Ma-Minda φ-classes studied in this paper:
Abstract
For starlike functions associated with a cardioid domain, we provide sharp bounds for the Zalcman functional
a2a3 −a4 and the second and third Hankel determinants. Our result on the third Hankel determinant gives the negative
answer to a conjecture given in [L. Shi et al., A study of third Hankel determinant problem for certain subfamilies of
analytic functions involving cardioid domain, Mathematics, 7(5):418, 2019].
MSC: 30C45, 30C50
Results & Lemmas (6)
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 [12,15].) Let p ∈P be given by (1.3). Then 2c2 = c2 1 + δ 4 −c2 1 , (1.4) 4c3 = c3 1 + 2 4 −c2 1 c1δ −
Lemma 1. (See [12,15].) Let p ∈P be given by (1.3). Then 2c2 = c2 1 + δ 4 −c2 1 , (1.4) 4c3 = c3 1 + 2 4 −c2 1 c1δ −
Lemma 2.
Lemma 2. (See [5].) Let D:= z ∈C: |z| ⩽1, and for real numbers A, B, C, let Y (A, B, C):= max A + Bz + Cz2 + 1 −|z|2, z ∈D . If AC…
Lemma 2. (See [5].) Let D := {z ∈C: |z| ⩽1}, and for real numbers A, B, C, let Y (A, B, C) := max A + Bz + Cz2 + 1 −|z|2, z ∈D . If AC ⩾0, then Y (A, B, C) =
Theorem 1.
Theorem 1. Let f ∈S∗ car be given by (1.1). Then H2(2)(f) ⩽4 9. The inequality is sharp for the function f1(z) = zez4/6+2z2/3 = z + 2…
Theorem 1. Let f ∈S∗ car be given by (1.1). Then H2(2)(f) ⩽4 9. The inequality is sharp for the function f1(z) = zez4/6+2z2/3 = z + 2 3z3 + 7 18z5 + · · · .
Lemma 2
Lemma 2 gives H2(2)(f) ⩽2 27c 4 −c2 |A| + |B| + |C| = 19c4 −144c2 + 1728 3888:= g(c), and so max g(c), c ∈[0, 2] = g(0) = 4/9.…
Lemma 2 gives H2(2)(f) ⩽2 27c 4 −c2 |A| + |B| + |C| = 19c4 −144c2 + 1728 3888 := g(c), and so max{g(c), c ∈[0, 2]} = g(0) = 4/9. Using Lemma 2, we have the required result. To see that the bound |H2(2)(f)| ⩽4/9 is sharp, consider the function f1 : D →C defined as f1(z) = zez4/6+2z2/3 = z + 2 3z3 + 7
Theorem 2.
Theorem 2. Let f ∈S∗ car be given by (1.1). Then H3(1)(f) ⩽16 81. (2.7) The inequality is sharp for the function f2 defined by f2(z) =…
Theorem 2. Let f ∈S∗ car be given by (1.1). Then H3(1)(f) ⩽16 81. (2.7) The inequality is sharp for the function f2 defined by f2(z) = ze4z3/9+z6/9 = z + 4 9z4 + · · · . (2.8)
Theorem 3.
Theorem 3. Let f ∈S∗ car be given by (1.1). Then |a2a3 −a4| ⩽64 81. The inequality is sharp for the function f3 defined by f3(z) =…
Theorem 3. Let f ∈S∗ car be given by (1.1). Then |a2a3 −a4| ⩽64 81. The inequality is sharp for the function f3 defined by f3(z) = ze4z/3+z2/3 = z + 4 3z2 + 11 9 z3 + 68 81z4 + · · · .
Function classes studied:
Related Papers