Abstract
In the paper we consider a general inequality |pn−1pn+1−pn
2| ≤
4 −|p1|2 involving coefficients of functions with a positive real part. We
prove this inequality for n = 2 and n = 3. Consequently, the relative
inequalities involving coefficients of Schwarz functions are obtained. As
an application, the two sharp estimates of the Hankel determinants H3,1
and H2,3 are proved for functions in S∗(1/2) and M, respectively.
Mathematics Subject Classification. 30C50, 30C45.
Results & Lemmas (8)
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Lemma 1.
Lemma 1. If p ∈P is of the form (1) with p1 ≥0, then 2p2 = p2 1 + (4 −p2 1)x, (3) 4p3 = p3 1 + (4 −p2 1)p1x(2 −x) + 2(4 −p2 1)(1 −|x|2)y…
Lemma 1. If p ∈P is of the form (1) with p1 ≥0, then 2p2 = p2 1 + (4 −p2 1)x, (3) 4p3 = p3 1 + (4 −p2 1)p1x(2 −x) + 2(4 −p2 1)(1 −|x|2)y (4) and 8p4 = p4 1 + (4 −p2 1)x
Theorem 2.
Theorem 2. If h ∈P is given by (1), then |p1p3 −p2 2| ≤4 −|p1|2. (6) Equality holds for rotations ht(εz) and gt(εz), |ε| = 1 of ht(z) = 1…
Theorem 2. If h ∈P is given by (1), then |p1p3 −p2 2| ≤4 −|p1|2 . (6) Equality holds for rotations ht(εz) and gt(εz), |ε| = 1 of ht(z) = 1 −z2 1 −2tz + z2 , t ∈[−1, 1] (7) and gt(z) = 1−t 2 h−1(z) + 1+t 2 h1(z) = 1 + 2tz + z2 1 −z2 , t ∈[−1, 1] .
Theorem 3.
Theorem 3. If h ∈P is given by (1), then |p2p4 −p3 2| ≤4 −|p1|2. (12) Equality holds for the same functions as in Theorem 2.
Theorem 3. If h ∈P is given by (1), then |p2p4 −p3 2| ≤4 −|p1|2 . (12) Equality holds for the same functions as in Theorem 2.
Theorem 4.
Theorem 4. If ω ∈B0 is given by (14), then c2c4 −c2 3 ≤(1 −|c1|2)2. (16) Equality holds for rotations ε−1ω(εz), |ε| = 1 of ω(z) = z(t…
Theorem 4. If ω ∈B0 is given by (14), then c2c4 −c2 3 ≤(1 −|c1|2)2 . (16) Equality holds for rotations ε−1ω(εz), |ε| = 1 of ω(z) = z(t + z2) 1 + tz2 = tz + (1 −t2)z3 −t(1 −t2)z5 + . . . t ∈[−1, 1]. Directly from Theorem 3 and (15) we get the following theorem.
Theorem 5.
Theorem 5. If ω ∈B0 is given by (14), then |c2c4 −c3 2 + c1 2c4 −2c1c2c3 + c2 3| ≤1 −|c1|2. (17) Equality holds for rotations ε−1ω(εz), |ε|…
Theorem 5. If ω ∈B0 is given by (14), then |c2c4 −c3 2 + c1 2c4 −2c1c2c3 + c2 3| ≤1 −|c1|2 . (17) Equality holds for rotations ε−1ω(εz), |ε| = 1 of ω(z) = (t + z)z 1 + tz = tz −(1−t2)z2 −t(1−t2)z3 −t2(1−t2)z4 +. . . , t ∈[−1, 1] . (18) Finally, we can combine the inequalities from Theorem 5 and Theorem 4 to obtain more general inequality.
Theorem 6.
Theorem 6. If ω ∈B0 is given by (14), then for all α ∈[0, 1] |c2c4 −c3 2 + α c1 2c4 −2c1c2c3 + c2 3 | ≤(1 −|c1|2)(1 −(1 −α)|c1|2). (19)
Theorem 6. If ω ∈B0 is given by (14), then for all α ∈[0, 1] |c2c4 −c3 2 + α c1 2c4 −2c1c2c3 + c2 3 | ≤(1 −|c1|2)(1 −(1 −α)|c1|2) . (19)
Theorem 7.
Theorem 7. If f ∈S∗(1/2) is given by (20), then |H2,3| ≤3 16. (23) Equality holds for f(z) = z √ 1 −z2 = z + 1 2z3 + 3 8z5 +....
Theorem 7. If f ∈S∗(1/2) is given by (20), then |H2,3| ≤3 16 . (23) Equality holds for f(z) = z √ 1 −z2 = z + 1 2z3 + 3 8z5 + . . . .
Theorem 8.
Theorem 8. If f ∈M is given by (20), then |H3,1| ≤3 16. (26) Equality holds for f(z) = z √ 1 −z2 = z + 1 2z3 + 3 8z5 +....
Theorem 8. If f ∈M is given by (20), then |H3,1| ≤3 16 . (26) Equality holds for f(z) = z √ 1 −z2 = z + 1 2z3 + 3 8z5 + . . . .
Function classes studied:
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