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Abstract
In this article, our aim is to study analytic functions related with
Salagean operator and associated with the right half of the lemniscate of Bernoulli.
We find the estimates of the third Hankel determinant for new family of analytic
functions. It is important to mention that our results generalize a number of exis-
tence results in the literature.
AMS Subject Classification (2020) 30C45; 30C50
Results & Lemmas (11)
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 2.1
Lemma 2.1 ( [14]). Let p (ω) = 1 + c1ω + c2ω2 +..., be in the class P. Then for any complex number µ c2 −µc2 1 = −4µ + 2 if µ ≤0,…
Lemma 2.1 ( [14]). Let p (ω) = 1 + c1ω + c2ω2 + ... , be in the class P. Then for any complex number µ c2 −µc2 1 = −4µ + 2 if µ ≤0, 2 if 0 ≤µ ≤1,
Lemma 2.2
Lemma 2.2 ( [11,12]). Let p (ω) = 1 + c1ω + c2ω2 +..., be in the class P. Then 2c2 = c2 1 + x 4 −c2 1 , for some x, (|x| ≤1) and 4c3 = c3…
Lemma 2.2 ( [11,12]). Let p (ω) = 1 + c1ω + c2ω2 + ... , be in the class P. Then 2c2 = c2 1 + x 4 −c2 1 , for some x, (|x| ≤1) and 4c3 = c3 1 + 2 4 −c2 1
Lemma 2.3
Lemma 2.3 ( [24]). Let p (ω) = 1 + c1ω + c2ω2 +..., be in the class P of functions with positive real part in U. Then |ck| ≤2, k ∈N, and…
Lemma 2.3 ( [24]). Let p (ω) = 1 + c1ω + c2ω2 + ... , be in the class P of functions with positive real part in U. Then |ck| ≤2, k ∈N, and the inequality is sharp. 3 Main Results
Theorem 3.1.
Theorem 3.1. Let λ ∈KL∗ n be of the form of (1.1). Then c3 −µc2 2 ≤ 1 2 1 −25µ 2n+3 + 1
Theorem 3.1. Let λ ∈KL∗ n be of the form of (1.1). Then c3 −µc2 2 ≤ 1 2 1 −25µ 2n+3 + 1
Corollary 3.2.
Corollary 3.2. For n = 0 the above class KL∗ n reduces to KL∗, in (3.1), we have c3 −µc2 2 ≤ 1 48 (62 −75µ), if µ < 38 75, 1
Corollary 3.2. For n = 0 the above class KL∗ n reduces to KL∗, in (3.1), we have c3 −µc2 2 ≤ 1 48 (62 −75µ) , if µ < 38 75, 1
Theorem 3.3.
Theorem 3.3. Let λ ∈KL∗ n be of the form (1.1). Then c2c4 −c2 3 ≤9 4. (3.18)
Theorem 3.3. Let λ ∈KL∗ n be of the form (1.1). Then c2c4 −c2 3 ≤9 4. (3.18)
Theorem 3.4.
Theorem 3.4. Let λ ∈KL∗ n be of the form (1.1). Then |c2c3 −c4| ≤ 7 4n+1 · 6.
Theorem 3.4. Let λ ∈KL∗ n be of the form (1.1). Then |c2c3 −c4| ≤ 7 4n+1 · 6.
Lemma 2.3
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Lemma 2.3, let p1 = p and assume without restriction that p ∈[0, 2]. Then, taking the absolute value and applying the triangle inequality…
Lemma 2.3, let p1 = p and assume without restriction that p ∈[0, 2]. Then, taking the absolute value and applying the triangle inequality with υ = |x|, we obtain: c2c3 −c4 = 5 3n+1 · 2n+1 · 4p1 1 3n+1 5 22n+2 −21 32 p2 1 + 3
Theorem 3.5.
Theorem 3.5. If a function λ of the form of (1.1) and (3.13)-(3.16) is in the class KL∗ n, then using (1.2) we have |c2| ≤ 5 2n+2, (3.22)…
Theorem 3.5. If a function λ of the form of (1.1) and (3.13)-(3.16) is in the class KL∗ n, then using (1.2) we have |c2| ≤ 5 2n+2, (3.22) |c3| ≤ 1 3n+1 5 22n −9 8
Corollary 3.6.
Corollary 3.6. (see [32]) For n = 0 the above class KL∗ n reduces to KL∗, and (3.22)- (3.25) reduce to |c2| ≤ 5 4, |c3| ≤ 31 24, |c4| ≤ 673…
Corollary 3.6. (see [32]) For n = 0 the above class KL∗ n reduces to KL∗, and (3.22)- (3.25) reduce to |c2| ≤ 5 4, |c3| ≤ 31 24, |c4| ≤ 673 192,
Theorem 3.7.
Theorem 3.7. Let λ ∈KL∗ n be of the form of (1.1). Then H3 (1) ≤ 3 4 · 3n 5 22n −9 8 + 4711 42n+2 · 288 + 1 5n+1
Theorem 3.7. Let λ ∈KL∗ n be of the form of (1.1). Then H3 (1) ≤ 3 4 · 3n 5 22n −9 8 + 4711 42n+2 · 288 + 1 5n+1
Function classes studied:
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