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: s1gabour@uqac.ca
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 operator Js,a,λ(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
LetA denote the class of functions f (z) normalized by
f(z) = z +
∞
k=
akzk, (.)
which are analytic in the open unit disk U =z: z∈ C and |z| < .
A function f (z) in the classA is said to be in the class S∗(α) of starlike functions of order
αinU 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 + ∞ k= + a k+ a s akzk, (.) 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 Com-mons Attribution License (http://creativecomCom-mons.org/licenses/by/4.0), which permits unrestricted use, distribution, and repro-duction in any medium, provided the original work is properly credited.
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 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∈ C when |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: (ρλ,...,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) and μj∈ 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), (κ = ; λ∈ C \ {}), (.)
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
Hp,qm,n(z) = Hp,qm,n z (a, A), . . . , (ap, Ap) (b, B), . . . , (bq, Bq) = π i 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, 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) 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,...,λ,...,ρpp;μ,σ,...,μ,...,σqq)(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
(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); ρj, σk∈ R+(j = , . . . , p; k = , . . . , q);
> – when s, z∈ C; = – and s∈ C when |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,a,λp),(μq),b(f ) :A → A, defined by
J(λs,a,λp),(μq),b(f )(z) = G(λs,a,λp),(μq),b(z)∗ f (z), (.) where∗ denotes the Hadamard product (or convolution) of analytic functions, and the function Gs(λ,a,λ p),(μq),b(z) is given by Gs(λ,a,λp),(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,a,λp),(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) and μj∈ C \ Z–(j = , . . . , q); p q + ; z ∈ U , (.)
with
min(a), (s)> ; λ> if(b) > and
s∈ C; a∈ C \ Z– if b = .
Remark It follows from (.) and (.) that the operator J(λs,a,λp),(μq),(f ) (special case of (.) when b = ) can be defined for a∈ C \ Z–by the following limit relationship:
J(λs,,λ p),(μq),(f )(z) := lima→ J(λs,a,λ p),(μq),(f )(z) . (.)
We can see that the operator J(λs,a,λp),(μq),bgeneralizes several recently investigated operators such as:
(i) If p = , q = and b = , then J(λs,a,λ
,λ),(μ),= J
s,a
λ,λ;μ, where J
s,a
λ,λ;μis the linear operator introduced by Prajapat and Bulboacă [, p., Eq. (.)].
(ii) J(γ –,),(ν),s,a,λ = Is
a,ν,γ, where Ias,ν,γ is the generalized operator recently studied by Noor
and Bukhari [, p., Eq. (.)]. (iii) J(γ –,),(ν),,,λ = Is
ν,γ, where Iνs,γ is the Choi-Saigo-Srivastava operator [].
(iv) J(γ ,),(γ ),s,a,λ = Js,a, where Js,ais the Srivastava-Attiya operator [].
(v) J(γ ,),(γ ),–r,a,λ = I(r, a)(a , r ∈ Z), where the operator I(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 Jung et
al.[].
(vii) J(γ ,),(γ ),,a,λ = Ja(a –), where Jadenotes the Bernardi operator [].
(viii) J(γ ,),(ν),,,λ =L(γ , ν), where L(γ , ν) is the well-known Carlson-Shaffer operator []. (ix) J(,),(–γ ),,,λ = γz ( γ < ), where γz is the fractional integral operator
investigated by Owa and Srivastava []. (x) J(λ,a,λ
–,...,λp–,),(μ–,...,μq–,),= H(λ, . . . , λp; μ, . . . , μq)(p q + ), where the operator H(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 function f ∈A is in the class S(λs,a,λ,∗p),(μq),b(α) if J(λs,a,λp),(μq),b(f ) is in the class
S∗(α), that is, if z(Js,a,λ (λp),(μq),b(f )) J(λs,a,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 = .
In this paper, we systematically investigate the classS(λs,a,λ,∗p),(μq),b(α) of analytic functions defined above by means of the new generalized Srivastava-Attiya convolution opera-tor J(λs,a,λp),(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 Let α∈ [, ). If f (z) ∈A satisfies the following equality
∞ k= (k – α) k! p j=(λj+ )k– q j=(μj+ )k– aa+ k+ s (a + k, b, s, λ)(a + , b, s, λ) |ak| – α, (.) then f ∈S(λs,a,λ,∗ p),(μq),b(α).
Proof Suppose that inequality (.) holds for α∈ [, ). Let us define the function F(z) by
F(z) :=z(J s,a,λ (λp),(μq),b) (f )(z) J(λs,a,λp),(μq),b(f )(z) – α f(z)∈A. (.)
It is sufficient to prove that
FF(z) – (z) + < (z ∈ U) (.) to prove that f (z)∈S(λs,a,λ,∗
p),(μq),b(α). In fact, we have that
F(z) := F(z) – F(z) + = z(Js,a,λ (λp),(μq),b)(f )(z) J(λp),(μq),bs,a,λ (f )(z) – α – z(J(λp),(μq),bs,a,λ )(f )(z) J(λp),(μq),bs,a,λ (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 functions f (z) belonging to the classS(λs,a,λ,∗p),(μq),b(α).
Theorem Let α∈ [, ). 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,λ (λp),(μq),b)(f )(z) J(λp),(μq),bs,a,λ (f )(z) – α – α = + cz+ cz +· · · .
Then p(z) is analytic and
p() = and p(z)> (z∈ U). We note easily that
z J(λs,a,λp),(μq),b(f )(z) =( – α)p(z) + αJ(λs,a,λp),(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! (.) for k∈ 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+ m+ s (a + m, b, s, λ) (a + , b, s, λ) |am| 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
|a| ( – α) q j=(μj+ ) p j=(λj+ ) aa+ + s (a + , b, s, λ) (a + , b, s, λ) . (.)
Now suppose that (.) is satisfied for k n. Then, from (.) and (.), we have that n (n + )! p j=(λj+ )n q j=(μj+ )n a+ n + a+ s (a + n + , b, s, λ) (a + , b, s, λ) · |an+| ( – α) + n m= p j=(λj+ )m– q j=(μj+ )m– aa+ m+ s (a + m, b, s, λ)(a + , b, s, λ) |am| m! ( – α) + n m= ( – α) m– m– j= +( – α) j– ( – α) n j= +( – α) j– , (.) whence |ak| k! ( – α) k– aa+ k+ 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 for the function f (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(λs,a,λ,∗
p),(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,a,λ,∗p),(μq),b(α) and α < . Then
r– ( – α)r ∞ k= k! k– aa+ k+ s (a + k, b, s, λ)(a + , b, s, λ) · q j=(μj+ )k– p j=(λj+ )k– k– j= +( – α) j– f(z) r + ( – α)r ∞ k= k! k– aa+ k+ s (a + k, b, s, λ)(a + , b, s, λ) · q j=(μj+ )k– p j=(λj+ )k– k– j= +( – α) j– |z| = r < (.)
and – ( – α)r ∞ k= k· k! k– aa+ k+ s (a + k, b, s, λ)(a + , b, s, λ) · q j=(μj+ )k– p j=(λj+ )k– k– j= +( – α) j– f(z) + ( – α)r ∞ k= k· k! k– aa+ k+ s (a + , b, s, λ) (a + k, b, s, λ) · q j=(μj+ )k– p j=(λj+ )k– k– j= +( – α) j– |z| = r < . (.)
Proof Let f (z)∈A be given by (.). Then, making use of Theorem , we find f(z) |z| + ∞ k= |ak| · zk r + ( – α)r ∞ k= k! k– aa+ k+ s (a + , b, s, λ)(a + k, b, s, λ) · 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+ k+ s (a + k, b, s, λ)(a + , b, s, λ) · 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+ k+ s (a + , b, s, λ) (a + k, b, s, λ) · q j=(μj+ )k– p j=(λj+ )k– k– j= +( – α) j– |z| = r < (.)
and f(z) – ∞ k= k· |ak| · zk – ( – α)r ∞ k= k· k! k– aa+ k+ s (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,a,λ,∗p),(μq),b(α). Let S(λs,a,λ,∗p),(μq),b(α) be the subclass ofS(λs,a,λ,∗p),(μq),b(α) that consists in all functions f (z)∈A, which satisfy inequality (.). Then the extreme points of S(λs,a,λ,∗p),(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+ k+ s (a + , b, s, λ) (a + k, b, s, λ) zk k∈ N \ {}. (.) Then f(z)∈ S(λs,a,λ,∗p),(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 (a + , b, s, λ)(a + k, b, s, λ) zk. (.)
Then ∞ k= (k – α) k! p j=(λj+ )k– q j=(μj+ )k– aa+ k+ s (a + k, b, s, λ)(a + , b, s, λ) · γk k!( – α) (k – α) p j=(μj+ )k– q j=(λj+ )k– aa+ k+ s (a + , b, s, λ) (a + k, b, s, λ) = ( – α) ∞ k= γk= ( – α)( – γ) – α. (.)
Thus, by the definition of the function class S(λs,a,λ,∗p),(μq),b(α), we have
f ∈ S(λs,a,λ,∗
p),(μq),b(α) ( α < ). Conversely, if
f ∈ S(λs,a,λ,∗p),(μq),b(α) ( α < ), then, by using (.), we may set
γk= (k – α) ( – α)k! p j=(λj+ )k– q j=(μj+ )k– aa+ k+ s (a + k, b, s, λ)(a + , b, s, λ) |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 inU 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). If f ∈S(λs,a,λ,∗ p),(μq),b(α), then a– τ a ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ( – α) q j=(μj+) p j=(λj+) · (a+ a+)s( (a+,b,s,λ) (a+,b,s,λ))(–ν + ) (τ σ), ( – α) q j=(μj+) p j=(λj+) · (a+ a+)s( (a+,b,s,λ) (a+,b,s,λ)) (σ τ σ), ( – α) q j=(μj+) p j=(λj+) · (a+ a+)s( (a+,b,s,λ) (a+,b,s,λ))(ν – ) (τ σ),
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 For f ∈S(λs,a,λ,∗p),(μq),b(α), let
p(z) = z(J(λp),(μq),bs,a,λ )(f )(z) Js,a,λ (λp),(μq),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– τ a= ( – 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) c– c( – α)
· 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– τ a= ( – 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, if τ < σor τ > σ, then the equality holds true if and only if
J(λs,a,λp),(τq),b(f )(z) = z
( – eiθz)(–α) (θ∈ R). (.)
For σ< τ < σ, the equality holds true if and only if
J(λs,a,λ
p),(τq),b(f )(z) =
z
( – eiθz)–α (θ∈ R). (.)
If τ = σ, then the equality holds true if and only if
J(λs,a,λp),(τq),b(f )(z) = z ( – eiθz)(–α) [(+ω)/] z ( + eiθz)(–α) [(–ω)/] = z [( – eiθz)+ω( + eiθz)–ω]–α (θ∈ R; ω ). (.)
Finally, when τ = σ, the equality holds true if and only if J(λs,a,λp),(τq),b(f )(z) satisfies the
fol-lowing condition: z(J(λs,a,λ p),(τq),b) (f )(z) J(λs,a,λp),(τq),b(f )(z) = ( – 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 in-volved, our main results (Theorem to Theorem ) would yield a number of (known or
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 References
1. Srivastava, HM, Attiya, AA: An integral operator associated with the Hurwitz-Lerch zeta function and differential subordination. Integral Transforms Spec. Funct. 18, 207-216 (2007)
2. Cho, NE, Kim, IH, Srivastava, HM: Sandwich-type theorems for multivalent functions associated with the Srivastava-Attiya operator. Appl. Math. Comput. 217, 918-928 (2010)
3. Rˇaducanu, D, Srivastava, HM: A new class of analytic functions defined by means of a convolution operator involving the Hurwitz-Lerch zeta function. Integral Transforms Spec. Funct. 18, 933-943 (2007)
4. Srivastava, HM, R˘aducanu, D, S˘al˘agean, GS: A new class of generalized close-to-starlike functions defined by the Srivastava-Attiya operator. Acta Math. Sin. Engl. Ser. 29, 833-840 (2013)
5. Srivastava, HM, Choi, J: Series Associated with Zeta and Related Functions. Kluwer Academic, Dordrecht (2001) 6. Srivastava, HM: Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Camb.
Philos. Soc. 129, 77-84 (2000)
7. Srivastava, HM, Choi, J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam (2012) 8. Srivastava, HM: A new family of theλ-generalized Hurwitz-Lerch zeta functions with applications. Appl. Math. Inf. Sci.
8, 1485-1500 (2014)
9. Srivastava, HM: Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 5, 390-444 (2011)
10. Srivastava, HM: Generating relations and other results associated with some families of the extended Hurwitz-Lerch zeta functions. SpringerPlus 2, Article ID 67 (2013)
11. Srivastava, HM, Gaboury, S: New expansion formulas for a family of theλ-generalized Hurwitz-Lerch zeta functions. Int. J. Math. Math. Sci. 2014, Article ID 131067 (2014)
12. Srivastava, HM, Jankov, D, Pogány, TK, Saxena, RK: Two-sided inequalities for the extended Hurwitz-Lerch zeta function. Comput. Math. Appl. 62, 516-522 (2011)
13. Srivastava, HM, Saxena, RK, Pogány, TK, Saxena, R: Integral and computational representations of the extended Hurwitz-Lerch zeta function. Integral Transforms Spec. Funct. 22, 487-506 (2011)
14. Srivastava, HM, Manocha, HL: A Treatise on Generating Functions. Halsted, New York (1984)
15. Mathai, AM, Saxena, RK, Haubold, HJ: The H-Function: Theory and Applications. Springer, New York (2010) 16. Srivastava, HM, Gupta, KC, Goyal, SP: The H-Functions of One and Two Variables with Applications. South Asian
Publishers, New Delhi (1982)
17. Prajapat, JK, Bulboac˘a, T: Double subordination preserving properties for a new generalized Srivastava-Attiya operator. Chin. Ann. Math. 33, 569-582 (2012)
18. Noor, KI, Bukhari, SZH: Some subclasses of analytic and spiral-like functions of complex order involving the Srivastava-Attiya integral operator. Integral Transforms Spec. Funct. 21, 907-916 (2010)
19. Choi, JH, Saigo, M, Srivastava, HM: Some inclusion properties of a certain family of integral operators. J. Math. Anal. Appl. 276, 432-445 (2002)
20. Cho, NE, Srivastava, HM: Argument estimation of certain analytic functions defined by a class of multiplier transformation. Math. Comput. Model. 37, 39-49 (2003)
21. Jung, IB, Kim, YC, Srivastava, HM: The Hardy space of analytic functions associated with certain one-parameter families of integral operators. J. Math. Anal. Appl. 176, 138-147 (1993)
22. Bernardi, SD: Convex and starlike univalent functions. Trans. Am. Math. Soc. 135, 429-446 (1969)
23. Carlson, BC, Shaffer, DB: Starlike and prestarlike hypergeometric functions. SIAM J. Math. Anal. 15, 737-745 (1984) 24. Owa, S, Srivastava, HM: Univalent and starlike generalized hypergeometric functions. Can. J. Math. 39, 1057-1077
(1987)
25. Dziok, J, Srivastava, HM: Classes of analytic functions associated with the generalized hypergeometric function. Appl. Math. Comput. 103, 1-13 (1999)
26. Dziok, J, Srivastava, HM: Certain subclasses of analytic functions associated with the generalized hypergeometric function. Integral Transforms Spec. Funct. 14, 7-18 (2003)
27. Hohlov, YE: Operators and operations in the class of univalent functions. Izv. Vysš. Uˇcebn. Zaved., Mat. 10, 83-89 (1978)
28. Ruscheweyh, S: New criteria for univalent functions. Proc. Am. Math. Soc. 49, 109-115 (1975)
29. Duren, PL: Univalent Functions. Grundlehren der Mathematischen Wissenschaften, vol. 259. Springer, Berlin (1983) 30. Ma, WC, Minda, D: A unified treatment of some special classes of functions. In: Proceedings of the Conference on
Complex Analysis (Tianjin, 1992). Conf. Proc. Lecture Notes in Anal., vol. 1, pp. 157-169. International Press, Cambridge (1994)
