R E S E A R C H
Open Access
A new class of analytic functions defined by
means of a generalization of the
Srivastava-Attiya operator
Hari M Srivastava
1and Sebastien Gaboury
2**Correspondence:
2Department 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 convolution operatorJs(λ,a,λ
p),(μq),bwhich 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
LetAdenote the class of functionsf(z) normalized by
f(z) =z+ ∞
k=
akzk, (.)
which are analytic in the open unit disk
U=z:z∈Cand|z|< .
A functionf(z) in the classAis said to be in the classS∗(α) of starlike functions of order
αinUif 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+
∞
k=
+a k+a
s
akzk, (.)
wherez∈U,a∈C\Z–
,s∈Candf ∈A.
In fact, the linear operatorJs,a(f) can be written as
Js,a(f)(z) :=Gs,a(z)∗f(z) (.)
in terms of the Hadamard product (or convolution), whereGs,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 Hurwitz-Lerch 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∈Cwhen|z|< ;(s) > when|z|= . (.)
Recently, a new family of λ-generalized Hurwitz-Lerch zeta functions was inves-tigated by Srivastava [] (see also [–]). Srivastava considered the following func-tion:
λ(ρ,...,λ,...,ρpp;μ,σ,...,μ,...,σqq)(z,s,a;b,λ)
=
λ(s)· ∞
n=
p j=(λj)nρj
(a+n)s·q j=(μj)nσj
H,,
(a+n)bλ (s, ),
,
λ
zn
n!
min(a),(s)> ;(b) > ;λ> , (.)
where
λj∈C(j= , . . . ,p) andμj∈C\Z–(j= , . . . ,q);ρj> (j= , . . . ,p);
σj> (j= , . . . ,q); + q
j=
σj– p
j=
ρj
and the equality in the convergence condition holds true for suitably bounded values of |z|given by
|z|<∇:=
p
j=
ρj–ρj
·
q
j=
σjσj
.
Here, and for the remainder of this paper, (λ)κdenotes the Pochhammer symbol defined, in terms of the gamma function, by
(λ)κ:=
(λ+κ)
(λ) =
⎧ ⎨ ⎩
λ(λ+ )· · ·(λ+n– ) (κ=n∈N;λ∈C),
it being understoodconventionallythat ():= and assumedtacitlythat the-quotient exists (see, for details, [, p.et seq.]).
Definition TheH-function involved in the right-hand side of (.) is the well-known Fox’sH-function [, Definition .] (see also [, ]) defined by
Hpm,q,n(z) =H
m,n
p,q
z(a,A), . . . , (ap,Ap) (b,B), . . . , (bq,Bq)
= πi
L(s)z
–sds z∈C\ {};arg(z)<π, (.)
where
(s) =
m
j=(bj+Bjs)· n
j=( –aj–Ajs) p
j=n+(aj+Ajs)· q
j=m+( –bj–Bjs)
,
an empty product is interpreted as ,m,n,pandqare integers such that mq, np,Aj> (j= , . . . ,p),Bj> (j= , . . . ,q),aj∈C(j= , . . . ,p),bj∈C(j= , . . . ,q) andL
is a suitable Mellin-Barnes type contour separating the poles of the gamma functions
(bj+Bjs) m
j=
from the poles of the gamma functions
( –aj+Ajs) n
j=.
It is worthy to mention that using the fact that [, p., Remark ]
lim
b→
H,,
(a+n)bλ (s, ),
,
λ
=λ(s) (λ> ), (.)
equation (.) reduces to
(ρλ,...,ρp,σ,...,σq)
,...,λp;μ,...,μq (z,s,a; ,λ) :=
(ρ,...,ρp,σ,...,σq)
λ,...,λp;μ,...,μq (z,s,a)
= ∞
n=
p j=(λj)nρj
(a+n)s·q j=(μj)nσj
zn
n!. (.)
Definition The function(ρλ,...,ρp,σ,...,σq)
,...,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 Srivastavaet al.[, p., Eq. (.)] defined by
(ρλ,...,ρp,σ,...,σq)
,...,λp;μ,...,μq (z,s,a) :=
∞
n=
p j=(λj)nρj
(a+n)s·q j=(μj)nσj
zn n!
p,q∈N;λj∈C(j= , . . . ,p);a,μj∈C\Z–(j= , . . . ,q);
> – whens,z∈C;
= – ands∈Cwhen|z|<∇∗;
= – and() >
when|z|=∇
∗ (.)
with
∇∗:=
p
j=
ρj–ρj
·
q
j=
σjσj
, (.)
:=
q
j=
σj– p
j=
ρj and :=s+ q
j=
μj– p
j=
λj+
p–q
. (.)
We propose to consider the following linear operator
J(λs,ap,λ),(μq),b(f) :A→A,
defined by
J(λs,ap,λ),(μq),b(f)(z) =G(λs,ap,λ),(μq),b(z)∗f(z), (.)
where∗denotes the Hadamard product (or convolution) of analytic functions, and the functionGs(λ,a,λ
p),(μq),b(z) is given by
Gs(λ,ap,λ),(μq),b(z)
:=λ
q
j=(μj)(s)(a+ )s p
j=(λj)
·(a+ ,b,s,λ)–
·
(,...,,,...,)λ
,...,λp;μ,...,μq(z,s,a;b,λ) –
a–s
λ(s)(a,b,s,λ)
=z+ ∞
k=
p
j=(λj+ )k–
q
j=(μj+ )k–
a+ a+k
s
(a+k,b,s,λ)
(a+ ,b,s,λ)
zk
k! (.)
with
(a,b,s,λ) :=H,,
abλ(s, ),
,
λ
.
Combining (.) and (.), we obtain
J(λs,ap,λ),(μq),b(f)(z)
=z+ ∞
k=
p
j=(λj+ )k–
q
j=(μj+ )k–
a+ a+k
s
(a+k,b,s,λ)
(a+ ,b,s,λ)
ak
zk
k!
λj∈C(j= , . . . ,p) andμj∈C\Z–(j= , . . . ,q);pq+ ;z∈U
with
min(a),(s)> ; λ> if(b) >
and
s∈C; a∈C\Z– ifb= .
Remark It follows from (.) and (.) that the operatorJ(λs,ap,λ),(μq),(f) (special case of (.) whenb= ) can be defined fora∈C\Z–by the following limit relationship:
J(λs,,λ
p),(μq),(f)(z) :=alim→
J(λs,a,λ
p),(μq),(f)(z)
. (.)
We can see that the operatorJ(λs,ap,λ),(μq),bgeneralizes several recently investigated operators such as:
(i) Ifp= ,q= andb= , thenJ(λs,a,λ
,λ),(μ),=J s,a
λ,λ;μ, whereJ s,a
λ,λ;μis the linear operator introduced by Prajapat and Bulboacă [, p., Eq. (.)].
(ii) J(γs,a–,),(ν),,λ =Is
a,ν,γ, whereIas,ν,γ is the generalized operator recently studied by Noor
and Bukhari [, p., Eq. (.)]. (iii) J(γ,,λ–,),(ν),=Is
ν,γ, whereIν,γs is the Choi-Saigo-Srivastava operator [].
(iv) J(γs,a,),(γ),,λ =Js,a, whereJs,ais the Srivastava-Attiya operator [].
(v) J(γ–r,),(γ),,a,λ =I(r,a)(a,r∈Z), where the operatorI(r,a)is the one introduced by Cho and Srivastava [].
(vi) J(β,),(α+β),,a,λ =Qα
β(α,β> –), where the operatorQαβwas studied by Junget
al.[].
(vii) J(γ,a,),(γ),,λ =Ja(a–), whereJadenotes the Bernardi operator [].
(viii) J(γ,,λ,),(ν),=L(γ,ν), whereL(γ,ν)is the well-known Carlson-Shaffer operator []. (ix) J(,),(–γ,,λ ),=γz (γ < ), whereγz is the fractional integral operator
investigated by Owa and Srivastava [].
(x) J(λ,a–,...,λ,λ p–,),(μ–,...,μq–,),=H(λ, . . . ,λp;μ, . . . ,μq)(pq+ ), where the operatorH(λ, . . . ,λp;μ, . . . ,μq)is the Dziok-Srivastava operator [, ] which contains as special cases the Hohlov operator [] and the Ruscheweyh
operator [].
We say that a functionf ∈Ais in the classS(λs,ap,λ,∗),(μq),b(α) ifJ(λs,ap,λ),(μq),b(f) is in the class
S∗(α), that is, if
z(Js,a,λ (λp),(μq),b(f))
J(λs,ap,λ),(μq),b(f)
>α
λj∈C(j= , . . . ,p) andμj∈C\Z–(j= , . . . ,q);
z∈U; α< ;pq+ , (.)
with
min(a),(s)> ; λ> if(b) >
and
In this paper, we systematically investigate the classS(λs,ap,λ,∗),(μq),b(α) of analytic functions defined above by means of the new generalized Srivastava-Attiya convolution opera-torJ(λs,ap,λ),(μq),b. Especially, we derive coefficient inequalities, distortion theorems, extreme points and the Fekete-Szegö problem for this new function class.
2 Coefficient inequalities
Theorem Letα∈[, ).If f(z)∈Asatisfies the following equality ∞
k= (k–α)
k!
p
j=(λj+ )k–
q
j=(μj+ )k–
aa++ ks((aa++ ,k,bb,,ss,,λλ))|ak| –α, (.)
then
f ∈S(λs,a,λ,∗
p),(μq),b(α).
Proof Suppose that inequality (.) holds forα∈[, ). Let us define the functionF(z) by
F(z) :=z(J
s,a,λ (λp),(μq),b)
(f)(z)
J(λs,ap,λ),(μq),b(f)(z) –α f(z)∈A
. (.)
It is sufficient to prove that
FF((zz) – ) + < (z∈U) (.)
to prove thatf(z)∈S(λs,a,λ,∗
p),(μq),b(α).
In fact, we have that
F(z):=F(z) – F(z) +
=
z(J(s,a,λp),(λμq),b)(f)(z)
J(s,a,λp),(λμq),b(f)(z) –α–
z(J(s,a,λp),(λμq),b)(f)(z)
J(s,a,λλ p),(μq),b(f)(z)
–α+
=
αz+∞k=
p j=(λj+)k– q
j=(μj+)k–( a+
a+k)
s((a+k,b,s,λ) (a+,b,s,λ))
(α+–k)
k! akz
k
( –α)z–∞k=
p j=(λj+)k– q
j=(μj+)k–( a+
a+k)s(
(a+k,b,s,λ) (a+,b,s,λ))
(α––k)
k! akzk
,
and thus
F(z)
α|z|+∞k=|
p j=(λj+)k– q
j=(μj+)k–||( a+
a+k)s||(
(a+k,b,s,λ) (a+,b,s,λ))||
(α+–k)
k! ||ak| · |z|k
( –α)|z|–∞k=|
p j=(λj+)k– q
j=(μj+)k–||( a+
a+k)s||(
(a+k,b,s,λ) (a+,b,s,λ))||
(α––k)
k! ||ak| · |z|k
<
α+∞k=|
p j=(λj+)k– q
j=(μj+)k–||( a+
a+k)
s||((a+k,b,s,λ) (a+,b,s,λ))|
(k–α–)
k! |ak|
( –α) –∞k=|
p j=(λj+)k– q
j=(μj+)k–||( a+
a+k)s||(
(a+k,b,s,λ) (a+,b,s,λ))|
(k–α+)
k! |ak|
,
The next theorem aims to provide coefficient inequalities for functionsf(z) belonging to the classS(λs,ap,λ,∗),(μq),b(α).
Theorem Letα∈[, ).If f(z)∈S(λs,a,λ,∗
p),(μq),b(α),then
|ak|k!
( –α) k–
a+k
a+
s
(a+ ,b,s,λ)
(a+k,b,s,λ)
·
q
j=(μj+ )k–
p
j=(λj+ )k–
k–
j=
+( –α) j–
k∈N\ {}. (.)
The result is sharp.
Proof Let
p(z) :=
z(Js,a,(λp),(λμ q),b)(f)(z) J(s,a,λp),(λμq),b(f)(z) –α
–α = +cz+cz
+· · ·.
Thenp(z) is analytic and
p() = and p(z)> (z∈U).
We note easily that
z J(λs,ap,λ),(μq),b(f)(z) =( –α)p(z) +αJ(λs,ap,λ),(μq),b(f)(z).
With the help of (.), we find
(k– ) k!
p
j=(λj+ )k–
q
j=(μj+ )k–
a+ a+k
s
(a+k,b,s,λ)
(a+ ,b,s,λ)
ak
= ( –α)·
ck–+
k–
m=
p
j=(λj+ )m–
q
j=(μj+ )m–
a+ a+m
s
(a+m,b,s,λ)
(a+ ,b,s,λ)
amck–m
m!
(.)
fork∈N\ {}.
By making use of the Carathéodory lemma [, p.], we have
(k– ) k!
p
j=(λj+ )k–
q
j=(μj+ )k–
aa+ +ks(a+k,b,s,λ)
(a+ ,b,s,λ)
· |ak|
( –α)
·
+
k–
m=
p
j=(λj+ )m–
q
j=(μj+ )m–
aa++ ms(a+m,b,s,λ)
(a+ ,b,s,λ)
|am|
m!
. (.)
We have to prove that inequality (.) holds true fork∈N\ {}. We will proceed by the principle of mathematical induction. Ifk= in (.), we obtain
|a|( –α)
q
j=(μj+ ) p
j=(λj+ )
aa+ + s(a+ ,b,s,λ)
(a+ ,b,s,λ)
Now suppose that (.) is satisfied forkn. Then, from (.) and (.), we have that
n (n+ )!
p
j=(λj+ )n q
j=(μj+ )n
a+a+ n+ s(a+n+ ,b,s,λ)
(a+ ,b,s,λ)
· |an+|
( –α)
+
n
m=
p
j=(λj+ )m–
q
j=(μj+ )m–
aa++ ms((aa++ ,m,bb,,ss,,λλ))|am|
m!
( –α)
+
n
m=
( –α) m–
m–
j=
+( –α) j–
( –α)
n
j=
+( –α) j–
, (.)
whence
|ak|k!
( –α) k–
aa++ ks(a+ ,b,s,λ)
(a+k,b,s,λ)
·
q
j=(μj+ )k–
p
j=(λj+ )k–
k–
j=
+( –α) j–
k∈N\ {}.
The result is sharp for the functionf(z) given by
f(z) =z+( –α) k–
a+k a+
s
(a+ ,b,s,λ)
(a+k,b,s,λ)
·
q
j=(μj+ )k–
p
j=(λj+ )k–
k–
j=
+( –α) j–
zk k∈N\ {}. (.)
3 Distortion inequalities for the function classS(sλp,a,),(λ,μq∗ ),b(
α
)In this section, we establish distortion inequalities for functions belonging to the class
Ss,a,λ,∗
(λp),(μq),b(α). These inequalities are given in the following theorem.
Theorem Let f(z)∈S(λs,ap,λ,∗),(μq),b(α)andα< .Then
r– ( –α)r ∞
k= k! k–
aa++ ks((aa+ ,+k,bb,,ss,,λλ))
·
q
j=(μj+ )k–
p
j=(λj+ )k–
k–
j=
+( –α) j–
f(z)
r+ ( –α)r ∞
k= k! k–
aa++ ks((aa+ ,+k,bb,,ss,,λλ))
·
q
j=(μj+ )k–
p
j=(λj+ )k–
k–
j=
+( –α) j–
and
– ( –α)r ∞
k= k·k! k–
aa++ ks((aa+ ,+k,bb,,ss,,λλ))
·
q
j=(μj+ )k–
p
j=(λj+ )k–
k–
j=
+( –α) j–
f(z)
+ ( –α)r ∞
k= k·k! k–
aa++ ks(a+ ,b,s,λ)
(a+k,b,s,λ)
·
q
j=(μj+ )k–
p
j=(λj+ )k–
k–
j=
+( –α) j–
|z|=r< . (.)
Proof Letf(z)∈Abe given by (.). Then, making use of Theorem , we find
f(z)|z|+ ∞
k=
|ak| ·zk
r+ ( –α)r ∞
k= k! k–
aa++ ks((aa++ ,k,bb,,ss,,λλ))
·
q
j=(μj+ )k–
p
j=(λj+ )k–
k–
j=
+( –α) j–
|z|=r< (.)
and
f(z)|z|– ∞
k=
|ak| ·zk
r– ( –α)r ∞
k= k! k–
aa++ ks((aa++ ,k,bb,,ss,,λλ))
·
q
j=(μj+ )k–
p
j=(λj+ )k–
k–
j=
+( –α) j–
|z|=r< . (.)
From (.), we also have that
f(z) + ∞
k=
k· |ak| ·zk–
+ ( –α)r ∞
k= k·k! k–
aa++ ks(a+ ,b,s,λ)
(a+k,b,s,λ)
·
q
j=(μj+ )k–
p
j=(λj+ )k–
k–
j=
+( –α) j–
and
f(z) – ∞
k=
k· |ak| ·zk
– ( –α)r ∞
k= k·k! k–
aa++ ks(a+ ,b,s,λ)
(a+k,b,s,λ)
·
q
j=(μj+ )k–
p
j=(λj+ )k–
k–
j=
+( –α) j–
|z|=r< . (.)
We thus obtain the results (.) and (.) asserted by Theorem .
4 Extreme points
This section is devoted to presenting the extreme points of the function classS(λs,ap,λ,∗),(μq),b(α). LetS(λs,ap,λ,∗),(μq),b(α) be the subclass ofS(λs,ap,λ,∗),(μq),b(α) that consists in all functionsf(z)∈A, which satisfy inequality (.). Then the extreme points ofS(λs,ap,λ,∗),(μq),b(α) are given by the following theorem.
Theorem Let
f(z) :=z (.)
and
fk(z) :=z+
k!( –α) (k–α)
pj=(μj+ )k–
q
j=(λj+ )k–
aa++ ks(a+ ,b,s,λ)
(a+k,b,s,λ)
zk
k∈N\ {}. (.)
Then
f(z)∈S(λs,ap,λ,∗),(μq),b(α) (α< )
if and only if it can be expressed in the following form:
f(z) = ∞
k=
γkfk(z)
γk> ;
∞
k=
γk=
. (.)
Proof Suppose that
f(z) = ∞
k=
γkfk(z)
=z+ ∞
k=
γk
k!( –α) (k–α)
p
j=(μj+ )k–
q
j=(λj+ )k–
·
a+k a+
s
Then
∞
k= (k–α)
k!
p
j=(λj+ )k–
q
j=(μj+ )k–
aa++ ks((aa++ ,k,bb,,ss,,λλ))
·γk
k!( –α) (k–α)
p
j=(μj+ )k–
q
j=(λj+ )k–
aa++ ks(a+ ,b,s,λ)
(a+k,b,s,λ)
= ( –α) ∞
k=
γk= ( –α)( –γ)
–α. (.)
Thus, by the definition of the function classS(λs,ap,λ,∗),(μq),b(α), we have
f ∈S(λs,a,λ,∗
p),(μq),b(α) (α< ).
Conversely, if
f ∈S(λs,ap,λ,∗),(μq),b(α) (α< ),
then, by using (.), we may set
γk=
(k–α) ( –α)k!
p
j=(λj+ )k–
q
j=(μj+ )k–
aa++ ks((aa++ ,k,bb,,ss,,λλ))|ak|
k∈N\ {}, (.)
which implies that
f(z) = ∞
k=
γkfk.
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
Ss,a,λ,∗
(λp),(μq),b(α) when
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
is an analytic function inUsuch that
p(z)> (z∈U),
then
c–νc
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–ν+ (ν),
(ν),
ν– (ν).
Whenν< orν> ,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.Ifν= ,then the equality holds true if and only if
p(z) =
+ω
+z –z
+
–ω
–z +z
(ω) (.)
or one of its rotations.Ifν= ,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 caseν= .
Theorem Let
s> , a> , α<
and
λj> – (j= , . . . ,p), μj> – (j= , . . . ,q).
If f ∈S(λs,a,λ,∗
p),(μq),b(α),then
a–τa
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
( –α)
q j=(μj+) p
j=(λj+)
·(aa++)s((a+,b,s,λ)
(a+,b,s,λ))(–ν+ ) (τ σ), ( –α)
q j=(μj+) p
j=(λj+)
·(aa++)s((a+,b,s,λ)
(a+,b,s,λ)) (στσ), ( –α)
q j=(μj+) p
j=(λj+)
·(aa++)s((a+,b,s,λ)
where
ν:= ( –α)
τ p j=
λj+ λj+
q
j=
μj+ μj+
a+ a+
s
a+ a+
s
·
(a+ ,b,s,λ)
(a+ ,b,s,λ)
(a+ ,b,s,λ)
(a+ ,b,s,λ)
–
, (.)
σ= p j=
λj+ λj+
q
j=
μj+ μj+
a+ a+
s
·
(a+ ,b,s,λ)
(a+ ,b,s,λ)
(a+ ,b,s,λ)
(a+ ,b,s,λ)
(.)
and
σ=
( –α) ( –α)
p
j=
λj+ λj+
q
j=
μj+ μj+
a+ a+
s
·
(a+ ,b,s,λ)
(a+ ,b,s,λ)
(a+ ,b,s,λ)
(a+ ,b,s,λ)
. (.)
The result is sharp.
Proof Forf ∈S(λs,ap,λ,∗),(μq),b(α), let
p(z) =
z(J(s,a,λλ
p),(μq),b)(f)(z) Js,a,(λp),(λμ
q),b(f)(z)
–α
–α = +cz+cz
+· · ·. (.)
Then, with the help of (.), we have
a= ( –α)c
q
j=(μj+ ) p
j=(λj+ )
a+ a+
s
(a+ ,b,s,λ)
(a+ ,b,s,λ)
(.)
and
a= ( –α)
q
j=(μj+ )
p
j=(λj+ )
a+ a+
s
(a+ ,b,s,λ)
(a+ ,b,s,λ)
c+ ( –α)c. (.)
Therefore, we find
a–τa= ( –α)
q
j=(μj+ )
p
j=(λj+ )
a+ a+
s
(a+ ,b,s,λ)
(a+ ,b,s,λ)
c+ ( –α)c
– τ( –α)c
q
j=(μj+ ) p
j=(λj+ )
a+ a+
s
(a+ ,b,s,λ)
(a+ ,b,s,λ)
= ( –α)
q
j=(μj+ )
p
j=(λj+ )
a+ a+
s
(a+ ,b,s,λ)
(a+ ,b,s,λ)
·
τ
p
j=
λ j+ λj+
q
j=
μ
j+ μj+
a+ a+
sa+
a+
s
·
(a+ ,b,s,λ)
(a+ ,b,s,λ)
(a+ ,b,s,λ)
(a+ ,b,s,λ)
–
. (.)
We thus write
a–τa= ( –α)
q
j=(μj+ )
p
j=(λj+ )
a+ a+
s
(a+ ,b,s,λ)
(a+ ,b,s,λ) c–νc
, (.)
where
ν: = ( –α)
τ
p
j=
λ j+ λj+
q
j=
μ
j+ μj+
a+ a+
s
a+ a+
s
·
(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, ifτ<σorτ>σ, then the equality holds true if and only if
J(λs,ap,λ),(τq),b(f)(z) = z
( – eiθz)(–α) (θ∈R). (.)
Forσ<τ<σ, the equality holds true if and only if
J(λs,a,λ
p),(τq),b(f)(z) =
z
( – eiθz)–α (θ∈R). (.)
Ifτ=σ, then the equality holds true if and only if
J(λs,ap,λ),(τq),b(f)(z) =
z ( – eiθz)(–α)
[(+ω)/] z ( + eiθz)(–α)
[(–ω)/]
= z
[( – eiθz)+ω( + eiθz)–ω]–α (θ∈R; ω). (.)
Finally, whenτ =σ, the equality holds true if and only ifJ(λs,ap,λ),(τq),b(f)(z) satisfies the fol-lowing condition:
z(J(λs,a,λ
p),(τq),b)
(f)(z)
J(λs,ap,λ),(τq),b(f)(z) = ( –α)p(z) +α, (.)
where
p(z)=
+ω
+z –z
+
–ω
–z +z
( <ω< ). (.)
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
1Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada. 2Department 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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