TITLE MARCH 2012 - Pakistan Academy of Sciences

30 Subordination Results for Certain Subclasses of Analytic Functions
and
2(β − 1)
+ 1 − λ + 3 − 2β −
λ(2β − 1) Γ 2(α 1 )
Re{f(z)} > −
[1 − λ + |3 − 2β − λ(2β − 1)|]Γ 2 (α 1 ) .
The constant factor
[1−λ+|3−2β−λ(2β−1)|]Γ 2 (α 1 )
2{2(β−1)+[1−λ+|3−2β−λ(2β−1)|]Γ 2 (α 1 )}
in the
(2.25)
subordination result (2.24) can not be replaced by
a larger one and the function
2(β − 1)
f 0 (z) = z −
[1 − λ + |3 − 2β − λ(2β − 1)|]Γ 2 (α 1 ) z2
gives the sharpness.
(2.26)
By taking b k = l+1+μ(k−1)
m (m ∈ N
l+1
0 , μ, l ≥
0) in Lemma 2 and Theorem 1, we obtain the
following corollary:
Corollary 6. Let the function f(z) defined by
(1.1) be in the class T ∗ (m, μ, l, λ, β) and satisfy
the condition
∞ (1 − λ)(k − 1)
l + 1 + μ(k − 1)
k − (2β − 1)
+
[1 + λ(k − 1)]
l + 1
k=2
Then for every function h ∈ K, we have:
m
|a k | ≤ 2(β − 1).
(2.27)
[1 − λ + |3 − 2β − λ(2β − 1)|](l + 1 + μ) m
(f ∗ h)(z) ≺ h(z)
2(l + 1) m (β − 1)
2
+ 1 − λ + 3 − 2β −
λ(2β − 1) (l + 1 + μ)m
and
(2.28)
2(l + 1) m (β − 1)
3 − 2β
+ 1 − λ +
(l + 1 + μ)m
−λ(2β − 1)
Re{f(z)} > −
[1 − λ + |3 − 2β − λ(2β − 1)|](l + 1 + μ) m.
(2.29)
The constant factor
[1−λ+|3−2β−λ(2β−1)|](l+1+μ) m
2{2(l+1) m (β−1)+[1−λ+|3−2β−λ(2β−1)|](l+1+μ) m }
in the subordination result (2.28) can not be
replaced by a larger one and the function
2(β − 1)(l + 1) m
f 0 (z) = z −
[1 − λ + |3 − 2β − λ(2β − 1)|](l + 1 + μ) m z2
(2.30)
gives the sharpness.
By taking b k = C k (b, μ), where C k (b, μ)
defined by (1.10), in Lemma 2 and Theorem 1, we
obtain the following corollary:
Corollary 7. Let the function f(z) defined by
(1.1) be in the class T ∗ (μ, b, λ, β) and satisfy the
condition
∞ (1 − λ)(k − 1)
k − (2β − 1)
+
[1 + λ(k − 1)] C k (b, μ)|a k | ≤ 2(β − 1).
k=2
Then for every function h ∈ K, we have:
(2.31)
[1 − λ + |3 − 2β − λ(2β − 1)|]C 2 (b, μ)
(f ∗ h)(z) ≺ h(z)
2(β − 1)
2
3 − 2β
+ 1 − λ +
−λ(2β − 1) C 2 (b, μ)
and
(2.32)
2(β − 1)
3 − 2β
+ 1 − λ +
−λ(2β − 1) C 2(b, μ)
Re{f(z)} > −
1 − λ + 3 − 2β −
.
λ(2β − 1) C 2(b, μ)
The constant factor
[1−λ+|3−2β−λ(2β−1)|]C 2 (b,μ)
2{2(β−1)+[1−λ+|3−2β−λ(2β−1)|]C 2 (b,μ)}
in the
(2.33)
subordination result (2.32) can not be replaced by
a larger one and the function
2(β − 1)
f 0 (z) = z −
[1 − λ + |3 − 2β − λ(2β − 1)|]C 2 (b, μ) z2
gives the sharpness.
3. ACKNOWLEDGEMENTS
(2.34)
The authors thank the anonymous referees of the paper
for their helpful suggestions.
4. REFERENCES
1. Aouf, M.K., A.A.. Shamandy, A.O. Mostafa &
E.A. Adwan. Subordination results for certain
class of analytic functions defined by convolution.