Ma-Minda φ-classes studied in this paper:
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 1.1.
Lemma 1.1. [11, 14] Let p ∈P, and be given by (1.3). Then 2c2 = c2 1 + δ(4 −c2 1), (1.4) 4c3 = c3 1 + 2(4 −c2 1)c1δ −(4 −c2 1)c1δ2 + 2(4…
Lemma 1.1. [11, 14] Let p ∈P, and be given by (1.3). Then 2c2 = c2 1 + δ(4 −c2 1), (1.4) 4c3 = c3 1 + 2(4 −c2 1)c1δ −(4 −c2 1)c1δ2 + 2(4 −c2 1)(1 −|δ|2)η, (1.5)
Lemma 1.2.
Lemma 1.2. [4]Let p ∈P, and be given by (1.3) with c1 ≥0. Then c1 = 2ζ1, (1.7) c2 = 2ζ2 1 + 2(1 −ζ2 1)ζ2, (1.8) c3 = 2ζ3 1 + 4(1 −ζ2
Lemma 1.2. [4]Let p ∈P, and be given by (1.3) with c1 ≥0. Then c1 = 2ζ1, (1.7) c2 = 2ζ2 1 + 2(1 −ζ2 1)ζ2, (1.8) c3 = 2ζ3 1 + 4(1 −ζ2
Lemma 1.3.
Lemma 1.3. [6]. 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
Lemma 1.3. [6]. 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
Theorem 2.1.
Theorem 2.1. Let f ∈S∗ car and be given by (1.1). Then |H2 (2) (f)| ≤4 9. The inequality is sharp for the function f1 (z) = ze z4 6 + 2z2 3…
Theorem 2.1. Let f ∈S∗ car and be given by (1.1). Then |H2 (2) (f)| ≤4 9. The inequality is sharp for the function f1 (z) = ze z4 6 + 2z2 3 = z + 2 3z3 + 7 18z5 + · · · .
Theorem 2.2.
Theorem 2.2. Let f ∈S∗ car and be given by (1.1). Then |H3 (1) (f)| ≤16 81. (2.8)
Theorem 2.2. Let f ∈S∗ car and be given by (1.1). Then |H3 (1) (f)| ≤16 81. (2.8)
Theorem 2.3.
Theorem 2.3. Let f ∈S∗ car and be given by (1.1). Then |a2a3 −a4| ≤64 81. The inequality is sharp for the function f3 defined by f3(z) = ze…
Theorem 2.3. Let f ∈S∗ car and be given by (1.1). Then |a2a3 −a4| ≤64 81. The inequality is sharp for the function f3 defined by f3(z) = ze 4z 3 + z2 3 = z + 4 3z2 + 11 9 z3 + 68 81z4 + · · · .
Function classes studied:
Related Papers