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Ma-Minda φ-classes studied in this paper:
Results & Lemmas (8)
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Lemma 1
Lemma 1 Let the function p ∈P have the series form (8). Then, for x,δ,ρ ∈D = D ∪ 1, 2c2 = c2 1 + 4 – c2 1
Lemma 1 Let the function p ∈P have the series form (8). Then, for x,δ,ρ ∈D = D ∪{1}, 2c2 = c2 1 + 4 – c2 1
Lemma 3
Lemma 3 If the function p ∈P has the series form (8), then |cn+k – μcnck| ≦2max 1,|2μ – 1|
Lemma 3 If the function p ∈P has the series form (8), then |cn+k – μcnck| ≦2max 1,|2μ – 1|
Theorem 5
Theorem 5 Let the function f of the form (1) be in the class S∗ tanh. Then |a2| ≦1, |a3| ≦1 2, |a4| ≦1 3. Each of these bounds is sharp.
Theorem 5 Let the function f of the form (1) be in the class S∗ tanh. Then |a2| ≦1, |a3| ≦1 2, |a4| ≦1 3. Each of these bounds is sharp.
Theorem 6
Theorem 6 Let the function f of the form (1) be in the class S∗ tanh. Then a3 – λa2 2 ≦max
Theorem 6 Let the function f of the form (1) be in the class S∗ tanh. Then a3 – λa2 2 ≦max
Theorem 7
Theorem 7 Let the function f of the form (1) be in the class S∗ tanh. Then |a2a3 – a4| ≦1 3. This result is sharp.
Theorem 7 Let the function f of the form (1) be in the class S∗ tanh. Then |a2a3 – a4| ≦1 3. This result is sharp.
Theorem 8
Theorem 8 Let the function f of the form (1) be in the class S∗ tanh. Then HD2,2(f ) ≦1 4. This inequality is sharp.
Theorem 8 Let the function f of the form (1) be in the class S∗ tanh. Then HD2,2(f ) ≦1 4. This inequality is sharp.
Theorem 9
Theorem 9 Let the function f of the form (1) be in the class S∗ tanh. Then HD3,1(f ) ≦1 9. This result is sharp.
Theorem 9 Let the function f of the form (1) be in the class S∗ tanh. Then HD3,1(f ) ≦1 9. This result is sharp.
Theorem 9
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Theorem 9 has thus been proved as asserted. □ 5 Concluding remarks and observations In the present article, we have introduced and studied…
Theorem 9 has thus been proved as asserted. □ 5 Concluding remarks and observations In the present article, we have introduced and studied a new subfamily of starlike func- tions in the open unit disk D, which involves the hyperbolic function tanhz. For functions belonging to such a class of starlike functions, we have considered some interesting prob- lems such as the bounds of the first three Taylor–Maclaurin coefficients, the estimates of the Fekete–Szegö type functional, and the estimates of th
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