Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
DOI 10.1186/s13660-015-0573-z
RESEARCH
Open Access
A new class of analytic functions defined by
means of a generalization of the
Srivastava-Attiya operator
Hari M Srivastava1 and Sebastien Gaboury2*
*
Correspondence:
[email protected]
2
Department of Mathematics and
Computer Science, University of
Québec at Chicoutimi, Chicoutimi,
Québec G7H 2B1, Canada
Full list of author information is
available at the end of the article
Abstract
In this paper, we introduce a new class of analytic functions defined by a new
λ
convolution operator J(s,a,
λp ),(μq ),b which generalizes the well-known Srivastava-Attiya
operator investigated by Srivastava and Attiya (Integral Transforms Spec. Funct.
18:207-216, 2007). We derive coefficient inequalities, distortion theorems, extreme
points and the Fekete-Szegö problem for this new function class.
MSC: Primary 30C45; 11M35; secondary 30C10
Keywords: analytic functions; starlike functions; generalized Hurwitz-Lerch zeta
function; Srivastava-Attiya operator; Hadamard product
1 Introduction
Let A denote the class of functions f (z) normalized by
f (z) = z +
∞
ak z k ,
(.)
k=
which are analytic in the open unit disk
U = z: z ∈ C and |z| <  .
A function f (z) in the class A is said to be in the class S ∗ (α) of starlike functions of order
α in U if it satisfies the following inequality:
zf (z)
f (z)
>α
(z ∈ U;  α < ).
(.)
The largely investigated Srivastava-Attiya operator is defined as [] (see also [–]):
Js,a (f )(z) = z +
∞ +a s
k=
k+a
ak z k ,
(.)
where z ∈ U, a ∈ C \ Z– , s ∈ C and f ∈ A.
© 2015 Srivastava and Gaboury; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
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In fact, the linear operator Js,a (f ) can be written as
Js,a (f )(z) := Gs,a (z) ∗ f (z)
(.)
in terms of the Hadamard product (or convolution), where Gs,a (z) is given by
Gs,a (z) := ( + a)s (z, s, a) – a–s (z ∈ U).
(.)
The function (z, s, a) involved in the right-hand side of (.) is the well-known HurwitzLerch zeta function defined by (see, for example, [, p. et seq.]; see also [] and [, p.
et seq.])
(z, s, a) :=
∞
n=
zn
(n + a)s
a ∈ C \ Z– ; s ∈ C when |z| < ; (s) >  when |z| =  .
(.)
Recently, a new family of λ-generalized Hurwitz-Lerch zeta functions was investigated by Srivastava [] (see also [–]). Srivastava considered the following function:
(ρ ,...,ρp ,σ ,...,σq )
λ,...,λp ;μ,...,μq (z, s, a; b, λ)
p
n
∞
z


 j= (λj )nρj
,
λ
(a + n)b (s, ), ,
·
H
=
λ(s) n= (a + n)s · qj= (μj )nσj ,
λ
n!
min (a), (s) > ; (b) > ; λ >  ,
(.)
where
λj ∈ C (j = , . . . , p) and μj ∈ C \ Z– (j = , . . . , q); ρj >  (j = , . . . , p);
σj >  (j = , . . . , q);  +
q
σj –
p
j=
ρj 
j=
and the equality in the convergence condition holds true for suitably bounded values of
|z| given by
|z| < ∇ :=
p
j=
–ρj
ρj
·
q
σj
σj
.
j=
Here, and for the remainder of this paper, (λ)κ denotes the Pochhammer symbol defined,
in terms of the gamma function, by
⎧
(λ + κ) ⎨λ(λ + ) · · · (λ + n – ) (κ = n ∈ N; λ ∈ C),
(λ)κ :=
=
⎩
(λ)
(κ = ; λ ∈ C \ {}),
(.)
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
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it being understood conventionally that () :=  and assumed tacitly that the -quotient
exists (see, for details, [, p. et seq.]).
Definition  The H-function involved in the right-hand side of (.) is the well-known
Fox’s H-function [, Definition .] (see also [, ]) defined by
m,n
m,n
(z) = Hp,q
Hp,q
=

πi
(a , A ), . . . , (a , A )
 
p
p
z
(b , B ), . . . , (bq , Bq )
L
z ∈ C \ {}; arg(z) < π ,
(s)z–s ds
(.)
where
m
j= (bj
+ Bj s) ·
j=n+ (aj
+ Aj s) ·
(s) = p
n
j= ( – aj
q
– Aj s)
j=m+ ( – bj
– Bj s)
,
an empty product is interpreted as , m, n, p and q are integers such that  m q,  n p, Aj >  (j = , . . . , p), Bj >  (j = , . . . , q), aj ∈ C (j = , . . . , p), bj ∈ C (j = , . . . , q) and L
is a suitable Mellin-Barnes type contour separating the poles of the gamma functions
m
(bj + Bj s) j=
from the poles of the gamma functions
n
( – aj + Aj s) j= .
It is worthy to mention that using the fact that [, p., Remark ]

 ,
(a + n)b λ (s, ), ,
lim H,
= λ(s) (λ > ),
b→
λ
(.)
equation (.) reduces to
(ρ ,...,ρp ,σ ,...,σq )
(ρ ,...,ρp ,σ ,...,σq )
λ,...,λp ;μ,...,μq (z, s, a; , λ) := λ,...,λp ;μ,...,μq (z, s, a)
p
∞
zn
j= (λj )nρj
.
=
q
(a + n)s · j= (μj )nσj n!
n=
(ρ ,...,ρp ,σ ,...,σq )
(.)
Definition  The function λ,...,λp ;μ,...,μq (z, s, a) involved in (.) is the multiparameter
extension and generalization of the Hurwitz-Lerch zeta function (z, s, a) introduced by
Srivastava et al. [, p., Eq. (.)] defined by
∞
(ρ ,...,ρ ,σ ,...,σ )
λ,...,λpp;μ,...,μqq (z, s, a) :=
n=
p
j= (λj )nρj
q
s
(a + n) · j= (μj )nσj
zn
n!
p, q ∈ N ; λj ∈ C (j = , . . . , p); a, μj ∈ C \ Z– (j = , . . . , q);
ρj , σk ∈ R+ (j = , . . . , p; k = , . . . , q);
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
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> – when s, z ∈ C;
= – and s ∈ C when |z| < ∇ ∗ ;
= – and (
) >

when |z| = ∇ ∗

(.)
with
∗
∇ :=
p
–ρj
ρj
·
j=
:=
q
σj –
j=
q
σj
σj
,
(.)
j=
p
and := s +
ρj
j=
q
μj –
j=
p
λj +
j=
p–q
.

(.)
We propose to consider the following linear operator
s,a,λ
J(λ
(f ) : A → A,
p ),(μq ),b
defined by
s,a,λ
s,a,λ
(f )(z) = G(λ
(z) ∗ f (z),
J(λ
p ),(μq ),b
p ),(μq ),b
(.)
where ∗ denotes the Hadamard product (or convolution) of analytic functions, and the
s,a,λ
function G(λ
(z) is given by
p ),(μq ),b
s,a,λ
G(λ
(z)
p ),(μq ),b
q
λ j= (μj )(s)(a + )s
· (a + , b, s, λ)–
:=
p
(λ
)
j
j=
a–s
· (,...,,,...,)
(z,
s,
a;
b,
λ)
–
(a,
b,
s,
λ)
λ ,...,λp ;μ ,...,μq
λ(s)
p
∞
a +  s (a + k, b, s, λ) zk
j= (λj + )k–
=z+
q
(a + , b, s, λ) k!
j= (μj + )k– a + k
(.)
k=
with
,
(a, b, s, λ) := H,

ab (s, ), ,
.
λ

λ
Combining (.) and (.), we obtain
s,a,λ
J(λ
(f )(z)
p ),(μq ),b
k
∞ p
a +  s (a + k, b, s, λ)
z
j= (λj + )k–
ak
=z+
q
a
+
k
(a
+
,
b,
s,
λ)
k!
(μ
+
)
j
k–
j=
k=
λj ∈ C (j = , . . . , p) and μj ∈ C \ Z– (j = , . . . , q); p q + ; z ∈ U ,
(.)
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
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with
min (a), (s) > ;
λ >  if (b) > 
and
s ∈ C;
a ∈ C \ Z–
if b = .
s,a,λ
(f ) (special case of
Remark  It follows from (.) and (.) that the operator J(λ
p ),(μq ),
–
(.) when b = ) can be defined for a ∈ C \ Z by the following limit relationship:
s,a,λ
s,,λ
J(λ
(f )(z) := lim J(λ
(f )(z) .
p ),(μq ),
p ),(μq ),
(.)
a→
s,a,λ
generalizes several recently investigated operators
We can see that the operator J(λ
p ),(μq ),b
such as:
s,a,λ
(i) If p = , q =  and b = , then J(λ
= Jλs,a ,λ ;μ , where Jλs,a ,λ ;μ is the linear
 ,λ ),(μ ),
operator introduced by Prajapat and Bulboacă [, p., Eq. (.)].
s
s
(ii) J(γs,a,λ
–,),(ν), = Ia,ν,γ , where Ia,ν,γ is the generalized operator recently studied by Noor
and Bukhari [, p., Eq. (.)].
s
s
(iii) J(γ,,λ
–,),(ν), = Iν,γ , where Iν,γ is the Choi-Saigo-Srivastava operator [].
s,a,λ
(iv) J(γ ,),(γ ), = Js,a , where Js,a is the Srivastava-Attiya operator [].
(v) J(γ–r,a,λ
,),(γ ), = I(r, a) (a , r ∈ Z), where the operator I(r, a) is the one introduced by
Cho and Srivastava [].
,a,λ
(vi) J(β,),(α+β),
= Qαβ (α , β > –), where the operator Qαβ was studied by Jung et
al. [].
(vii) J(γ,a,λ
,),(γ ), = Ja (a –), where Ja denotes the Bernardi operator [].
(viii) J(γ,,λ
,),(ν), = L(γ , ν), where L(γ , ν) is the well-known Carlson-Shaffer operator [].
γ
γ
,,λ
(ix) J(,),(–γ
), = z ( γ < ), where z is the fractional integral operator
investigated by Owa and Srivastava [].
,a,λ
(x) J(λ
= H (λ , . . . , λp ; μ , . . . , μq ) (p q + ), where the
 –,...,λp –,),(μ –,...,μq –,),
operator H (λ , . . . , λp ; μ , . . . , μq ) is the Dziok-Srivastava operator [, ] which
contains as special cases the Hohlov operator [] and the Ruscheweyh
operator [].
s,a,λ,∗
s,a,λ
We say that a function f ∈ A is in the class S(λ
(α) if J(λ
(f ) is in the class
p ),(μq ),b
p ),(μq ),b
∗
S (α), that is, if
z(J s,a,λ
(λp ),(μq ),b (f ))
s,a,λ
J(λp ),(μq ),b (f )
>α
λj ∈ C (j = , . . . , p) and μj ∈ C \ Z– (j = , . . . , q);
z ∈ U;  α < ; p q +  ,
with
min (a), (s) > ;
λ >  if (b) > 
and
s ∈ C;
a ∈ C \ Z–
if b = .
(.)
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
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s,a,λ,∗
In this paper, we systematically investigate the class S(λ
(α) of analytic functions
p ),(μq ),b
defined above by means of the new generalized Srivastava-Attiya convolution operas,a,λ
. Especially, we derive coefficient inequalities, distortion theorems, extreme
tor J(λ
p ),(μq ),b
points and the Fekete-Szegö problem for this new function class.
2 Coefficient inequalities
Theorem  Let α ∈ [, ). If f (z) ∈ A satisfies the following equality
p
∞
(k – α) j= (λj + )k– a +  s (a + k, b, s, λ) |ak |  – α,
k! q (μ + ) a + k (a + , b, s, λ) j
j=
k=
(.)
k–
then
s,a,λ,∗
f ∈ S(λ
(α).
p ),(μq ),b
Proof Suppose that inequality (.) holds for α ∈ [, ). Let us define the function F(z) by
F(z) :=
s,a,λ
z(J(λ
) (f )(z)
p ),(μq ),b
s,a,λ
J(λ
(f )(z)
p ),(μq ),b
–α
f (z) ∈ A .
(.)
It is sufficient to prove that
F(z) –  F(z) +  < 
(z ∈ U)
(.)
s,a,λ,∗
(α).
to prove that f (z) ∈ S(λ
p ),(μq ),b
In fact, we have that
s,a,λ
z(J(λ
),(μ ),b ) (f )(z)
p
q
– α – 
(f )(z)
F(z) –  J(λs,a,λ
),(μ
F (z) := = p q ),b
F(z) +  z(J s,a,λ
(f )(z)
)
(λp ),(μq ),b
–α+
s,a,λ
J(λ ),(μ ),b (f )(z)
p
q
= p
∞
j= (λj +)k– a+ s (a+k,b,s,λ) (α+–k)
( ) ( (a+,b,s,λ) ) k! ak zk k= q (μj +)
k– a+k
j=
,
∞ pj= (λj +)k– a+ s (a+k,b,s,λ) (α––k) k ( – α)z – k= q (μ +) ( a+k ) ( (a+,b,s,λ) ) k! ak z
j
k–
αz +
j=
and thus
F (z) α|z| +
p
∞
(λj +)k–
j=
k= | q (μ +)
j
j=
( – α)|z| –
α+
∞
( – α) –
a+ s
||( a+k
) ||( (a+k,b,s,λ)
)|| (α+–k)
||ak | · |z|k
(a+,b,s,λ)
k!
p
j= (λj +)k–
a+ s
||( a+k
) ||( (a+k,b,s,λ)
)|| (α––k)
||ak | · |z|k
k= | q (μj +)
(a+,b,s,λ)
k!
k–
j=
∞
p
(λj +)k–
j=
k= | q (μ +)
j=
<
k–
∞
p
j
k–
(λj +)k–
j=
k= | q (μ +)
j=
provided that (.) is satisfied.
a+ s
||( a+k
) ||( (a+k,b,s,λ)
)| (k–α–)
|ak |
(a+,b,s,λ)
k!
j
k–
,
a+ s
||( a+k
) ||( (a+k,b,s,λ)
)| (k–α+)
|ak |
(a+,b,s,λ)
k!
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
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The next theorem aims to provide coefficient inequalities for functions f (z) belonging
s,a,λ,∗
(α).
to the class S(λ
p ),(μq ),b
s,a,λ,∗
Theorem  Let α ∈ [, ). If f (z) ∈ S(λ
(α), then
p ),(μq ),b
( – α) a + k s (a + , b, s, λ) k –  a +  (a + k, b, s, λ) k– q
j= (μj + )k– ( – α)
+
k ∈ N \ {} .
· p
j–
j= (λj + )k– j=
|ak | k!
(.)
The result is sharp.
Proof Let
s,a,λ
z(J(λ
) (f )(z)
p ),(μq ),b
p(z) :=
s,a,λ
J(λ
(f )(z)
p ),(μq ),b
–α
=  + c z + c z + · · · .
–α
Then p(z) is analytic and
p() =  and
p(z) >  (z ∈ U).
We note easily that
s,a,λ
s,a,λ
z J(λ
(f )(z) = ( – α)p(z) + α J(λ
(f )(z).
p ),(μq ),b
p ),(μq ),b
With the help of (.), we find
p
(k – ) j= (λj + )k– a +  s (a + k, b, s, λ)
ak
q
k!
(a + , b, s, λ)
j= (μj + )k– a + k
k– p
a +  s (a + m, b, s, λ) am ck–m
j= (λj + )m–
(.)
= ( – α) · ck– +
q
(a + , b, s, λ)
m!
j= (μj + )m– a + m
m=
for k ∈ N \ {}.
By making use of the Carathéodory lemma [, p.], we have
p
(k – ) j= (λj + )k– a +  s (a + k, b, s, λ) · |ak |
k! qj= (μj + )k– a + k (a + , b, s, λ) ( – α)
k– p
j= (λj + )m– a +  s (a + m, b, s, λ) |am |
· +
q (μ + ) a + m (a + , b, s, λ) m! .
m=
j=
j
(.)
m–
We have to prove that inequality (.) holds true for k ∈ N \ {}. We will proceed by the
principle of mathematical induction. If k =  in (.), we obtain
q
j= (μj + ) a +  s (a + , b, s, λ) |a | ( – α) p
a +  (a + , b, s, λ) .
j= (λj + )
(.)
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
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Now suppose that (.) is satisfied for k n. Then, from (.) and (.), we have that
p
s (a + n + , b, s, λ) n j= (λj + )n a+
· |an+ |
(n + )! qj= (μj + )n a + n +  (a + , b, s, λ)
n p
j= (λj + )m– a +  s (a + m, b, s, λ) |am |
q
( – α)  +
a + m (a + , b, s, λ) m!
j= (μj + )m–
m=
n
m–
( – α)
( – α) +
( – α)  +
m –  j=
j–
m=
( – α)
n ( – α)
+
,
j–
j=
(.)
whence
( – α) a + k s (a + , b, s, λ) |ak | k!
k –  a +  (a + k, b, s, λ) q
k– j= (μj + )k– ( – α)
· p
k ∈ N \ {} .

+
j–
j= (λj + )k– j=
The result is sharp for the function f (z) given by
( – α) a + k s (a + , b, s, λ)
f (z) = z +
k–
a+
(a + k, b, s, λ)
q
k– ( – α) k j= (μj + )k–
k ∈ N \ {} .
+
z
· p
j–
j= (λj + )k– j=
(.)
λ,∗
3 Distortion inequalities for the function class S(s,a,
λp ),(μq ),b (α )
In this section, we establish distortion inequalities for functions belonging to the class
s,a,λ,∗
S(λ
(α). These inequalities are given in the following theorem.
p ),(μq ),b
s,a,λ,∗
(α) and  α < . Then
Theorem  Let f (z) ∈ S(λ
p ),(μq ),b
r – ( – α)r
∞
k! a + k s (a + , b, s, λ) k –  a +  (a + k, b, s, λ) k=
k– q
j= (μj + )k– ( – α)

+
· p
j–
j= (λj + )k– j=
f (z)
∞
k! a + k s (a + , b, s, λ) r + ( – α)r
k –  a +  (a + k, b, s, λ) 
k=
k– q
j= (μj + )k– ( – α)

+
|z| = r < 
· p
j–
j= (λj + )k– j=
(.)
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
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and
∞
k · k! a + k s (a + , b, s, λ)  – ( – α)r
k –  a +  (a + k, b, s, λ) k=
k– q
j= (μj + )k– ( – α)

+
· p
j–
j= (λj + )k– j=
f (z)
∞
k · k! a + k s (a + , b, s, λ)  + ( – α)r
k –  a +  (a + k, b, s, λ) k=
q
k– j= (μj + )k– ( – α)
· p
+
|z| = r <  .
j
–

(λ
+
)
k– j=
j= j
(.)
Proof Let f (z) ∈ A be given by (.). Then, making use of Theorem , we find
∞
f (z) |z| +
|ak | · zk k=
r + ( – α)r
∞
k! a + k s (a + , b, s, λ) k –  a +  (a + k, b, s, λ) k=
k– q
j= (μj + )k– ( – α)
|z| = r < 

+
· p
j–
j= (λj + )k– j=
(.)
and
∞
f (z) |z| –
|ak | · zk k=
r – ( – α)r
∞
k! a + k s (a + , b, s, λ) k –  a +  (a + k, b, s, λ) k=
k– q
j= (μj + )k– ( – α)
+
|z| = r <  .
· p
j
–

j= (λj + )k– j=
(.)
From (.), we also have that
∞
f (z)  +
k · |ak | · zk– k=
 + ( – α)r
∞
k · k! a + k s (a + , b, s, λ) k –  a +  (a + k, b, s, λ) k=
k– q
j= (μj + )k– ( – α)
|z| = r < 

+
· p
j–
j= (λj + )k– j=
(.)
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
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and
∞
f (z)  –
k · |ak | · zk k=
∞
k · k! a + k s (a + , b, s, λ)  – ( – α)r
k –  a +  (a + k, b, s, λ) k=
k– q
j= (μj + )k– ( – α)

+
|z| = r <  .
· p
j–
j= (λj + )k– j=
(.)
We thus obtain the results (.) and (.) asserted by Theorem .
4 Extreme points
s,a,λ,∗
This section is devoted to presenting the extreme points of the function class S(λ
(α).
p ),(μq ),b
s,a,λ,∗
s,a,λ,∗
Let S(λp ),(μq ),b (α) be the subclass of S(λp ),(μq ),b (α) that consists in all functions f (z) ∈ A,
s,a,λ,∗
which satisfy inequality (.). Then the extreme points of S(λ
(α) are given by the
p ),(μq ),b
following theorem.
Theorem  Let
f (z) := z
(.)
p
k!( – α) j= (μj + )k– a + k s (a + , b, s, λ) k
fk (z) := z +
z
(k – α) qj= (λj + )k– a +  (a + k, b, s, λ) k ∈ N \ {} .
(.)
and
Then
s,a,λ,∗
(α) ( α < )
f (z) ∈ S(λ
p ),(μq ),b
if and only if it can be expressed in the following form:
f (z) =
∞
γk fk (z)
k=
γk > ;
∞
γk =  .
(.)
k=
Proof Suppose that
f (z) =
∞
γk fk (z)
k=
p
k!( – α) j= (μj + )k– γk
=z+
(k – α) qj= (λj + )k– k=
a + k s (a + , b, s, λ) k
z .
· a +  (a + k, b, s, λ) ∞
(.)
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
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Then
p
∞
(k – α) j= (λj + )k– a +  s (a + k, b, s, λ) k! q (μ + ) a + k (a + , b, s, λ) k=
j=
j
k–
p
k!( – α) j= (μj + )k– a + k s (a + , b, s, λ) · γk
(k – α) qj= (λj + )k– a +  (a + k, b, s, λ) = ( – α)
∞
γk = ( – α)( – γ )
k=
 – α.
(.)
s,a,λ,∗
Thus, by the definition of the function class S(λ
(α), we have
p ),(μq ),b
s,a,λ,∗
(α) ( α < ).
f ∈ S(λ
p ),(μq ),b
Conversely, if
s,a,λ,∗
f ∈ S(λ
(α) ( α < ),
p ),(μq ),b
then, by using (.), we may set
p
(k – α) j= (λj + )k– a +  s (a + k, b, s, λ) |ak |
γk =
( – α)k! qj= (μj + )k– a + k (a + , b, s, λ) k ∈ N \ {} ,
(.)
which implies that
f (z) =
∞
γk fk .
k=
The proof of Theorem  is thus completed.
5 The Fekete-Szegö problem
In this section, we shall obtain the Fekete-Szegö inequality for functions in the class
s,a,λ,∗
S(λ
(α) when
p ),(μq ),b
s > ,
a>
and  α < .
We need to recall an important lemma due to Ma and Minda [] in order to prove our
result involving Fekete-Szegö inequality.
Lemma  If
p(z) =  + c z + c z + · · ·
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
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is an analytic function in U such that
p(z) >  (z ∈ U),
then
c – νc 
⎧
⎪
⎪
⎨–ν +  (ν ),
( ν ),

⎪
⎪
⎩
ν – 
(ν ).
When ν <  or ν > , the equality holds true if and only if
p(z) =
+z
–z
(.)
or one of its rotations. If  < ν < , then the equality holds true if and only if
p(z) =
 + z
 – z
(.)
or one of its rotations. If ν = , then the equality holds true if and only if
p(z) =
+ω

+z
–z
–ω
–z
+

+z
( ω )
(.)
or one of its rotations. If ν = , then the equality holds true if and only if p(z) is the reciprocal
of one of the functions such that the equality holds true in the case ν = .
Theorem  Let
s > ,
a > ,
α<
and
λj > – (j = , . . . , p),
μj > – (j = , . . . , q).
s,a,λ,∗
If f ∈ S(λ
(α), then
p ),(μq ),b
a – τ a 
⎧
q
⎪
j= (μj +)
⎪
p
(
–
α)
⎪
⎪
⎪
j= (λj +)
⎪
⎪
⎪
a+ s (a+,b,s,λ)
⎪
⎪ · ( a+ ) ( (a+,b,s,λ) )(–ν + ) (τ σ ),
⎪
⎪
q
⎪
⎪
(μj +)
⎪
⎨( – α) j=
p
(λ +)
j=
j

⎪
(a+,b,s,λ)
⎪
· ( a+
)s ( (a+,b,s,λ)
)
⎪
⎪
a+
⎪
⎪
q
⎪
(μ
+)
⎪

j
⎪
⎪
( – α) j=
p
⎪
⎪
j= (λj +)
⎪
⎪
⎪
⎩ · ( a+ )s ( (a+,b,s,λ) )(ν – )
a+
(a+,b,s,λ)
(σ τ σ ),
(τ σ ),
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
Page 13 of 15
where
ν := ( – α)
q p τ λj +  μj + 
a+ s a+ s
 j= λj +  j= μj + 
a+
a+
(a + , b, s, λ)
·
(a + , b, s, λ)
(a + , b, s, λ)
– ,
(a + , b, s, λ)
(.)
q p  λj +  μj + 
a+ s
σ =
 j= λj +  j= μj + 
a+
(a + , b, s, λ)
·
(a + , b, s, λ)
(a + , b, s, λ)
(a + , b, s, λ)
(.)
and
σ =
q p a+ s
( – α) λj +  μj + 
( – α) j= λj +  j= μj + 
a+
·
(a + , b, s, λ)
(a + , b, s, λ)
(a + , b, s, λ)
.
(a + , b, s, λ)
(.)
=  + c z + c z + · · · .
(.)
The result is sharp.
s,a,λ,∗
(α), let
Proof For f ∈ S(λ
p ),(μq ),b
s,a,λ
z(J(λ
) (f )(z)
p ),(μq ),b
p(z) =
s,a,λ
J(λ
(f )(z)
p ),(μq ),b
–α
–α
Then, with the help of (.), we have
q
+ ) a +  s (a + , b, s, λ)
(a + , b, s, λ)
j= (λj + ) a + 
j= (μj
a = ( – α)c p
(.)
and
q
+ ) a +  s (a + , b, s, λ) c + ( – α)c .
a+
(a + , b, s, λ)
j= (λj + )
j= (μj
a = ( – α) p
Therefore, we find
q
a – τ a
=
j= (μj + )
( – α) p
j= (λj + )
a+
a+
s (a + , b, s, λ) c + ( – α)c
(a + , b, s, λ)
q
– τ ( – α) c
q
=

a +  s (a + , b, s, λ) 
j= (μj + )
p

a+
(a + , b, s, λ)
j= (λj + )
j= (μj + )
( – α) p
j= (λj + )
a+
a+
s (a + , b, s, λ)
(a + , b, s, λ)
c – c ( – α)
(.)
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
·
Page 14 of 15
q p a+ s a+ s
τ λj +  μj + 
 j= λj +  j= μj + 
a+
a+
(a + , b, s, λ)
·
(a + , b, s, λ)
(a + , b, s, λ)
– .
(a + , b, s, λ)
(.)
We thus write
q
+ ) a +  s (a + , b, s, λ) c – νc ,
a+
(a + , b, s, λ)
j= (λj + )
j= (μj
a – τ a = ( – α) p
(.)
where
q p τ λj +  μj + 
a+ s a+ s
ν : = ( – α)
 j= λj +  j= μj + 
a+
a+
(a + , b, s, λ)
·
(a + , b, s, λ)
(a + , b, s, λ)
– .
(a + , b, s, λ)
(.)
The result asserted by Theorem  follows by applying Lemma .
Moreover, if τ < σ or τ > σ , then the equality holds true if and only if
s,a,λ
J(λ
(f )(z) =
p ),(τq ),b
z
( – eiθ z)(–α)
(θ ∈ R).
(.)
For σ < τ < σ , the equality holds true if and only if
s,a,λ
J(λ
(f )(z) =
p ),(τq ),b
z
( – eiθ z)–α
(θ ∈ R).
(.)
If τ = σ , then the equality holds true if and only if
[(+ω)/] [(–ω)/]
z
z
( – eiθ z)(–α)
( + eiθ z)(–α)
z
=
(θ ∈ R;  ω ).
iθ
+ω
[( – e z) ( + eiθ z)–ω ]–α
s,a,λ
J(λ
(f )(z) =
p ),(τq ),b
(.)
s,a,λ
(f )(z) satisfies the folFinally, when τ = σ , the equality holds true if and only if J(λ
p ),(τq ),b
lowing condition:
s,a,λ
) (f )(z)
z(J(λ
p ),(τq ),b
s,a,λ
J(λ
(f )(z)
p ),(τq ),b
= ( – α)p(z) + α,
(.)
where

=
p(z)
+ω

+z
–z
–ω
–z
+

+z
( < ω < ).
(.)
We conclude this paper by mentioning that, by suitably specializing the parameters involved, our main results (Theorem  to Theorem ) would yield a number of (known or
Srivastava and Gaboury Journal of Inequalities and Applications (2015) 2015:39
Page 15 of 15
new) results for much simpler function classes, which were investigated in several earlier
works by employing many special cases of the new generalized Srivastava-Attiya operator.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors completed the paper together. All authors read and approved the final manuscript.
Author details
1
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada.
2
Department of Mathematics and Computer Science, University of Québec at Chicoutimi, Chicoutimi, Québec G7H 2B1,
Canada.
Received: 30 September 2014 Accepted: 22 January 2015
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