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Abstract
The purpose of the present paper is to introduce the classes Mα(β) and Qα(β),
respectively, of normalized strongly α-convex and α-quasiconvex functions of order β in the
open unit disk and to obtain sharp Fekete-Szeg¨o inequalities for functions belonging to the
classes Mα(β) and Qα(β).
Results & Lemmas (5)
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Lemma 2.1.
Lemma 2.1. Let p be analytic in U and satisfy Re p(z) > 0 for z ∈U, with p(z) = 1 + p1z + p2z2 + · · ·. Then |pn| ≤2 (n ≥1) (2.1) and p2…
Lemma 2.1. Let p be analytic in U and satisfy Re{p(z)} > 0 for z ∈U, with p(z) = 1 + p1z + p2z2 + · · ·. Then |pn| ≤2 (n ≥1) (2.1) and p2 −p2 1 2 ≤2 −|p1|2 2 . (2.2) The inequality (2.1) was first proved by Carath´eodory [3] (also, see Duren [4, p.41]) and
Theorem 2.1.
Theorem 2.1. Let f ∈Mα(β) and be given by (1.1). The for complex number µ, |a3 −µa2 2| ≤ β 1 + 2α max 1, |α2 + 8α + 3 −4µ(1 + 2α)|β (1 +…
Theorem 2.1. Let f ∈Mα(β) and be given by (1.1). The for complex number µ, |a3 −µa2 2| ≤ β 1 + 2α max 1, |α2 + 8α + 3 −4µ(1 + 2α)|β (1 + α)2 . For each µ, there is a function in Mα(β) such that equality holds.
Theorem 2.2.
Theorem 2.2. Let f ∈Mα(β) and be given by (1.1). Then for real number µ, |a3 −µa2 2|≤
Theorem 2.2. Let f ∈Mα(β) and be given by (1.1). Then for real number µ, |a3 −µa2 2|≤
Theorem 2.3.
Theorem 2.3. Let f ∈Qα(β) and be given by (1.1). Then for α ≥0 and β ≥0, we have 3(2α + 1)|a3 −µa2 2|
Theorem 2.3. Let f ∈Qα(β) and be given by (1.1). Then for α ≥0 and β ≥0, we have 3(2α + 1)|a3 −µa2 2|
Corollary 2.1.
Corollary 2.1. Let f ∈Q1(β) and be given by (1.1). Then for β ≥0, we have 9|a3 −µa2 2| ≤
Corollary 2.1. Let f ∈Q1(β) and be given by (1.1). Then for β ≥0, we have 9|a3 −µa2 2| ≤
Function classes studied:
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