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Abstract
In the present paper, we give the bounds for the second Hankel determinant of the
logarithmic coefficients of a certain subclass of normalized univalent functions, which we have
introduced here. Relevant connections of the results, which we have presented here, with those
available in the existing literature are also described briefly.
2010 Mathematics Subject Classification: Primary 30C45, 30C50; Secondary 30C80
Results & Lemmas (4)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1.
Lemma 1. (see [5]) If the function p ∈P is given by the series (1.2), then |ck| ≤2 (k ∈N). (2.1) This inequality is sharp for each k.
Lemma 1. (see [5]) If the function p ∈P is given by the series (1.2), then |ck| ≤2 (k ∈N). (2.1) This inequality is sharp for each k.
Lemma 2.
Lemma 2. (see [14,15]) Let the function p ∈P be given by the series (1.2). Then there exist x,z ∈C with |x| ≤1 and |z| ≤1 such that 2c2 =…
Lemma 2. (see [14,15]) Let the function p ∈P be given by the series (1.2). Then there exist x,z ∈C with |x| ≤1 and |z| ≤1 such that 2c2 = c2 1 +x 4−c2 1 (2.2) and 4c3 = c3 1 +2 4−c2 1 c1x−c1
Theorem 1.
Theorem 1. Let f(z) ∈R (λ,α). Then γ1γ3 −γ2 2 ≤
Theorem 1. Let f(z) ∈R (λ,α). Then γ1γ3 −γ2 2 ≤
Corollary 1.
Corollary 1. (see [10]) Let f(z) ∈S ∗(α). Then γ1γ3 −γ2 2 ≤(1−α)2 4. 4. CONCLUSION The present investigation is motivated essentially by…
Corollary 1. (see [10]) Let f(z) ∈S ∗(α). Then γ1γ3 −γ2 2 ≤(1−α)2 4 . 4. CONCLUSION The present investigation is motivated essentially by several recent developments. We have given the bounds for the second Hankel determinant: H2,1 Ff 2 = γ1γ3 −γ2 2
Definitions (3)
Def 1.
Definition 1. The Hankel determinant Hq,n(f) (q,n ∈N) is defined for a function f ∈A of form (1.1) by Hq,n(f) =
Definition 1. The Hankel determinant Hq,n(f) (q,n ∈N) is defined for a function f ∈A of form (1.1) by Hq,n(f) =
Def 2.
Definition 2. The Hankel determinant Hq,n Ff 2 involving the logarithmic coeffi- cients of f is defined by Hq,n Ff 2
Definition 2. The Hankel determinant Hq,n Ff 2 involving the logarithmic coeffi- cients of f is defined by Hq,n Ff 2
Def 3.
Definition 3. For 0 ≤α < 1, a function f ∈A is said to be in the class R (λ,α) (λ ≥ 0) if it satisfies the following condition: ℜ z f…
Definition 3. For 0 ≤α < 1, a function f ∈A is said to be in the class R (λ,α) (λ ≥ 0) if it satisfies the following condition: ℜ z f ′(z)+λz2 f ′′(z) f(z) > α (z ∈U). (1.6) 2. A SET OF LEMMAS
Function classes studied:
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