (Dα,β m f(z))′′ (1 − γ)(Dα,β m f(z))′ − γDα,β m f(z) < 0 , z ∈ U ∗ . (9) Consider the analytic function p(z) ∈ U, given by p(z) = − z(Dm α,β f(z))′ D m α,β f(z) . (10) Then, the inequality (9) becomes [ R p(z) − zp′ (z) p(z) + (1 − ] γ)zp′ (z) > 0 , z ∈ U. (11) (1 − γ)p(z) + γ

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