Paper-4-20-2009. 129KB Jun 04 2011 12:03:14 AM
ACTA UNIVERSITATIS APULENSIS
No 20/2009
INTEGRAL MEANS FOR CERTAIN SUBCLASSES OF
UNIFORMLY STARLIKE AND CONVEX FUNCTIONS DEFINED
BY CONVOLUTION
M. K. Aouf, R. M. El-Ashwah and S. M. El-Deeb
Abstract. We introduce some generalized subclasses T Sγ (f, g; α, β) of uniformly starlike and convex functions, we settle the Silverman’s conjecture for the
integral means inequality. In particular, we obtain integral means inequalities for
various classes of uniformly β−starlike and uniformly β−convex functions in the
unit disc.
2000 Mathematics Subject Classification: 30C45.
1.Introduction
Let S denote the class of functions of the form:
f (z) = z +
∞
X
ak z k
(1.1)
k=2
that are analytic and univalent in the open unit disk U = {z : |z| < 1} . Let f (z) ∈
S be given by (1.1) and Φ(z) ∈ S be given by
Φ(z) = z +
∞
X
ck z k ,
(1.2)
k=2
then for analytic functions f and Φ with f (0) = Φ(0), f is said to be subordinate
to Φ, denoted by f ≺ Φ, if there exists an analytic function w such that w(0) = 0,
|w(z)| < 1 and f (z) = Φ(w(z)), for all z ∈ U.
The Hadamard product (or convolution) f ∗ Φ of f and Φ is defined (as usual) by
(f ∗ Φ)(z) = z +
∞
X
ak ck z k = (Φ ∗ f )(z).
k=2
31
(1.3)
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
Following Goodman ([7] and [8]), Ronning ([17] and [18]) introduced and studied
the following subclasses:
(i) A function f (z) of the form (1.1) is said to be in the class Sp (α, β) of uniformly
β−starlike functions if it satisfies the condition:
′
( ′
)
zf (z)
zf (z)
Re
−α >β
− 1
(z ∈ U ),
(1.4)
f (z)
f (z)
where −1 ≤ α < 1 and β ≥ 0.
(ii) A function f (z) of the form (1.1) is said to be in the class U CV (α, β) of
uniformly β−convex functions if it satisfies the condition:
′′
)
(
′′
zf (z)
zf (z)
(z ∈ U ),
(1.5)
−α >β ′
Re 1 + ′
f (z)
f (z)
where −1 ≤ α < 1 and β ≥ 0. We also observe that
Sp (α, 0) = T ∗ (α),
U CV (α, 0) = C(α)
are, respectively, well-known subclasses of starlike functions of order α and convex
functions of order α. Indeed it follows from (1.4) and (1.5) that
′
f (z) ∈ U CV (α, β) ⇐⇒ zf (z) ∈ Sp (α, β).
(1.6)
For −1 ≤ α < 1, 0 ≤ γ ≤ 1 and β ≥ 0, we let Sγ (f, g; α, β) be the subclass of S
consisting of functions f (z) of the form (1.1) and functions g(z) given by
g(z) = z +
∞
X
bk z k
(bk ≥ 0),
(1.7)
k=2
and satisfying the analytic criterion:
)
(
′
′′
z(f ∗ g) (z) + γz 2 (f ∗ g) (z)
−α
Re
(1 − γ) (f ∗ g)(z) + γz(f ∗ g)′ (z)
z(f ∗ g)′ (z) + γz 2 (f ∗ g)′′ (z)
>β
− 1 .
′
(1 − γ) (f ∗ g)(z) + γz(f ∗ g) (z)
(1.8)
Let T denote the subclass of S consisting of functions of the form:
f (z) = z −
∞
X
ak z k
k=2
32
(ak ≥ 0) .
(1.9)
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
Further, we define the class T Sγ (f, g; α, β) by
T Sγ (f, g; α, β) = Sγ (f, g; α, β) ∩ T.
(1.10)
We note that:
z
z
; α, 1) = Sp T (α) and T S0 (f, (1−z)
(i) T S0 (f, (1−z)
2 ; α, 1) =
z
; α, 1) = U CT (α) (−1 ≤ α < 1) (see Bharati et al. [4]);
T S1 (f, (1−z)
z
(ii) T S1 (f, (1−z)
; 0, β) = U CT (β) (β ≥ 0) (see Subramanian et al. [24]);
∞ (a)
P
k−1 k
z ; α, β) = T S (α, β) (−1 ≤ α < 1, β ≥ 0, c 6=
(iii) T S0 (f, z +
(c)
k−1
k=2
0, −1, −2, ...) (see Murugusundaramoorthy and Magesh [12] and [13]);
∞
P
k n z k ; α, β) = T S (n, α, β) (−1 ≤ α < 1, β ≥ 0, n ∈ N0 =
(iv) T S0 (f, z +
k=2
N ∪ {0}, N = {1, 2, ...})
(see Rosy
and Murugusundaramoorthy [19]);
∞
P
k+λ−1 k
z ; α, β) = D (α, β, λ) (−1 ≤ α < 1, β ≥ 0,
(v) T S0 (f, z +
λ
k=2
λ > −1) (see Shams et al. [23]);
(vi) T S0 (f, z +
∞
P
[1 + λ(k − 1)]n z k ; α, β) = T Sλ (n, α, β) (−1 ≤ α < 1,
k=2
β ≥ 0, λ ≥ 0, n ∈ N0 ) (see Aouf and Mostafa [2]);
∞ (a)
P
k−1 k
(vii) T Sγ (f, z +
z ; α, β) = T S (γ, α, β) (−1 ≤ α < 1, β ≥ 0,
(c)
k=2
k−1
0 ≤ γ ≤ 1, c 6= 0, −1, −2, ...) (see Murugusundaramoorthy et al. [14]);
(viii) T S0 (f, g; α, β) = HT (g, α, β) (−1 ≤ α < 1, β ≥ 0) (see Raina and Bansal
[16]);
∞
P
Γk z k ; α, β) = T Sqs (γ, α, β) (see Ahuja et al. [1]), where
(viii) T Sγ (f, z +
k=2
Γk =
(α1 )k−1 ...(αq )k−1
1
(β1 )k−1... (βs )k−1 (k − 1)!
(1.11)
(αi > 0, i = 1, ..., q; βj > 0, j = 1, ..., s; q ≤ s + 1; q, s ∈ N0 ) .
Also we note that
∞
P
k n z k ; α, β) = T Sγ (n, α, β)
(i) T Sγ (f, z +
k=2
=
(
f ∈ T : Re
(
′
′
(1 − γ)z(Dn f (z)) + γz(Dn+1 f (z))
−α
(1 − γ)Dn f (z) + γDn+1 f (z)
33
)
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
)
(1 − γ)z(Dn f (z))′ + γz(Dn+1 f (z))′
> β
− 1 , (−1 ≤ α < 1, β ≥ 0, n ∈ N0 , z ∈ U )
(1 − γ)Dn f (z) + γDn+1 f (z)
(1.12)
∞
P
c+1
k
(ii) T Sγ (f, z +
c+k z ; α, β) =
k=2
= T Sγ (c, α, β) =
(
f ∈ T : Re
(
′
′′
z(Jc f (z)) + γz 2 (Jc f (z))
−α
(1 − γ)Jc f (z) + γz(Jc f (z))′
)
)
z(J f (z))′ + γz 2 (J f (z))′′
c
c
>β
− 1 , 0 ≤ γ ≤ 1, −1 ≤ α < 1, β ≥ 0, c > −1, z ∈ U ;
(1 − γ)Jc f (z) + γz(Jc f (z))′
(1.13)
where Jc is a Bernardi operator [3], defined by
c+1
Jc f (z) =
zc
Zz
t
c−1
f (t)dt = z +
∞
X
c+1
k=2
0
c+k
ak z k .
Note that the operator J1 f (z) was studied earlier by Libera [9] and Livingston
[11];
∞
P
(µ)k−1
k
(iii) T Sγ (f, z +
(λ+1)k−1 z ; α, β) = T Sγ (µ, λ; α, β)
k=2
′
′′
z(Iλ,µ f (z)) +γz 2 (Iλ,µ f (z))
′
(1−γ)Iλ,µ f (z)+γz(Iλ,µ f (z))
f ∈ T : Re
−α
z(I f (z))′ + γz 2 (I f (z))′′
λ,µ
λ,µ
> β
′ − 1 ,
(1 − γ)Iλ,µ f (z) + γz(Iλ,µ f (z))
=
(0 ≤ γ ≤ 1, − 1 ≤ α < 1, β ≥ 0, λ > −1, µ > 0, z ∈ U )
where Iλ,µ is a Choi-Saigo-Srivastava operator [6], defined by
Iλ,µ f (z) = z +
∞
X
k=2
(µ)k−1
ak z k (λ > −1; µ > 0);
(λ + 1)k−1
34
;
(1.14)
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
(iv) T Sγ (f, z +
∞
P
k=2
(c)k−1 (λ+1)k−1 k
(a)k−1 (1)k−1 z ; α, β)
=
f ∈ T : Re
= T Sγ (a, c, λ; α, β)
′
′′
z(I λ (a,c)f (z)) +γz 2 (I λ (a,c)f (z))
(1−γ)I λ (a,c)f (z)+γz(I λ (a,c)f (z))′
−α
z(I λ (a, c)f (z))′ + γz 2 (I λ (a, c)f (z))′′
−
1
> β
,
′
λ
λ
(1 − γ)I (a, c)f (z) + γz(I (a, c)f (z))
0 ≤ γ ≤ 1, − 1 ≤ α < 1, β ≥ 0, a, c ∈ R\Z0− , λ > −1, z ∈ U
;
(1.15)
where I λ (a, c) is a Cho-Kwon-Srivastava operator [5], defined by
I λ (a, c)f (z) = z +
∞
X
(c)k−1 (λ + 1)k−1
ak z k ;
(a)k−1 (1)k−1
k=2
(v) T Sγ (f, z +
∞
P
k=2
=
(2)k−1
k
(n+1)k−1 z ; α, β)
(
f ∈ T : Re
(
= T Sγ (n; α, β)
′
′′
z(In f (z)) + γz 2 (In f (z))
−α
(1 − γ)In f (z) + γz(In f (z))′
)
)
z(I f (z))′ + γz 2 (I f (z))′′
n
n
>β
− 1 , 0 ≤ γ ≤ 1, −1 ≤ α < 1, β ≥ 0, n > −1, z ∈ U ;
(1 − γ)In f (z) + γz(In f (z))′
(1.16)
where In is a Noor integral operator [15], defined by
In f (z) = z +
∞
X
k=2
(2)k−1
ak z k (n > −1).
(n + 1)k−1
2
In [20], Silverman found that the function f2 (z) = z − z2 is often extremal
over the family T . He applied this function to resolve his integral means inequality,
conjectured in [21] and settled in [22], that
Z2π
Z2π
η
η
iθ
iθ
f
(re
)
dθ
≤
f
(re
)
dθ,
2
0
0
35
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
for all f ∈ T , η > 0 and 0 < r < 1. In [22], he also proved his conjecture for the
subclasses T ∗ (α) and C(α) of T.
In this paper, we prove Silverman’s conjecture for the functions in the class
T Sγ (f, g; α, β). By taking appropriate choises of the function g, we obtain the integral means inequalities for several known as well as new subclasses of uniformly
convex and uniformly starlike functions in U. In fact, these results also settle the
Silverman’s conjecture for several other subclasses of T.
2.Lemmas and their proofs
To prove our main results, we need the following lemmas.
Lemma 1. A function f (z) of the form (1.1) is in the class T Sγ (f, g; α, β) if
∞
X
[k(1 + β) − (α + β)] [1 + γ(k − 1)] |ak | bk ≤ 1 − α,
(2.1)
k=2
where −1 ≤ α < 1, β ≥ 0 and 0 ≤ γ ≤ 1.
Proof. It suffices to show that
(
)
′
′′
z(f ∗ g)′ (z) + γz 2 (f ∗ g)′′ (z)
z(f ∗ g) (z) + γz 2 (f ∗ g) (z)
β
− 1 −Re
−1
(1 − γ)(f ∗ g)(z) + γz(f ∗ g)′ (z)
(1 − γ)(f ∗ g)(z) + γz(f ∗ g)′ (z)
≤ 1 − α.
We have
(
)
′
′′
z(f ∗ g)′ (z) + γz 2 (f ∗ g)′′ (z)
z(f ∗ g) (z) + γz 2 (f ∗ g) (z)
− 1 −Re
−1
β
(1 − γ)(f ∗ g)(z) + γz(f ∗ g)′ (z)
(1 − γ)(f ∗ g)(z) + γz(f ∗ g)′ (z)
∞
(1 + β) P (k − 1) [1 + γ(k − 1)] |ak | bk
z(f ∗ g)′ (z) + γz 2 (f ∗ g)′′ (z)
k=2
.
≤ (1+β)
− 1 ≤
∞
P
(1 − γ)(f ∗ g)(z) + γz(f ∗ g)′ (z)
[1 + γ(k − 1)] |ak | bk
1−
k=2
This last expression is bounded above by (1 − α) if
∞
X
[k(1 + β) − (α + β)] [1 + γ(k − 1)] |ak | bk ≤ 1 − α,
k=2
36
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
and hence the proof is completed.
Lemma 2. A necessary and sufficient condition for f (z) of the form (1.9) to be in
the class T Sγ (f, g; α, β) is that
∞
X
[k(1 + β) − (α + β)] [1 + γ(k − 1)] ak bk ≤ 1 − α,
(2.2)
k=2
Proof. In view of Lemma 1, we need only to prove the necessity. If f (z) ∈ T Sγ (f, g; α, β)
and z is real, then
1−
∞
P
k [1 + γ(k − 1)] ak bk
k=2
∞
P
1−
z k−1
[1 + γ(k − 1)] ak bk z k−1
k=2
P
∞
k−1
(k − 1) [1 + γ(k − 1)] ak bk z
k=2
.
−α≥β
∞
1 − P [1 + γ(k − 1)] ak bk z k−1
k=2
Letting z → 1− along the real axis, we obtain the desired inequality
∞
X
[k(1 + β) − (α + β)] [1 + γ(k − 1)] ak bk ≤ 1 − α.
k=2
Corollary 1.Let the function f (z) be defined by (1.9) be in the class T Sγ (f, g; α, β).
Then
ak ≤
1−α
[k(1 + β) − (α + β)] [1 + γ(k − 1)] bk
(k ≥ 2).
(2.3)
The result is sharp for the function
f (z) = z −
1−α
zk
[k(1 + β) − (α + β)] [1 + γ(k − 1)] bk
(k ≥ 2).
(2.4)
Lemma 3. The extreme points of T Sγ (f, g; α, β) are
1−α
z k , for k = 2, 3, ... .
[k(1 + β) − (α + β)] [1 + γ(k − 1)] bk
(2.5)
The proof of Lemma 3 is similar to the proof of the theorem on extreme points given
in [20] and therefore are omit it.
In 1925, Littlewood [10] proved the following subordination theorem.
Lemma 4. If the functions f and g are analytic in U with g ≺ f, then for η > 0,
and 0 < r < 1,
f1 (z) = z and fk (z) = z−
37
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
Z2π
Z2π
η
η
iθ
iθ
g(re ) dθ ≤ f (re ) dθ.
0
(2.6)
0
3.Main theorem
Applying Lemma 4, Lemma 2 and Lemma 3, we prove the following result.
Theorem 1. Suppose f ∈ T Sγ (f, g; α, β), η > 0, −1 ≤ α < 1, 0 ≤ γ ≤ 1, β ≥ 0
and f2 (z) is defined by
f2 (z) = z −
1−α
z2.
(2 + β − α)(1 + γ)b2
Then for z = reiθ , 0 < r < 1, we have
Z2π
η
|f (z)| dθ ≤
Proof. For f (z) = z −
|f2 (z)|η dθ.
(3.1)
0
0
∞
P
Z2π
ak z k (ak ≥ 0), (3.1) is equivalent to proving that
k=2
η
η
Z2π
Z2π
∞
X
1−α
k−1
ak z
z dθ.
1 −
dθ ≤ 1 −
(2 + β − α)(1 + γ)b2
0
k=2
0
By Lemma 4, it suffices to show that
1−
∞
X
ak z k−1 ≺ 1 −
k=2
Setting
1−
∞
X
ak z k−1 = 1 −
k=2
1−α
z.
(2 + β − α)(1 + γ)b2
1−α
w(z),
(2 + β − α)(1 + γ)b2
(3.2)
and using (2.2), we obtain
∞
∞
X (2 + β − α)(1 + γ)b
X
(2 + β − α)(1 + γ)b2
2
k−1
|w(z)| =
ak z
ak ≤ |z| .
≤ |z|
1−α
1−α
k=2
k=2
This completes the proof of Theorem 1.
By taking different choices of g(z), α, β and γ in the above theorem, we can state
38
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
the following integral means results for various subclasses.
∞
P
Γk z k , where Γk is given by (1.11) in Theorem
Remarks. (i) Taking g(z) = z +
k=2
1, we obtain the result obtained by Ahuja et al. [1, Theorem 3.1];
z
(ii) Taking g(z) = 1−z
and γ = 0 in Theorem 1, we obtain the result obtained
by Ahuja et al. [1, Corollary 3.2];
z
(iii) Taking g(z) = 1−z
and γ = 1 in Theorem 1, we obtain the result obtained
by Ahuja et al. [1, Corollary 3.4];
∞
P
k+λ−1 k
(iv) Taking g(z) = z +
z (λ > −1) and γ = 0 in Theorem 1, we
λ
k=2
obtain the result obtained by Ahuja et al. [1, Corollary 3.6];
∞
P
c+1
k
(v) Taking g(z) = z +
c+k z (c > −1) and γ = 0 in Theorem 1, we obtain
k=2
the result obtained by Ahuja et al. [1, Corollary 3.7];
∞ (a)
P
k−1 k
z (a > 0; c > 0) and γ = 0 in Theorem 1, we
(vi) Taking g(z) = z +
(c)
k=2
k−1
obtain the result obtained by Ahuja et al. [1, Corollary 3.8].
∞
P
k n z k ; α, β) = T S (n, α, β) (−1 ≤ α < 1, β ≥ 0,
Corollary 2. If f ∈ T S0 (f, z +
k=2
n ∈ N0 and η > 0), then the assertion (3.1) holds true, where
f2 (z) = z −
Corollary 3. If f ∈ T S0 (f, z +
∞
P
(1 − α)
z2.
2n (2 + β − α)
[1 + λ(k − 1)]n z k ; α, β) = T Sλ (n, α, β) (−1 ≤
k=2
α < 1, β ≥ 0, λ ≥ 0, n ∈ N0 and η > 0), then the assertion (3.1) holds true, where
(1 − α)
z2.
(2 + β − α) (1 + λ)n
∞ (a)
P
k−1 k
z ; α, β) = T S (γ, α, β) (0 ≤ γ ≤ 1,
Corollary 4. If f ∈ T Sγ (f, z +
(c)
f2 (z) = z −
k=2
k−1
−1 ≤ α < 1, β ≥ 0, a > 0, c > 0 and η > 0), then the assertion (3.1) holds true,
where
f2 (z) = z −
c (1 − α)
z2.
a(2 + β − α)(1 + γ)
Corollary 5. If f ∈ T S0 (f, g(z); α, β) = HT (g, α, β) (−1 ≤ α < 1, β ≥ 0 and
η > 0), then the assertion (3.1) holds true, where
f2 (z) = z −
(1 − α)
z2.
(2 + β − α)b2
39
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
Corollary 6. If f ∈ T Sγ (f, z +
∞
P
k n z k ; α, β) = T Sγ (n, α, β) (0 ≤ γ ≤ 1, −1 ≤
k=2
α < 1, β ≥ 0, n ∈ N0 and η > 0), then the assertion (3.1) holds true, where
(1 − α)
z2.
+ β − α)(1 + γ)
∞
P
c+1
k
Corollary 7. If f ∈ T Sγ (f, z +
c+k z ; α, β) = T Sγ (c, α, β) (0 ≤ γ ≤ 1,
f2 (z) = z −
2n (2
k=2
−1 ≤ α < 1, β ≥ 0, c > −1 and η > 0), then the assertion (3.1) holds true, where
f2 (z) = z −
Corollary 8. If f ∈ T Sγ (f, z +
(1 − α) (c + 2)
z2.
(2 + β − α)(1 + γ)(c + 1)
∞
P
k=2
(µ)k−1
k
(λ+1)k−1 z ; α, β)
= T Sγ (µ, λ; α, β) (0 ≤ γ ≤ 1,
−1 ≤ α < 1, β ≥ 0, λ > −1, µ > 0 and η > 0), then the assertion (3.1) holds true,
where
f2 (z) = z −
Corollary 9.
If f ∈ T Sγ (f, z +
(1 − α) (λ + 1)
z2.
(2 + β − α)(1 + γ)µ
∞
P
k=2
(c)k−1 (λ+1)k−1 k
(a)k−1 (1)k−1 z ; α, β)
= T Sγ (a, c, λ; α, β)
(0 ≤ γ ≤ 1, −1 ≤ α < 1, β ≥ 0, a, c ∈ R\Z0− , λ > −1 and η > 0), then the assertion
(3.1) holds true, where
f2 (z) = z −
a (1 − α)
z2.
c(2 + β − α)(1 + γ)(λ + 1)
Corollary 10. If f ∈ T Sγ (f, z +
∞
P
k=2
(2)k−1
k
(n+1)k−1 z ; α, β)
= T Sγ (n, α, β) (0 ≤ γ ≤ 1,
−1 ≤ α < 1, β ≥ 0, n > −1 and η > 0), then the assertion (3.1) holds true, where
f2 (z) = z −
(1 − α) (n + 1)
z2.
2(2 + β − α)(1 + γ)
References
[1] O. Ahuja, G. Murugusundaramoorthy and N. Magesh, Integral means for
uniformly convex and starlike functions associated with generalized hypergeometric
functions, J. Inequal. Pure Appl. Math. 8(2007), no. 4, Art. 118, 1-9.
40
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
[2] M. K. Aouf and A. O. Mostafa, Some properties of a subclass of uniformly
convex functions with negative coefficients, Demonstratio Math., 2 (2008), 353-370.
[3] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math.
Soc., 135(1969), 429-446.
[4] R. Bharati, R. Parvatham and A. Swaminathan, On subclasses of uniformly
convex functions and corresponding class of starlike functions, Tamakang J. Math.
28(1997), 17-32.
[5] N. E. Cho, O. S. Kwon and H. M. Srivastava, Inclusion relationships and
argument properties for certain subclasses of multivalent functions associated with a
family of linear operators, J. Math. Anal. Appl. 292(2004), 470-483.
[6] J. H. Choi, M. Saigo, H. M. Srivastava, Some inclusion properties of a certain
family of integral operators, J. Math. Anal. Appl. 276(2002), 432-445.
7] A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56(1991),
87-92.
[8] A. W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl.
155(1991), 364-370.
[9] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math.
Soc. 16(1965), 755-758.
[10] J. E. Littlewood, On inequalities in theory of functions, Proc. London Math.
Soc. 23(1925), 481-519.
[11] A. E. Livingston, On the radius of univalence of certain analytic functions,
Proc. Amer. Math. Soc. 17(1966), 352-357.
[12] G. Murugusundaramoorthy and N. Magesh, A new subclass of uniformly
convex functions and corresponding subclass of starlike functions with fixed second
coefficient, J. Inequal. Pure Appl. Math. 5(2004), no. 4, Art. 85, 1-10.
[13] G. Murugusundaramoorthy and N. Magesh, Linear operators associated with
a subclass of uniformly convex functions, Internat. J. Pure Appl. Math. Sci.
3(2006), no. 2, 113-125.
[14] G. Murugusundaramoorthy, T. Rosy and K. Muthunagai, Carlson-Shaffer
operator and their applications to certain subclass of uniformly convex function,
General Math. 15(2007), no. 4, 131-143.
[15] K. I. Noor, On new classes of integral operators, J. Natur. Geem. 16(1999),
71-80.
[16] R. K. Raina and D. Bansal, Some properties of a new class of analytic
functions defined in terms of a Hadamard product, J. Inequal. Pure Appl. Math.
9(2008), no. 1, Art. 22, 1-9.
[17] F. Ronning, On starlike functions associated with parabolic regions, Ann.
Univ. Mariae-Curie-Sklodowska, Sect. A 45(1991), 117-122.
[18] F. Ronning, Uniformly convex functions and a corresponding class of starlike
41
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
functions, Proc. Amer. Math. Soc. 118(1993), 189-196.
[19] T. Rosy and G. Murugusundaramoorthy, Fractional calculus and their applications to certain subclass of uniformly convex functions, Far East J. Math. Sci.
(FJMS), 115(2004), no. 2, 231-242.
[20] H. Silverman, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc. 51(1975), 109-116.
[21] H. Silverman, A survey with open problems on univalent functions whose
coefficients are negative, Rocky Mountain J. Math. 21(1991), 1099-1125.
[22] H. Silverman, Integral means for univalent functions with negative coefficients, Houston J. Math. 23(1997), 169-174.
[23] S. Shams, S.R. Kulkarni and J. M. Jahangiri, Classes of Ruscheweyh-type
analytic univalent functions, Southwest J. Pure Appl. Math., 2(2003), 105-110.
[24] K. G. Subramanian, G. Murugusundaramoorthy, P. Balasubrahma and H.
Silverman, Subclasses of uniformly convex and uniformly starlike functions, Math.
Japon. 42(1995), no. 3, 517-522.
M. K. Aouf
Department of Mathematics
Faculty of Science
University of Mansoura
Mansoura 33516, Egypt
e-mail: [email protected]
R. M. El-Ashwah Department of Mathematics
Faculty of Science at Damietta
University of Mansoura
New Damietta 34517, Egypt
e-mail: r [email protected]
S. M. El-Deeb Department of Mathematics
Faculty of Science at Damietta
University of Mansoura
New Damietta 34517, Egypt
e-mail: [email protected]
42
No 20/2009
INTEGRAL MEANS FOR CERTAIN SUBCLASSES OF
UNIFORMLY STARLIKE AND CONVEX FUNCTIONS DEFINED
BY CONVOLUTION
M. K. Aouf, R. M. El-Ashwah and S. M. El-Deeb
Abstract. We introduce some generalized subclasses T Sγ (f, g; α, β) of uniformly starlike and convex functions, we settle the Silverman’s conjecture for the
integral means inequality. In particular, we obtain integral means inequalities for
various classes of uniformly β−starlike and uniformly β−convex functions in the
unit disc.
2000 Mathematics Subject Classification: 30C45.
1.Introduction
Let S denote the class of functions of the form:
f (z) = z +
∞
X
ak z k
(1.1)
k=2
that are analytic and univalent in the open unit disk U = {z : |z| < 1} . Let f (z) ∈
S be given by (1.1) and Φ(z) ∈ S be given by
Φ(z) = z +
∞
X
ck z k ,
(1.2)
k=2
then for analytic functions f and Φ with f (0) = Φ(0), f is said to be subordinate
to Φ, denoted by f ≺ Φ, if there exists an analytic function w such that w(0) = 0,
|w(z)| < 1 and f (z) = Φ(w(z)), for all z ∈ U.
The Hadamard product (or convolution) f ∗ Φ of f and Φ is defined (as usual) by
(f ∗ Φ)(z) = z +
∞
X
ak ck z k = (Φ ∗ f )(z).
k=2
31
(1.3)
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
Following Goodman ([7] and [8]), Ronning ([17] and [18]) introduced and studied
the following subclasses:
(i) A function f (z) of the form (1.1) is said to be in the class Sp (α, β) of uniformly
β−starlike functions if it satisfies the condition:
′
( ′
)
zf (z)
zf (z)
Re
−α >β
− 1
(z ∈ U ),
(1.4)
f (z)
f (z)
where −1 ≤ α < 1 and β ≥ 0.
(ii) A function f (z) of the form (1.1) is said to be in the class U CV (α, β) of
uniformly β−convex functions if it satisfies the condition:
′′
)
(
′′
zf (z)
zf (z)
(z ∈ U ),
(1.5)
−α >β ′
Re 1 + ′
f (z)
f (z)
where −1 ≤ α < 1 and β ≥ 0. We also observe that
Sp (α, 0) = T ∗ (α),
U CV (α, 0) = C(α)
are, respectively, well-known subclasses of starlike functions of order α and convex
functions of order α. Indeed it follows from (1.4) and (1.5) that
′
f (z) ∈ U CV (α, β) ⇐⇒ zf (z) ∈ Sp (α, β).
(1.6)
For −1 ≤ α < 1, 0 ≤ γ ≤ 1 and β ≥ 0, we let Sγ (f, g; α, β) be the subclass of S
consisting of functions f (z) of the form (1.1) and functions g(z) given by
g(z) = z +
∞
X
bk z k
(bk ≥ 0),
(1.7)
k=2
and satisfying the analytic criterion:
)
(
′
′′
z(f ∗ g) (z) + γz 2 (f ∗ g) (z)
−α
Re
(1 − γ) (f ∗ g)(z) + γz(f ∗ g)′ (z)
z(f ∗ g)′ (z) + γz 2 (f ∗ g)′′ (z)
>β
− 1 .
′
(1 − γ) (f ∗ g)(z) + γz(f ∗ g) (z)
(1.8)
Let T denote the subclass of S consisting of functions of the form:
f (z) = z −
∞
X
ak z k
k=2
32
(ak ≥ 0) .
(1.9)
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
Further, we define the class T Sγ (f, g; α, β) by
T Sγ (f, g; α, β) = Sγ (f, g; α, β) ∩ T.
(1.10)
We note that:
z
z
; α, 1) = Sp T (α) and T S0 (f, (1−z)
(i) T S0 (f, (1−z)
2 ; α, 1) =
z
; α, 1) = U CT (α) (−1 ≤ α < 1) (see Bharati et al. [4]);
T S1 (f, (1−z)
z
(ii) T S1 (f, (1−z)
; 0, β) = U CT (β) (β ≥ 0) (see Subramanian et al. [24]);
∞ (a)
P
k−1 k
z ; α, β) = T S (α, β) (−1 ≤ α < 1, β ≥ 0, c 6=
(iii) T S0 (f, z +
(c)
k−1
k=2
0, −1, −2, ...) (see Murugusundaramoorthy and Magesh [12] and [13]);
∞
P
k n z k ; α, β) = T S (n, α, β) (−1 ≤ α < 1, β ≥ 0, n ∈ N0 =
(iv) T S0 (f, z +
k=2
N ∪ {0}, N = {1, 2, ...})
(see Rosy
and Murugusundaramoorthy [19]);
∞
P
k+λ−1 k
z ; α, β) = D (α, β, λ) (−1 ≤ α < 1, β ≥ 0,
(v) T S0 (f, z +
λ
k=2
λ > −1) (see Shams et al. [23]);
(vi) T S0 (f, z +
∞
P
[1 + λ(k − 1)]n z k ; α, β) = T Sλ (n, α, β) (−1 ≤ α < 1,
k=2
β ≥ 0, λ ≥ 0, n ∈ N0 ) (see Aouf and Mostafa [2]);
∞ (a)
P
k−1 k
(vii) T Sγ (f, z +
z ; α, β) = T S (γ, α, β) (−1 ≤ α < 1, β ≥ 0,
(c)
k=2
k−1
0 ≤ γ ≤ 1, c 6= 0, −1, −2, ...) (see Murugusundaramoorthy et al. [14]);
(viii) T S0 (f, g; α, β) = HT (g, α, β) (−1 ≤ α < 1, β ≥ 0) (see Raina and Bansal
[16]);
∞
P
Γk z k ; α, β) = T Sqs (γ, α, β) (see Ahuja et al. [1]), where
(viii) T Sγ (f, z +
k=2
Γk =
(α1 )k−1 ...(αq )k−1
1
(β1 )k−1... (βs )k−1 (k − 1)!
(1.11)
(αi > 0, i = 1, ..., q; βj > 0, j = 1, ..., s; q ≤ s + 1; q, s ∈ N0 ) .
Also we note that
∞
P
k n z k ; α, β) = T Sγ (n, α, β)
(i) T Sγ (f, z +
k=2
=
(
f ∈ T : Re
(
′
′
(1 − γ)z(Dn f (z)) + γz(Dn+1 f (z))
−α
(1 − γ)Dn f (z) + γDn+1 f (z)
33
)
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
)
(1 − γ)z(Dn f (z))′ + γz(Dn+1 f (z))′
> β
− 1 , (−1 ≤ α < 1, β ≥ 0, n ∈ N0 , z ∈ U )
(1 − γ)Dn f (z) + γDn+1 f (z)
(1.12)
∞
P
c+1
k
(ii) T Sγ (f, z +
c+k z ; α, β) =
k=2
= T Sγ (c, α, β) =
(
f ∈ T : Re
(
′
′′
z(Jc f (z)) + γz 2 (Jc f (z))
−α
(1 − γ)Jc f (z) + γz(Jc f (z))′
)
)
z(J f (z))′ + γz 2 (J f (z))′′
c
c
>β
− 1 , 0 ≤ γ ≤ 1, −1 ≤ α < 1, β ≥ 0, c > −1, z ∈ U ;
(1 − γ)Jc f (z) + γz(Jc f (z))′
(1.13)
where Jc is a Bernardi operator [3], defined by
c+1
Jc f (z) =
zc
Zz
t
c−1
f (t)dt = z +
∞
X
c+1
k=2
0
c+k
ak z k .
Note that the operator J1 f (z) was studied earlier by Libera [9] and Livingston
[11];
∞
P
(µ)k−1
k
(iii) T Sγ (f, z +
(λ+1)k−1 z ; α, β) = T Sγ (µ, λ; α, β)
k=2
′
′′
z(Iλ,µ f (z)) +γz 2 (Iλ,µ f (z))
′
(1−γ)Iλ,µ f (z)+γz(Iλ,µ f (z))
f ∈ T : Re
−α
z(I f (z))′ + γz 2 (I f (z))′′
λ,µ
λ,µ
> β
′ − 1 ,
(1 − γ)Iλ,µ f (z) + γz(Iλ,µ f (z))
=
(0 ≤ γ ≤ 1, − 1 ≤ α < 1, β ≥ 0, λ > −1, µ > 0, z ∈ U )
where Iλ,µ is a Choi-Saigo-Srivastava operator [6], defined by
Iλ,µ f (z) = z +
∞
X
k=2
(µ)k−1
ak z k (λ > −1; µ > 0);
(λ + 1)k−1
34
;
(1.14)
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
(iv) T Sγ (f, z +
∞
P
k=2
(c)k−1 (λ+1)k−1 k
(a)k−1 (1)k−1 z ; α, β)
=
f ∈ T : Re
= T Sγ (a, c, λ; α, β)
′
′′
z(I λ (a,c)f (z)) +γz 2 (I λ (a,c)f (z))
(1−γ)I λ (a,c)f (z)+γz(I λ (a,c)f (z))′
−α
z(I λ (a, c)f (z))′ + γz 2 (I λ (a, c)f (z))′′
−
1
> β
,
′
λ
λ
(1 − γ)I (a, c)f (z) + γz(I (a, c)f (z))
0 ≤ γ ≤ 1, − 1 ≤ α < 1, β ≥ 0, a, c ∈ R\Z0− , λ > −1, z ∈ U
;
(1.15)
where I λ (a, c) is a Cho-Kwon-Srivastava operator [5], defined by
I λ (a, c)f (z) = z +
∞
X
(c)k−1 (λ + 1)k−1
ak z k ;
(a)k−1 (1)k−1
k=2
(v) T Sγ (f, z +
∞
P
k=2
=
(2)k−1
k
(n+1)k−1 z ; α, β)
(
f ∈ T : Re
(
= T Sγ (n; α, β)
′
′′
z(In f (z)) + γz 2 (In f (z))
−α
(1 − γ)In f (z) + γz(In f (z))′
)
)
z(I f (z))′ + γz 2 (I f (z))′′
n
n
>β
− 1 , 0 ≤ γ ≤ 1, −1 ≤ α < 1, β ≥ 0, n > −1, z ∈ U ;
(1 − γ)In f (z) + γz(In f (z))′
(1.16)
where In is a Noor integral operator [15], defined by
In f (z) = z +
∞
X
k=2
(2)k−1
ak z k (n > −1).
(n + 1)k−1
2
In [20], Silverman found that the function f2 (z) = z − z2 is often extremal
over the family T . He applied this function to resolve his integral means inequality,
conjectured in [21] and settled in [22], that
Z2π
Z2π
η
η
iθ
iθ
f
(re
)
dθ
≤
f
(re
)
dθ,
2
0
0
35
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
for all f ∈ T , η > 0 and 0 < r < 1. In [22], he also proved his conjecture for the
subclasses T ∗ (α) and C(α) of T.
In this paper, we prove Silverman’s conjecture for the functions in the class
T Sγ (f, g; α, β). By taking appropriate choises of the function g, we obtain the integral means inequalities for several known as well as new subclasses of uniformly
convex and uniformly starlike functions in U. In fact, these results also settle the
Silverman’s conjecture for several other subclasses of T.
2.Lemmas and their proofs
To prove our main results, we need the following lemmas.
Lemma 1. A function f (z) of the form (1.1) is in the class T Sγ (f, g; α, β) if
∞
X
[k(1 + β) − (α + β)] [1 + γ(k − 1)] |ak | bk ≤ 1 − α,
(2.1)
k=2
where −1 ≤ α < 1, β ≥ 0 and 0 ≤ γ ≤ 1.
Proof. It suffices to show that
(
)
′
′′
z(f ∗ g)′ (z) + γz 2 (f ∗ g)′′ (z)
z(f ∗ g) (z) + γz 2 (f ∗ g) (z)
β
− 1 −Re
−1
(1 − γ)(f ∗ g)(z) + γz(f ∗ g)′ (z)
(1 − γ)(f ∗ g)(z) + γz(f ∗ g)′ (z)
≤ 1 − α.
We have
(
)
′
′′
z(f ∗ g)′ (z) + γz 2 (f ∗ g)′′ (z)
z(f ∗ g) (z) + γz 2 (f ∗ g) (z)
− 1 −Re
−1
β
(1 − γ)(f ∗ g)(z) + γz(f ∗ g)′ (z)
(1 − γ)(f ∗ g)(z) + γz(f ∗ g)′ (z)
∞
(1 + β) P (k − 1) [1 + γ(k − 1)] |ak | bk
z(f ∗ g)′ (z) + γz 2 (f ∗ g)′′ (z)
k=2
.
≤ (1+β)
− 1 ≤
∞
P
(1 − γ)(f ∗ g)(z) + γz(f ∗ g)′ (z)
[1 + γ(k − 1)] |ak | bk
1−
k=2
This last expression is bounded above by (1 − α) if
∞
X
[k(1 + β) − (α + β)] [1 + γ(k − 1)] |ak | bk ≤ 1 − α,
k=2
36
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
and hence the proof is completed.
Lemma 2. A necessary and sufficient condition for f (z) of the form (1.9) to be in
the class T Sγ (f, g; α, β) is that
∞
X
[k(1 + β) − (α + β)] [1 + γ(k − 1)] ak bk ≤ 1 − α,
(2.2)
k=2
Proof. In view of Lemma 1, we need only to prove the necessity. If f (z) ∈ T Sγ (f, g; α, β)
and z is real, then
1−
∞
P
k [1 + γ(k − 1)] ak bk
k=2
∞
P
1−
z k−1
[1 + γ(k − 1)] ak bk z k−1
k=2
P
∞
k−1
(k − 1) [1 + γ(k − 1)] ak bk z
k=2
.
−α≥β
∞
1 − P [1 + γ(k − 1)] ak bk z k−1
k=2
Letting z → 1− along the real axis, we obtain the desired inequality
∞
X
[k(1 + β) − (α + β)] [1 + γ(k − 1)] ak bk ≤ 1 − α.
k=2
Corollary 1.Let the function f (z) be defined by (1.9) be in the class T Sγ (f, g; α, β).
Then
ak ≤
1−α
[k(1 + β) − (α + β)] [1 + γ(k − 1)] bk
(k ≥ 2).
(2.3)
The result is sharp for the function
f (z) = z −
1−α
zk
[k(1 + β) − (α + β)] [1 + γ(k − 1)] bk
(k ≥ 2).
(2.4)
Lemma 3. The extreme points of T Sγ (f, g; α, β) are
1−α
z k , for k = 2, 3, ... .
[k(1 + β) − (α + β)] [1 + γ(k − 1)] bk
(2.5)
The proof of Lemma 3 is similar to the proof of the theorem on extreme points given
in [20] and therefore are omit it.
In 1925, Littlewood [10] proved the following subordination theorem.
Lemma 4. If the functions f and g are analytic in U with g ≺ f, then for η > 0,
and 0 < r < 1,
f1 (z) = z and fk (z) = z−
37
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
Z2π
Z2π
η
η
iθ
iθ
g(re ) dθ ≤ f (re ) dθ.
0
(2.6)
0
3.Main theorem
Applying Lemma 4, Lemma 2 and Lemma 3, we prove the following result.
Theorem 1. Suppose f ∈ T Sγ (f, g; α, β), η > 0, −1 ≤ α < 1, 0 ≤ γ ≤ 1, β ≥ 0
and f2 (z) is defined by
f2 (z) = z −
1−α
z2.
(2 + β − α)(1 + γ)b2
Then for z = reiθ , 0 < r < 1, we have
Z2π
η
|f (z)| dθ ≤
Proof. For f (z) = z −
|f2 (z)|η dθ.
(3.1)
0
0
∞
P
Z2π
ak z k (ak ≥ 0), (3.1) is equivalent to proving that
k=2
η
η
Z2π
Z2π
∞
X
1−α
k−1
ak z
z dθ.
1 −
dθ ≤ 1 −
(2 + β − α)(1 + γ)b2
0
k=2
0
By Lemma 4, it suffices to show that
1−
∞
X
ak z k−1 ≺ 1 −
k=2
Setting
1−
∞
X
ak z k−1 = 1 −
k=2
1−α
z.
(2 + β − α)(1 + γ)b2
1−α
w(z),
(2 + β − α)(1 + γ)b2
(3.2)
and using (2.2), we obtain
∞
∞
X (2 + β − α)(1 + γ)b
X
(2 + β − α)(1 + γ)b2
2
k−1
|w(z)| =
ak z
ak ≤ |z| .
≤ |z|
1−α
1−α
k=2
k=2
This completes the proof of Theorem 1.
By taking different choices of g(z), α, β and γ in the above theorem, we can state
38
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
the following integral means results for various subclasses.
∞
P
Γk z k , where Γk is given by (1.11) in Theorem
Remarks. (i) Taking g(z) = z +
k=2
1, we obtain the result obtained by Ahuja et al. [1, Theorem 3.1];
z
(ii) Taking g(z) = 1−z
and γ = 0 in Theorem 1, we obtain the result obtained
by Ahuja et al. [1, Corollary 3.2];
z
(iii) Taking g(z) = 1−z
and γ = 1 in Theorem 1, we obtain the result obtained
by Ahuja et al. [1, Corollary 3.4];
∞
P
k+λ−1 k
(iv) Taking g(z) = z +
z (λ > −1) and γ = 0 in Theorem 1, we
λ
k=2
obtain the result obtained by Ahuja et al. [1, Corollary 3.6];
∞
P
c+1
k
(v) Taking g(z) = z +
c+k z (c > −1) and γ = 0 in Theorem 1, we obtain
k=2
the result obtained by Ahuja et al. [1, Corollary 3.7];
∞ (a)
P
k−1 k
z (a > 0; c > 0) and γ = 0 in Theorem 1, we
(vi) Taking g(z) = z +
(c)
k=2
k−1
obtain the result obtained by Ahuja et al. [1, Corollary 3.8].
∞
P
k n z k ; α, β) = T S (n, α, β) (−1 ≤ α < 1, β ≥ 0,
Corollary 2. If f ∈ T S0 (f, z +
k=2
n ∈ N0 and η > 0), then the assertion (3.1) holds true, where
f2 (z) = z −
Corollary 3. If f ∈ T S0 (f, z +
∞
P
(1 − α)
z2.
2n (2 + β − α)
[1 + λ(k − 1)]n z k ; α, β) = T Sλ (n, α, β) (−1 ≤
k=2
α < 1, β ≥ 0, λ ≥ 0, n ∈ N0 and η > 0), then the assertion (3.1) holds true, where
(1 − α)
z2.
(2 + β − α) (1 + λ)n
∞ (a)
P
k−1 k
z ; α, β) = T S (γ, α, β) (0 ≤ γ ≤ 1,
Corollary 4. If f ∈ T Sγ (f, z +
(c)
f2 (z) = z −
k=2
k−1
−1 ≤ α < 1, β ≥ 0, a > 0, c > 0 and η > 0), then the assertion (3.1) holds true,
where
f2 (z) = z −
c (1 − α)
z2.
a(2 + β − α)(1 + γ)
Corollary 5. If f ∈ T S0 (f, g(z); α, β) = HT (g, α, β) (−1 ≤ α < 1, β ≥ 0 and
η > 0), then the assertion (3.1) holds true, where
f2 (z) = z −
(1 − α)
z2.
(2 + β − α)b2
39
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
Corollary 6. If f ∈ T Sγ (f, z +
∞
P
k n z k ; α, β) = T Sγ (n, α, β) (0 ≤ γ ≤ 1, −1 ≤
k=2
α < 1, β ≥ 0, n ∈ N0 and η > 0), then the assertion (3.1) holds true, where
(1 − α)
z2.
+ β − α)(1 + γ)
∞
P
c+1
k
Corollary 7. If f ∈ T Sγ (f, z +
c+k z ; α, β) = T Sγ (c, α, β) (0 ≤ γ ≤ 1,
f2 (z) = z −
2n (2
k=2
−1 ≤ α < 1, β ≥ 0, c > −1 and η > 0), then the assertion (3.1) holds true, where
f2 (z) = z −
Corollary 8. If f ∈ T Sγ (f, z +
(1 − α) (c + 2)
z2.
(2 + β − α)(1 + γ)(c + 1)
∞
P
k=2
(µ)k−1
k
(λ+1)k−1 z ; α, β)
= T Sγ (µ, λ; α, β) (0 ≤ γ ≤ 1,
−1 ≤ α < 1, β ≥ 0, λ > −1, µ > 0 and η > 0), then the assertion (3.1) holds true,
where
f2 (z) = z −
Corollary 9.
If f ∈ T Sγ (f, z +
(1 − α) (λ + 1)
z2.
(2 + β − α)(1 + γ)µ
∞
P
k=2
(c)k−1 (λ+1)k−1 k
(a)k−1 (1)k−1 z ; α, β)
= T Sγ (a, c, λ; α, β)
(0 ≤ γ ≤ 1, −1 ≤ α < 1, β ≥ 0, a, c ∈ R\Z0− , λ > −1 and η > 0), then the assertion
(3.1) holds true, where
f2 (z) = z −
a (1 − α)
z2.
c(2 + β − α)(1 + γ)(λ + 1)
Corollary 10. If f ∈ T Sγ (f, z +
∞
P
k=2
(2)k−1
k
(n+1)k−1 z ; α, β)
= T Sγ (n, α, β) (0 ≤ γ ≤ 1,
−1 ≤ α < 1, β ≥ 0, n > −1 and η > 0), then the assertion (3.1) holds true, where
f2 (z) = z −
(1 − α) (n + 1)
z2.
2(2 + β − α)(1 + γ)
References
[1] O. Ahuja, G. Murugusundaramoorthy and N. Magesh, Integral means for
uniformly convex and starlike functions associated with generalized hypergeometric
functions, J. Inequal. Pure Appl. Math. 8(2007), no. 4, Art. 118, 1-9.
40
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
[2] M. K. Aouf and A. O. Mostafa, Some properties of a subclass of uniformly
convex functions with negative coefficients, Demonstratio Math., 2 (2008), 353-370.
[3] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math.
Soc., 135(1969), 429-446.
[4] R. Bharati, R. Parvatham and A. Swaminathan, On subclasses of uniformly
convex functions and corresponding class of starlike functions, Tamakang J. Math.
28(1997), 17-32.
[5] N. E. Cho, O. S. Kwon and H. M. Srivastava, Inclusion relationships and
argument properties for certain subclasses of multivalent functions associated with a
family of linear operators, J. Math. Anal. Appl. 292(2004), 470-483.
[6] J. H. Choi, M. Saigo, H. M. Srivastava, Some inclusion properties of a certain
family of integral operators, J. Math. Anal. Appl. 276(2002), 432-445.
7] A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56(1991),
87-92.
[8] A. W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl.
155(1991), 364-370.
[9] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math.
Soc. 16(1965), 755-758.
[10] J. E. Littlewood, On inequalities in theory of functions, Proc. London Math.
Soc. 23(1925), 481-519.
[11] A. E. Livingston, On the radius of univalence of certain analytic functions,
Proc. Amer. Math. Soc. 17(1966), 352-357.
[12] G. Murugusundaramoorthy and N. Magesh, A new subclass of uniformly
convex functions and corresponding subclass of starlike functions with fixed second
coefficient, J. Inequal. Pure Appl. Math. 5(2004), no. 4, Art. 85, 1-10.
[13] G. Murugusundaramoorthy and N. Magesh, Linear operators associated with
a subclass of uniformly convex functions, Internat. J. Pure Appl. Math. Sci.
3(2006), no. 2, 113-125.
[14] G. Murugusundaramoorthy, T. Rosy and K. Muthunagai, Carlson-Shaffer
operator and their applications to certain subclass of uniformly convex function,
General Math. 15(2007), no. 4, 131-143.
[15] K. I. Noor, On new classes of integral operators, J. Natur. Geem. 16(1999),
71-80.
[16] R. K. Raina and D. Bansal, Some properties of a new class of analytic
functions defined in terms of a Hadamard product, J. Inequal. Pure Appl. Math.
9(2008), no. 1, Art. 22, 1-9.
[17] F. Ronning, On starlike functions associated with parabolic regions, Ann.
Univ. Mariae-Curie-Sklodowska, Sect. A 45(1991), 117-122.
[18] F. Ronning, Uniformly convex functions and a corresponding class of starlike
41
Aouf, El-Ashwah, El-Deeb - Integral means for certain subclasses of uniformly...
functions, Proc. Amer. Math. Soc. 118(1993), 189-196.
[19] T. Rosy and G. Murugusundaramoorthy, Fractional calculus and their applications to certain subclass of uniformly convex functions, Far East J. Math. Sci.
(FJMS), 115(2004), no. 2, 231-242.
[20] H. Silverman, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc. 51(1975), 109-116.
[21] H. Silverman, A survey with open problems on univalent functions whose
coefficients are negative, Rocky Mountain J. Math. 21(1991), 1099-1125.
[22] H. Silverman, Integral means for univalent functions with negative coefficients, Houston J. Math. 23(1997), 169-174.
[23] S. Shams, S.R. Kulkarni and J. M. Jahangiri, Classes of Ruscheweyh-type
analytic univalent functions, Southwest J. Pure Appl. Math., 2(2003), 105-110.
[24] K. G. Subramanian, G. Murugusundaramoorthy, P. Balasubrahma and H.
Silverman, Subclasses of uniformly convex and uniformly starlike functions, Math.
Japon. 42(1995), no. 3, 517-522.
M. K. Aouf
Department of Mathematics
Faculty of Science
University of Mansoura
Mansoura 33516, Egypt
e-mail: [email protected]
R. M. El-Ashwah Department of Mathematics
Faculty of Science at Damietta
University of Mansoura
New Damietta 34517, Egypt
e-mail: r [email protected]
S. M. El-Deeb Department of Mathematics
Faculty of Science at Damietta
University of Mansoura
New Damietta 34517, Egypt
e-mail: [email protected]
42