Abstract
In the present paper we introduce a new class of analytic functions f
in the open unit disk normalized by f(0) = f ′(0)−1 = 0, associated with
exponential functions. The aim of the present paper is to investigate the
third-order Hankel determinant H3(1) for this function class and obtain
the upper bound of the determinant H3(1).
1
Results & Lemmas (7)
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Lemma 1.
Lemma 1. ( [5]) If p ∈P and has the form (5) then |cn| ≤2, n = 1, 2,... (11) and the inequality is sharp.
Lemma 1. ( [5]) If p ∈P and has the form (5) then |cn| ≤2, n = 1, 2, . . . (11) and the inequality is sharp.
Lemma 2.
Lemma 2. ([21], [9]) If p ∈P and has the form (5) then |cn+k −µcnck| < 2 for 0 ≤µ ≤1; (12) |cmcn −ckcl| ≤ 4 for m + n = k + l; (13) cn+2k…
Lemma 2. ([21], [9]) If p ∈P and has the form (5) then |cn+k −µcnck| < 2 for 0 ≤µ ≤1; (12) |cmcn −ckcl| ≤ 4 for m + n = k + l; (13) cn+2k −µcnc2 k
Lemma 3.
Lemma 3. ([10], [11]) If the function p ∈P is given by (5), then exists some x, z with |x| ≤1, |z| ≤1 such that 2c2 = c2 1 + x 4 −c2 1 ;…
Lemma 3. ([10], [11]) If the function p ∈P is given by (5) , then exists some x, z with |x| ≤1, |z| ≤1 such that 2c2 = c2 1 + x 4 −c2 1 ; (17) 4c3 = c3 1 + 2c1x 4 −c2 1 −
Theorem 1.
Theorem 1. If the function f ∈SC∗ α, where f is given by f(z) = z + P∞ n=2 anzn, z ∈C then we have a3 −a2 2 ≤ 1 2 (1 + 2α). (19)
Theorem 1. If the function f ∈SC∗ α, where f is given by f(z) = z + P∞ n=2 anzn, z ∈C then we have a3 −a2 2 ≤ 1 2 (1 + 2α). (19)
Theorem 2.
Theorem 2. If the function f ∈SC∗ α, where f is given by f(z) = z + P∞ n=2 anzn, z ∈C then we have |a2a3 −a4| ≤ 4 −ec2 ec 24 + 3 4 −ec2…
Theorem 2. If the function f ∈SC∗ α, where f is given by f(z) = z + P∞ n=2 anzn, z ∈C then we have |a2a3 −a4| ≤ 4 −ec2 ec 24 + 3 4 −ec2 ec 8 + 4 −ec2 12 + ec3ε(ρ)
Theorem 3.
Theorem 3. If the function f ∈SC∗ α, where f is given by f(z) = z + P∞ n=2 anzn, z ∈C then we have a2a4 −a2 3 ≤2. (28)
Theorem 3. If the function f ∈SC∗ α, where f is given by f(z) = z + P∞ n=2 anzn, z ∈C then we have a2a4 −a2 3 ≤2. (28)
Theorem 4.
Theorem 4. If the function f ∈SC∗ α, then we have |H3 (1)| ≤18, 001. (29)
Theorem 4. If the function f ∈SC∗ α, then we have |H3 (1)| ≤18, 001. (29)
Function classes studied:
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