Ma-Minda φ-classes studied in this paper:
Abstract
The purpose of the present work is to determine the possible upper bound of third order Hankel determi-
nantforthefunctionsstarlikeandconvexwithrespecttosymmetricpointsassociatedwithexponentialfunctions.
Key–Words: Analytic function, Univalent function, Subordination, Fekete-Szeg¨o inequality, Hankel determinant,
Symmetricpoints.
Received: October 23, 2019. Revised: April 4, 2020. Accepted: April 17, 2020. Published: April 29, 2020.
1
Results & Lemmas (13)
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Lemma 3
Lemma 3 [34] If p ∈P, then |pn| ≤2, ∀n ∈N.
Lemma 3 [34] If p ∈P, then |pn| ≤2, ∀n ∈N.
Lemma 4
Lemma 4 [22] If p(z) = 1+p1z+p2z2+p3z3+· · ·, is such that Re(p(z)) > 0 in E, then for some x, z with |x| ≤1, |z| ≤1, we have 2p2 = p2 1 +…
Lemma 4 [22] If p(z) = 1+p1z+p2z2+p3z3+· · · , is such that Re(p(z)) > 0 in E, then for some x, z with |x| ≤1, |z| ≤1, we have 2p2 = p2 1 + x(4 −p2 1), for some x, |x| ≤1 (11) 4p3 = p3 1 + 2p1(4 −p2 1)x −p1(4 −p2 1)x2 + 2(4 −p2 1)(1 −|x|2)z (12)
Lemma 5
Lemma 5 [40] If p ∈P, then |p2 −νp2 1| ≤max| 1, |2ν −1| for any ν ∈C. 2 Mains Results
Lemma 5 [40] If p ∈P, then |p2 −νp2 1| ≤max|{1, |2ν −1|} for any ν ∈C. 2 Mains Results
Theorem 6
Theorem 6 If f ∈S∗ s(ez) then |a2| ≤1 2, |a3| ≤1 2, |a4| ≤19 48, |a5| ≤13 24.
Theorem 6 If f ∈S∗ s(ez) then |a2| ≤1 2, |a3| ≤1 2, |a4| ≤19 48, |a5| ≤13 24.
Theorem 7
Theorem 7 If f ∈S∗ s(ez) then |a3 −a2 2| ≤1 2.
Theorem 7 If f ∈S∗ s(ez) then |a3 −a2 2| ≤1 2.
Theorem 8
Theorem 8 If f ∈S∗ s(ez) then |a2a3 −a4| ≤765+59 √ 118 3468.
Theorem 8 If f ∈S∗ s(ez) then |a2a3 −a4| ≤765+59 √ 118 3468 .
Theorem 9
Theorem 9 If f ∈S∗ s(ez) then |a2a4 −a2 3| ≤3 8.
Theorem 9 If f ∈S∗ s(ez) then |a2a4 −a2 3| ≤3 8.
Theorem 10
Theorem 10 If f ∈S∗ s(ez) then |H3(1)| ≤90831+1121 √ 118 166464 = 0.618.
Theorem 10 If f ∈S∗ s(ez) then |H3(1)| ≤90831+1121 √ 118 166464 = 0.618.
Theorem 11
Theorem 11 If f ∈Cs(ez) then |a2| ≤1 4, |a3| ≤1 6, |a4| ≤19 192, |a5| ≤13 120.
Theorem 11 If f ∈Cs(ez) then |a2| ≤1 4, |a3| ≤1 6, |a4| ≤19 192, |a5| ≤13 120.
Theorem 12
Theorem 12 If f ∈Cs(ez) then |a3 −a2 2| ≤1 6.
Theorem 12 If f ∈Cs(ez) then |a3 −a2 2| ≤1 6.
Theorem 13
Theorem 13 If f ∈Cs(ez) then |a2a3 −a4| ≤157446−5575 √ 104 1168032 = 0.08612.
Theorem 13 If f ∈Cs(ez) then |a2a3 −a4| ≤157446−5575 √ 104 1168032 = 0.08612.
Theorem 14
Theorem 14 If f ∈Cs(ez) then |a2a4 −a2 3| ≤25 576.
Theorem 14 If f ∈Cs(ez) then |a2a4 −a2 3| ≤25 576.
Theorem 15
Theorem 15 If f ∈Cs(ez) then |H3(1)| ≤0.0338. References: [1] R.M. Ali, S. K. Lee, V. Ravichandran, S. Supra- maniam, The Fekete-Szeg¨o…
Theorem 15 If f ∈Cs(ez) then |H3(1)| ≤0.0338. References: [1] R.M. Ali, S. K. Lee, V. Ravichandran, S. Supra- maniam, The Fekete-Szeg¨o coefficient func- tional for transforms of analytic functions. Bull. Iranian Math. Soc. Vol.35, No. 2, 2009,pp. 119- 142. [2] R.M. Ali, V. Ravichandran, N. Seenivasagan, Coefficient bounds for p-valent functions. Appl. Math. Comput. Vol.187, No. 1,2007, pp. 35-46. [3] S. Altinkaya, S. Yalcin, Third Hankel determi- nant for Bazilevic functions, Advances in Math., 5
Function classes studied:
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