MATEMATIQKI VESNIK
originalni nauqni rad
research paper
63, 4 (2011), 235–246
December 2011
THE QUASI-HADAMARD PRODUCTS OF UNIFORMLY CONVEX
FUNCTIONS DEFINED BY DZIOK-SRIVASTAVA OPERATOR
M. K. Aouf, A. Shamandy, A. O. Mostafa and E. A. Adwan
Abstract. The purpose of this paper is to obtain many interesting results about the quasiHadamard products of uniformly convex functions defined by Dziok-Srivastava operator belonging
to the class Tq,s ([α1 ]; α, β).
1. Introduction
Let T denote the class of functions of the form
∞
P
an z n (an ≥ 0) ,
f (z) = z −
(1.1)
n=2
which are analytic in the open unit disc U = {z : |z| < 1}. Let C(γ) and T ∗ (γ)
denote the subclasses of T which are, respectively, convex and starlike functions of
order γ, 0 ≤ γ < 1. For convenience, we write C(0) = C and T ∗ (0) = T ∗ (see [9]).
A function f ∈ T is said to be in U ST (β, γ), the class of β-uniformly starlike
functions of order γ, −1 ≤ γ < 1, if it satisfies the condition
¯ 0
¯
¾
½ 0
¯ zf (z)
¯
zf (z)
− γ > β ¯¯
− 1¯¯ , (β ≥ 0) .
(1.2)
Re
f (z)
f (z)
0
Replacing f (z) in (1.2) by zf (z) we have the condition
¯ 00
¯
¾
½
¯ zf (z) ¯
zf 00 (z)
¯
¯
−γ >β¯ 0
Re 1 + 0
¯ (β ≥ 0) ,
¯ f (z) ¯
f (z)
required for the function f to be in the subclass U CT (β, γ) of β-uniformly convex
functions of order γ (see [2]).
Let fj (z) ∈ T (j = 1, . . . , t) be given by
fj (z) = z −
∞
P
n=2
an,j z n
(an,j > 0; j = 1, 2, . . . , t).
(1.3)
2010 AMS Subject Classification: 30C45.
Keywords and phrases: Analytic functions; Dziok-Srivastava operator; uniformly convex
functions; quasi-Hadamard product.
235
236
M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan
Then the quasi-Hadamard product (or convolution) of these functions is defined by
(f1 ∗ f2 ∗ · · · ∗ ft )(z) = z −
∞ ³Q
t
P
n=2 j=1
´
an,j z n .
(1.4)
For positive real parameters α1 , . . . , αq and β1 , . . . , βs , βi ∈ C \ Z − ; Z − =
{0, −1, −2, . . . }; i = 1, 2, . . . , s, the Dziok-Srivastava operator (see [3] and [4])
Hq,s (α1 ) : T → T is given by
Hq,s (α1 )f (z) = z q Fs (α1 , . . . αq ; β1 , . . . βs ) ∗ f (z) = z −
∞
P
n=2
Ψn a n z n ,
(1.5)
where
(α1 )n−1 · · · (αq )n−1
,
(β1 )n−1 · · · (βs )n−1 (n − 1)!
½
1,
n=0
Γ(θ + n)
(θ)n =
=
Γ(θ)
θ(θ + 1) · · · (θ + n − 1), n ∈ N.
Ψn =
(1.6)
and q Fs (α1 , . . . , αq ; β1 , . . . , βs ; z) (q ≤ s + 1; s, q ∈ N0 = N ∪ {0}, N = {1, 2, . . . };
z ∈ U ) is the generalized hypergeometric function.
q Fs (α1 , . . . , αq ;
β1 , . . . , βs ; z) =
∞ (α ) . . . (α )
P
1 n
q n n
z .
n=0 (β1 )n . . . (βs )n n!
For −1 ≤ γ < 1, β ≥ 0, and for all z ∈ U , Aouf and Murugusundaramoorthy
[1] defined the subclass Tq,s ([α1 ]; γ, β) of functions of T which satisfy:
¯
¯
Ã
!
0
¯ z(H (α )f (z))0
¯
z(Hq,s (α1 )f (z))
¯
¯
q,s
1
Re
−γ >β¯
− 1¯ , z ∈ U.
(1.7)
¯ Hq,s (α1 )f (z)
¯
Hq,s (α1 )f (z)
They also proved [1] that the necessary and sufficient condition for functions f (z)
of the form (1.1) to be in the class Tq,s ([α1 ]; γ, β) is that:
∞
P
n=2
[n(1 + β) − (γ + β)]Ψn an ≤ 1 − γ.
(1.8)
We note that for suitable choices of q, s, γ and β, we obtain the following subclasses
studied by various authors.
(1) For q = 2 and s = α1 = α2 = β1 = 1 in (1.7), the class T2,1 ([1]; γ, β)
reduces to the class Sp T (γ, β) (−1 ≤ γ < 1, β ≥ 0) and the class Sp T (γ, 1) which
for β = 1 reduces to the class Sp T (γ) (see [2]).
(2) For q = 2, s = 1, α1 = a (a > 0) , α2 = 1 and β1 = c (c > 0) in (1.7), the
class T2,1 ([a]; γ, β) reduces to the class Sp T (a, c; γ, β) (−1 ≤ γ < 1, β ≥ 0) (see [5]).
(3) For q = 2, s = 1, α1 = λ + 1 (λ > −1), α2 = 1 and β1 = 1 in (1.7), the
class T2,1 (λ + 1, 1; 1; γ, β) reduces to the class Sp T (λ; γ, β) (−1 ≤ γ < 1, β ≥ 0)
(see [8]).
Quasi-Hadamard products of uniformly convex functions
237
(4) For q = 2, s = 1, α1 = v + 1 (v > −1), α2 = 1 and β1 = v + 2 in (1.7), the
class T2,1 (v + 1, 1; v + 2; γ, β) reduces to the class Sp T (v; γ, β) (−1 ≤ γ < 1, β ≥ 0)
(see [1]).
(5) For q = 2, s = 1, α1 = 2, α2 = 1 and β1 = 2 − µ (µ 6= 2, 3, . . . ) in (1.7), the
class T2,1 (2, 1; 2 − µ; γ, β) reduces to the class Sp T (µ; γ, β) (−1 ≤ γ < 1, β ≥ 0)
(see [1]).
2. Main results
Unless otherwise mentioned, we shall assume in the remainder of this paper
that the parameters α1 , . . . , αq and β1 , . . . , βs are positive real numbers, −1 ≤ γ <
1, β ≥ 0, z ∈ U , Ψn is defined by (1.6), Ψn ≥ 1 and j = 1, 2, . . . , t.
Theorem 1. Let the functions fj (z) defined by (1.3) be in the class
Tq,s ([α1 ]; γj , β). Then we have (f1 ∗ · · · ∗ ft )(z) ∈ Tq,s ([α1 ]; δ, β), where
Qt
(1 + β) j=1 (1 − γj )
δ = 1 − Qt
.
Qt
t−1
− j=1 (1 − γj )
j=1 (2 + β − γj )Ψ2
(2.1)
The result is sharp for the functions fj (z) given by
fj (z) = z −
(1 − γj )
z2.
(2 + β − γj )Ψ2
(2.2)
Proof. Employing the technique used earlier by Schild and Silverman [7] and
Owa [6], we prove Theorem 1 by using mathematical induction on t. For t = 2,
(1.8) gives
∞ [n(1 + β) − (γ + β)]Ψ
P
j
n
an,j ≤ 1 (j = 1, 2).
(2.3)
(1 − γj )
n=2
By the Cauchy-Schwarz inequality, we have
s
∞
2
P
Q
[n(1+β)−(γj +β)]Ψn √
an,2 an,2 ≤ 1.
(2.4)
(1−γj )
n=2
j=1
To prove the case when t = 2, we need to find the largest δ(−1 ≤ δ < 1) such that
∞ [n(1 + β) − (δ + β)]Ψ
P
n
an,1 an,2 ≤ 1,
(1 − δ)
n=2
thus, it suffices to show that
[n(1 + β) − (δ + β)]Ψn
an,1 an,2 ≤
(1 − δ)
or, equivalently, to
√
an,1 an,2 ≤
(1 − δ)
qQ
2
qQ
j=1 [n(1
(2.5)
+ β) − (γj + β)]Ψn √
qQ
an,1 an,2
2
(1
−
γ
)
j
j=1
2
j=1 [n(1
+ β) − (γj + β)]Ψn
qQ
.
2
[n(1 + β) − (δ + β)]Ψn
(1
−
γ
)
j
j=1
238
M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan
By noting that
√
qQ
2
j=1 (1
an,1 an,2 ≤ qQ
2
j=1 [n(1
− γj )
,
+ β) − (γj + β)]Ψn
consequently, we need only to prove that
Q2
(1 − δ)
j=1 (1 − γj )
≤
Q2
[n(1
+
β) − (δ + β)]
[n(1
+
β)
−
(γ
+
β)]Ψ
j
n
j=1
which is equivalent to
(n − 1)(1 + β)
Q2
− γj )
.
Q2
j=1 [n(1 + β) − (γj + β)]Ψn −
j=1 (1 − γj )
δ ≤ 1 − Q2
j=1 (1
Since
Q2
− γj )
,
Q2
j=1 [n(1 + β) − (γj + β)]Ψn −
j=1 (1 − γj )
B(n) = 1 − Q2
(n − 1)(1 + β)
j=1 (1
is an increasing function of n (n > 2), then
Q2
− γj )
.
Q2
j=1 (1 − γj )
j=1 (2 + β − γj )Ψ2 −
δ ≤ B(2) = 1 − Q2
(1 + β)
j=1 (1
Therefore, the result is true for t = 2.
Suppose that the result is true for any positive integer t = k. Then we have
(f1 ∗ · · · ∗ fk ∗ fk+1 )(z) ∈ Tq,s ([α1 ]; λ, β),
where
λ=1−
(1 + β)(1 − γk+1 )(1 − δ)
,
(2 + β − γk+1 )(2 + β − δ)Ψ2 − (1 − γk+1 )(1 − δ)
and δ is given by (2.2). After simple calculations, we have
Qk+1
(1 + β) j=1 (1 − γj )
.
λ = 1 − Qk+1
Qk+1
k
j=1 (1 − γj )
j=1 (2 + β − γj )Ψ2 −
(2.6)
This shows that the result is true for t = k + 1. Therefore, by mathematical
induction, the result is true for any positive integer t (t ≥ 2).
Taking the functions fj (z) given by (2.2), we have
(f1 ∗ · · · ∗ ft )(z) = z −
t
Y
(1 − γj )
z 2 = z − H2 z 2 ,
(2
+
β
−
γ
)Ψ
j
2
j=1
which shows that
t
∞ [n(1 + β) − (δ + β)]Ψ
P
(2 + β − δ]Ψ2 Q
(1 − γj )
n
H2 =
·
= 1;
(1
−
δ)
(1
−
δ)
(2
+
β − γj )Ψ2
n=2
j=1
(2.7)
Quasi-Hadamard products of uniformly convex functions
239
Consequently, the result is sharp for functions fj (z) given by (2.2). This completes
the proof of Theorem 1.
Letting γj = γ in Theorem 1, we obtain the following corollary.
Corollary 1. Let the functions fj (z) defined by (1.3) be in the class
Tq,s ([α1 ]; γ, β). Then we have (f1 ∗ · · · ∗ ft )(z) ∈ Tq,s ([α1 ]; δ, β),
δ =1−
(1 + β)(1 − γ)t
.
(2 + β − γ)t Ψt−1
− (1 − γ)t
2
(2.8)
The result is sharp for the functions fj (z) given by
fj (z) = z −
(1 − γ)
z2.
(2 + β − γ)Ψ2
(2.9)
Putting t = 2 in Corollary 1, we obtain the following corollary.
Corollary 2. Let the functions fj (z) (j = 1, 2) defined by (1.3) be in the
class Tq,s ([α1 ]; γ, β). Then we have (f1 ∗ f2 )(z) ∈ Tq,s ([α1 ]; δ, β), where
δ =1−
(1 + β)(1 − γ)2
.
(2 + β − γ)2 Ψ2 − (1 − γ)2
The result is sharp.
Next, similarly by applying the method of proof of Theorem 1, we easily get
the following result.
Theorem 2. Let the functions fj (z) defined by (1.3) be in the class
Tq,s ([α1 ]; γ, ζj ), ζj ≥ 0. Then we have (f1 ∗ · · · ∗ ft )(z) ∈ Tq,s ([α1 ]; γ, η), where
Qt
t−1
j=1 (2 + ζj − γ)Ψ2
η=
+ γ − 2.
(2.10)
(1 − γ)t−1
The result is sharp for the functions fj (z) given by
fj (z) = z −
(1 − γ)
z2.
(2 + ζj − γ)Ψ2
(2.11)
Let ζj = β (j = 1, 2, . . . , t) in Theorem 2, we obtain the following corollary.
Corollary 3. Let the functions fj (z) defined by (1.3) be in the class
Tq,s ([α1 ]; γ, β). Then we have (f1 ∗ · · · ∗ ft )(z) ∈ Tq,s ([α1 ]; γ, η), where
η=
(2 + β − γ)t Ψt−1
2
+ γ − 2.
(1 − γ)t−1
The result is sharp for the functions fj (z) given by (2.9).
240
M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan
Putting t = 2 in Corollary 3, we obtain the following corollary.
Corollary 4. Let the functions fj (z) (j = 1, 2) defined by (1.3) be in the
class Tq,s ([α1 ]; γ, β). Then we have (f1 ∗ f2 )(z) ∈ Tq,s ([α1 ]; γ, η), where
η=
(2 + β − γ)2 Ψ2
+ γ − 2.
(1 − γ)
The result is sharp.
Theorem 3. Let the functions fj (z) defined by (1.3) be in the class
Tq,s ([α1 ]; γj , β). Then the function
F (z) = z −
∞ ³P
t
P
n=2 j=1
´
n
am
n,j z
(m > 1)
(2.12)
belongs to the class Tq,s ([α1 ]; δt , β), where
δt = 1 −
t(1 + β)(1 − γ)m
(γ = min {γj }),
1≤j≤t
(2 + β − γ)m Ψm−1
− t(1 − γ)m
2
(2.13)
and (2 + β − γ)m Ψm−1
≥ t(2 + β)(1 − γ)m . The result is sharp for the functions
2
fj (z) (j = 1, 2, . . . , t) given by (2.2).
Proof. By virtue of (1.8), we have
∞ [n(1 + β) − (γ + β)]Ψ
P
j
n
an,j ≤ 1.
(1 − γj )
n=2
By the Cauchy-Schwarz inequality, we have
³P
´m
∞ ³ [n(1 + β) − (γ + β)]Ψ ´m
∞ [n(1 + β) − (γ + β)]Ψ
P
j
n
j
n
am
≤
a
≤ 1.
n,j
n,j
(1 − γj )
(1 − γj )
n=2
n=2
(2.14)
It follows from (2.14) that
Ã
!
µ
¶m
∞
t
P
1 P
[n(1 + β) − (γj + β)]Ψn
m
an,j ≤ 1.
t j=1
(1 − γj )
n=2
By setting γ = min1≤j≤t {γj }, the last inequality gives
à µ
!
¶m t
∞
P
P m
1 [n(1 + β) − (γ + β)]Ψn (α1 )
an,j ≤ 1.
t
(1 − γ)
n=2
j=1
Therefore, to prove our result we need to fined the largest δt such that
∞ [n(1 + β) − (δ + β)]Ψ (α ) P
t
P
t
n
1
am
n,j ≤ 1,
(1 − δt )
n=2
j=1
241
Quasi-Hadamard products of uniformly convex functions
that is, that
[n(1 + β) − (δt + β)]Ψn
1
≤
(1 − δt )
t
µ
[n(1 + β) − (γ + β)]Ψn
(1 − γ)
¶m
which leads to
δt ≤ 1 −
t(n − 1)(1 + β)(1 − γ)m
m−1
[n(1 + β) − (γ + β)]m (Ψn )
− t(1 − γ)m
.
Now let
R(n) = 1 −
t(n − 1)(1 + β)(1 − γ)m
[n(1 + β) − (γ + β)]m Ψnm−1 − t(1 − γ)m
(n ≥ 2).
Since R(n) is an increasing function of n (n > 2), then we have
δt ≤ R(2) = 1 −
t(1 + β)(1 − γ)m
,
(2 + β − γ)m Ψm−1
− t(1 − γ)m
2
and by noting that (2+β −γ)m Ψm−1
≥ t(2+β)(1−γ)m , we can see that 0 ≤ δt < 1.
2
The result is sharp for the functions fj (z) (j = 1, 2, . . . , t) given by (2.2). This
completes the proof of Theorem 3.
Putting m = 2 and γj = γ (j = 1, . . . , t) in Theorem 3, we obtain the following
corollary.
Corollary 5. Let the functions fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in
the class Tq,s ([α1 ]; γ, β). Then the function
G(z) = z −
∞ ³P
t
P
n=2 j=1
´
a2n,j z n ,
(2.15)
belongs to the class Tq,s ([α1 ]; δt , β), where
δt = 1 −
t(1 + β)(1 − γ)2
(2 + β − γ)2 Ψ2 − t(1 − γ)2
(2.16)
and (2 + β − γ)2 Ψ2 ≥ t(2 + β)(1 − γ)2 . The result is sharp for the functions
fj (z) (j = 1, 2, . . . , t) given by (2.9).
Similarly by applying the method of proof of Theorem 3, we easily get the
following result.
Theorem 4. Let the functions fj (z) defined by (1.3) be in the class
Tq,s ([α1 ]; γ, ζj ), ζj ≥ 0. Then the function F (z) defined by (2.12) belongs to the
class Tq,s ([α1 ]; γ, ηt ), where
ηt =
(2 + β − γ)m Ψ2m−1
+γ−2
t(1 − γ)m−1
(β = min {ζj }),
1≤j≤t
242
M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan
and (2 + β − γ)m Ψm−1
≥ t(2 − γ)(1 − γ)m−1 . The result is sharp for the functions
2
fj (z) (j = 1, 2, . . . , t) given by (2.11).
Putting m = 2 and ζj = β (j = 1, 2, . . . ., t) in Theorem 4, we obtain the
following corollary.
Corollary 6. Let the functions fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in
the class Tq,s ([α1 ]; γ, β). Then the function G(z) defined by (2.15) belongs to the
class Tq,s ([α1 ]; γ, ηt ), where
(2 + β − γ)2 Ψ2
+γ−2
t(1 − γ)
ηt =
and (2 + β − γ)2 Ψ2 ≥ t(2 − γ)(1 − γ). The result is sharp for the functions fj (z)
given by (2.9).
Theorem 5. Let the functions fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the
class Tq,s ([α1 ]; γj , β) (j = 1, 2, . . . , t) and let the functions gm (z) defined by
gm (z) = z −
∞
P
bn,m z n (bn,m > 0; m = 1, 2, . . . , s),
n=2
(2.17)
be in the class Tq,s ([α1 ]; γm , β) (m = 1, 2, . . . , s), then
(f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Tq,s ([α1 ]; Ω, β),
where
(1 + β)
Ω=1−
t
Q
j=1
t
Q
(2 + β − γj )
j=1
s
Q
(2 + β −
m=1
(1 − γj )
s
Q
(1 − γm )
m=1
γm )Ψ2t+s−1
−
t
Q
(1 − γj )
j=1
s
Q
m=1
.
(1 − γm )
(2.18)
The result is sharp for the functions fj (z) given by (2.2) and the functions gm (z)
given by
(1 − γm )
z 2 (m = 1, 2, . . . , s).
(2.19)
gm (z) = z −
(2 + β − γm )Ψ2
Proof. From Theorem 1 we note that, if f (z) ∈ Tq,s ([α1 ]; δ, β) and g(z) ∈
Tq,s ([α1 ]; µ, β), then (f ∗ g)(z) ∈ Tq,s ([α1 ]; Ω, β), where
Ω=1−
(1 + β)(1 − δ)(1 − µ)
.
(2 + β − δ)(2 + β − µ)Ψ2 − (1 − δ)(1 − µ)
(2.20)
Since Theorem 1 leads to (f1 ∗ f2 ∗ · · · ∗ ft )(z) ∈ Tq,s ([α1 ]; δ, β), where δ is defined
by (2.1) and (g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Tq,s ([α1 ]; µ, β) with
Qs
(1 + β) m=1 (1 − γm )
µ = 1 − Qs
.
(2.21)
Qs
s−1
− m=1 (1 − γm )
m=1 (2 + β − γm )Ψ2
Quasi-Hadamard products of uniformly convex functions
243
Then, we have (f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Tq,s ([α1 ]; Ω, β), where Ω is
given by (2.18), this completes the proof of Theorem 5.
Letting γj = γ (j = 1, 2, . . . , t) and γm = γ (m = 1, 2, . . . , s) in Theorem 5, we
obtain the following corollary.
Corollary 7. Let the functions fj (z) (j = 1, 2, . . . , t) defined by (1.3) and let
the functions gm (z) defined by (2.17) be in the class Tq,s ([α1 ]; γ, β). Then we have
(f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ . . . . ∗ gs )(z) ∈ Tq,s ([α1 ]; Ω, β), where
Ω=1−
(1 + β)(1 − γ)t+s
.
(2 + β − γ)t+s Ψt+s−1
− (1 − γ)t+s
2
The result is sharp for the functions fj (z) given by (2.9) and the functions gm (z)
given by
(1 − γ)
gm (z) = z −
z 2 (m = 1, 2, . . . , s).
(2 + β − γ)Ψ2
Letting t = s = 2 in Corollary 7, we obtain the following corollary.
Corollary 8. Let the functions fj (z) (j = 1, 2) defined by (1.3) and let the
functions gm (z) (m = 1, 2) defined by (2.17) be in the class Tq,s ([α1 ]; γ, β). Then
we have (f1 ∗ f2 ∗ g1 ∗ g2 )(z) ∈ Tq,s ([α1 ]; Ω, β), where
Ω=1−
(1 + β)(1 − γ)4
.
(2 + β − γ)4 Ψ32 − (1 − γ)4
The result is sharp.
Putting q = 2, s = 1 and α1 = α2 = β1 = 1 in Theorem 5, we obtain the
following corollary.
Corollary 9. Let fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the class
Sp T (γj , β) and let the functions gm (z) (m = 1, 2, . . . , s) defined by (2.17) be in the
class Sp T (γm , β) (m = 1, 2, . . . , s), then
(f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Sp T (τ, β),
where
Qs
Qt
(1 + β) j=1 (1 − γj ) m=1 (1 − γm )
.
τ = 1 − Qt
Qs
Qt
Qs
m=1 (2 + β − γm ) −
j=1 (1 − γj )
m=1 (1 − γm )
j=1 (2 + β − γj )
The result is sharp for the functions fj (z) given by
fj (z) = z −
(1 − γj )
z2
(2 + β − γj )
(j = 1, 2, . . . , t)
and the functions gm (z) given by
gm (z) = z −
(1 − γm )
z2
(2 + β − γm )
(m = 1, 2, . . . , s).
244
M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan
Putting q = 2, s = 1, α1 = a(a > 0), α2 = 1 and β1 = c (c > 0) in Theorem 5,
we obtain the following corollary.
Corollary 10. Let fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the class
Sp T (a, c; γj , β) and let the functions gm (z) (m = 1, 2, . . . , s) defined by (2.17) be in
the class Sp T (a, c; γm , β) (m = 1, 2, . . . , s), then
(f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Sp T (a, c; ζ, β),
where
ct+s−1 (1 + β)
t
Q
j=1
ζ = 1−
at+s−1
t
Q
j=1
(2 + β − γj )
s
Q
m=1
(1 − γj )
(2 + β − γm ) −
s
Q
(1 − γm )
m=1
ct+s−1
t
Q
(1 − γj )
j=1
.
s
Q
m=1
(1 − γm )
The result is sharp for the functions fj (z) given by
fj (z) = z −
(1 − γj )c
z2
(2 + β − γj )a
(j = 1, 2, . . . , t)
and the functions gm (z) given by
gm (z) = z −
(1 − γm )c
z 2 (m = 1, 2, . . . , s).
(2 + β − γm )a
Putting q = 2, s = 1, α1 = λ + 1(λ > −1), α2 = 1 and β1 = 1 in Theorem 5,
we obtain the following corollary.
Corollary 11. Let fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the class
Sp T (λ; γj , β) and let the functions gm (z) (m = 1, 2, . . . , s) defined by (2.17) be in
the class Sp T (λ; γm , β) (m = 1, 2, . . . , s), then
(f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Sp T (λ; ν, β),
where
t
Q
(1 + β)
ν =1−
(λ + 1)t+s−1
t
Q
j=1
(1 − γj )
j=1
s
Q
(2 + β − γj )
s
Q
(1 − γm )
m=1
(2 + β − γm ) −
m=1
t
Q
(1 − γj )
j=1
The result is sharp for the functions fj (z) given by
fj (z) = z −
(1 − γj )
z2
(2 + β − γj )(λ + 1)
(j = 1, 2, . . . , t)
s
Q
.
(1 − γm )
m=1
245
Quasi-Hadamard products of uniformly convex functions
and the functions gm (z) given by
(1 − γm )
z2
(2 + β − γm )(λ + 1)
gm (z) = z −
(m = 1, 2, . . . , s).
Putting q = 2, s = 1, α1 = v + 1(v > −1), α2 = 1 and β1 = v + 2 in Theorem
5, we obtain the following corollary.
Corollary 12. Let fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the class
Sp T (v; γj , β) and let the functions gm (z) (m = 1, 2, . . . , s) defined by (2.17) be in
the class Sp T (v; γm , β) (m = 1, 2, . . . , s), then
(f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Sp T (v; σ, β),
where
σ = 1−
t
Q
(v + 2)t+s−1 (1 + β)
j=1
(v + 1)t+s−1
t
Q
j=1
(2 + β − γj )
s
Q
j=1
(1 − γj )
s
Q
(1 − γm )
j=1
(2 + β − γm ) − (v + 2)t+s−1
t
Q
j=1
(1 − γj )
s
Q
.
(1 − γm )
j=1
The result is sharp for the functions fj (z) given by
fj (z) = z −
(1 − γj )(v + 2)
z2
(2 + β − γj )(v + 1)
(j = 1, 2, . . . , t)
and the functions gm (z) given by
(1 − γm )(v + 2)
z2
(2 + β − γm )(v + 1)
gm (z) = z −
(m = 1, 2, . . . , s).
Putting q = 2, s = 1, α1 = 2, α2 = 1 and β1 = 2 − µ (µ 6= 2, 3, . . . ) in Theorem
5, we obtain the following corollary.
Corollary 13. Let fj (z) (j = 1, 2, . . . , t) defined by (1.3) be in the class
Sp T (µ; γj , β) and let the functions gm (z) (m = 1, 2, . . . , s) defined by (2.17) be in
the class Sp T (µ; γm , β) (m = 1, 2, . . . , s), then
(f1 ∗ f2 ∗ · · · ∗ ft ∗ g1 ∗ g2 ∗ · · · ∗ gs )(z) ∈ Sp T (µ; κ, β),
where
κ = 1−
(2 − µ)t+s−1 (1 + β)
t
Q
j=1
2t+s−1
t
Q
(2 + β − γj )
j=1
s
Q
(1 − γj )
(2 + β − γm ) − (2 −
m=1
s
Q
(1 − γm )
m=1
µ)t+s−1
t
Q
j=1
(1 − γj )
s
Q
.
(1 − γm )
m=1
246
M. K. Aouf, A. Shamandy, A. O. Mostafa, E. A. Adwan
The result is sharp for the functions fj (z) given by
fj (z) = z −
(2 − µ)(1 − γj ) 2
z
2(2 + β − γj )
(j = 1, 2, . . . , t)
and the functions gm (z) given by
gm (z) = z −
(2 − µ)(1 − γm ) 2
z
2(2 + β − γm )
(m = 1, 2, . . . , s).
Acknowledgements. The authors would like to thank the referees of the
paper for their helpful suggestions.
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(received 17.05.2010; in revised form 24.09.2010)
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
E-mail: mkaouf127@yahoo.com, shamandy16@hotmail.com, adelaeg254@yahoo.com,
eman.a2009@yahoo.com