- Research
- Open access
- Published:
A new class of analytic functions defined by means of a generalization of the Srivastava-Attiya operator
Journal of Inequalities and Applications volume 2015, Article number: 39 (2015)
Abstract
In this paper, we introduce a new class of analytic functions defined by a new convolution operator \(J_{(\lambda_{p}),(\mu _{q}),b}^{s,a,\lambda}\) 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.
1 Introduction
Let \(\mathcal{A}\) denote the class of functions \(f(z)\) normalized by
which are analytic in the open unit disk
A function \(f(z)\) in the class \(\mathcal{A}\) is said to be in the class \(\mathcal{S}^{*}(\alpha)\) of starlike functions of order α in \(\mathbb{U}\) if it satisfies the following inequality:
The largely investigated Srivastava-Attiya operator is defined as [1] (see also [2–4]):
where \(z \in\mathbb{U}\), \(a \in\mathbb{C}\setminus\mathbb {Z}^{-}_{0}\), \(s \in\mathbb{C}\) and \(f \in\mathcal{A}\).
In fact, the linear operator \(J_{s,a}(f)\) can be written as
in terms of the Hadamard product (or convolution), where \(G_{s,a}(z)\) is given by
The function \(\Phi(z,s,a)\) involved in the right-hand side of (1.5) is the well-known Hurwitz-Lerch zeta function defined by (see, for example, [5, p.121 et seq.]; see also [6] and [7, p.194 et seq.])
Recently, a new family of λ-generalized Hurwitz-Lerch zeta functions was investigated by Srivastava [8] (see also [9–13]). Srivastava considered the following function:
where
and the equality in the convergence condition holds true for suitably bounded values of \(|z|\) given by
Here, and for the remainder of this paper, \((\lambda)_{\kappa}\) denotes the Pochhammer symbol defined, in terms of the gamma function, by
it being understood conventionally that \((0)_{0}:=1\) and assumed tacitly that the Γ-quotient exists (see, for details, [14, p.21 et seq.]).
Definition 1
The H-function involved in the right-hand side of (1.7) is the well-known Fox’s H-function [15, Definition 1.1] (see also [14, 16]) defined by
where
an empty product is interpreted as 1, m, n, \(\mathfrak{p}\) and \(\mathfrak{q}\) are integers such that \(1\leqq m \leqq\mathfrak{q}\), \(0\leqq n \leqq\mathfrak{p}\), \(A_{j}>0\) (\(j=1,\ldots,\mathfrak{p}\)), \(B_{j}>0\) (\(j=1,\ldots,\mathfrak{q}\)), \(a_{j} \in\mathbb{C}\) (\(j=1,\ldots,\mathfrak{p}\)), \(b_{j} \in\mathbb{C}\) (\(j=1,\ldots ,\mathfrak{q}\)) and ℒ is a suitable Mellin-Barnes type contour separating the poles of the gamma functions
from the poles of the gamma functions
It is worthy to mention that using the fact that [8, p.1496, Remark 7]
equation (1.7) reduces to
Definition 2
The function \(\Phi_{\lambda_{1},\ldots,\lambda_{p};\mu_{1},\ldots,\mu _{q}}^{(\rho_{1},\ldots,\rho_{p},\sigma_{1},\ldots,\sigma_{q})}(z,s,a)\) involved in (1.11) is the multiparameter extension and generalization of the Hurwitz-Lerch zeta function \(\Phi(z,s,a)\) introduced by Srivastava et al. [13, p.503, Eq. (6.2)] defined by
with
We propose to consider the following linear operator
defined by
where ∗ denotes the Hadamard product (or convolution) of analytic functions, and the function \(G_{(\lambda_{p}),(\mu_{q}),b}^{s,a,\lambda }(z)\) is given by
with
Combining (1.15) and (1.16), we obtain
with
and
Remark 1
It follows from (1.15) and (1.17) that the operator \(J_{(\lambda_{p}),(\mu_{q}),0}^{s,a,\lambda}(f)\) (special case of (1.17) when \(b=0\)) can be defined for \(a \in\mathbb{C}\setminus \mathbb{Z}^{-}\) by the following limit relationship:
We can see that the operator \(J_{(\lambda_{p}),(\mu_{q}),b}^{s,a,\lambda }\) generalizes several recently investigated operators such as:
-
(i)
If \(p=2\), \(q=1\) and \(b=0\), then \(J_{(\lambda_{1},\lambda _{2}),(\mu_{1}),0}^{s,a,\lambda}=J_{\lambda_{1},\lambda_{2};\mu _{1}}^{s,a}\), where \(J_{\lambda_{1},\lambda_{2};\mu_{1}}^{s,a}\) is the linear operator introduced by Prajapat and Bulboacă [17, p.571, Eq. (1.8)].
-
(ii)
\(J_{(\gamma-1,1),(\nu),0}^{s,a,\lambda}=I_{a,\nu,\gamma }^{s}\), where \(I_{a,\nu,\gamma}^{s}\) is the generalized operator recently studied by Noor and Bukhari [18, p.2, Eq. (1.3)].
-
(iii)
\(J_{(\gamma-1,1),(\nu),0}^{0,0,\lambda}=I_{\nu,\gamma }^{s}\), where \(I_{\nu,\gamma}^{s}\) is the Choi-Saigo-Srivastava operator [19].
-
(iv)
\(J_{(\gamma,1),(\gamma),0}^{s,a,\lambda}= J_{s,a}\), where \(J_{s,a}\) is the Srivastava-Attiya operator [1].
-
(v)
\(J_{(\gamma,1),(\gamma),0}^{-r,a,\lambda}= I(r,a)\) (\(a\geqq 0\), \(r \in\mathbb{Z}\)), where the operator \(I(r,a)\) is the one introduced by Cho and Srivastava [20].
-
(vi)
\(J_{(\beta,1),(\alpha+\beta),0}^{0,a,\lambda}= \mathcal {Q}_{\beta}^{\alpha}\) (\(\alpha\geqq0\), \(\beta>-1\)), where the operator \(\mathcal{Q}_{\beta}^{\alpha}\) was studied by Jung et al. [21].
-
(vii)
\(J_{(\gamma,1),(\gamma),0}^{1,a,\lambda}= J_{a}\) (\(a\geqq -1\)), where \(J_{a}\) denotes the Bernardi operator [22].
-
(viii)
\(J_{(\gamma,1),(\nu),0}^{0,0,\lambda}= \mathcal{L}(\gamma ,\nu)\), where \(\mathcal{L}(\gamma,\nu)\) is the well-known Carlson-Shaffer operator [23].
-
(ix)
\(J_{(2,1),(2-\gamma),0}^{0,0,\lambda}= \Omega_{z}^{\gamma}\) (\(0\leqq\gamma<1\)), where \(\Omega_{z}^{\gamma}\) is the fractional integral operator investigated by Owa and Srivastava [24].
-
(x)
\(J_{(\lambda_{1}-1,\ldots,\lambda_{p}-1,1),(\mu_{1}-1,\ldots,\mu _{q}-1,0),0}^{0,a,\lambda}=H_{1}(\lambda_{1},\ldots,\lambda_{p}; \mu _{1},\ldots,\mu_{q})\) (\(p\leqq q+1\)), where the operator \(H_{1}(\lambda _{1},\ldots,\lambda_{p}; \mu_{1},\ldots,\mu_{q})\) is the Dziok-Srivastava operator [25, 26] which contains as special cases the Hohlov operator [27] and the Ruscheweyh operator [28].
We say that a function \(f \in\mathcal{A}\) is in the class \(\mathcal {S}_{(\lambda_{p}),(\mu_{q}),b}^{s,a,\lambda,*}(\alpha)\) if \(J_{(\lambda _{p}),(\mu_{q}),b}^{s,a,\lambda}(f)\) is in the class \(\mathcal {S}^{*}(\alpha)\), that is, if
with
and
In this paper, we systematically investigate the class \(\mathcal {S}_{(\lambda_{p}),(\mu_{q}),b}^{s,a,\lambda,*}(\alpha)\) of analytic functions defined above by means of the new generalized Srivastava-Attiya convolution operator \(J_{(\lambda_{p}),(\mu _{q}),b}^{s,a,\lambda}\). Especially, we derive coefficient inequalities, distortion theorems, extreme points and the Fekete-Szegö problem for this new function class.
2 Coefficient inequalities
Theorem 1
Let \(\alpha\in[0,1)\). If \(f(z) \in\mathcal{A}\) satisfies the following equality
then
Proof
Suppose that inequality (2.1) holds for \(\alpha\in[0,1)\). Let us define the function \(F(z)\) by
It is sufficient to prove that
to prove that \(f(z) \in\mathcal{S}_{(\lambda_{p}),(\mu _{q}),b}^{s,a,\lambda,*}(\alpha)\).
In fact, we have that
and thus
provided that (2.1) is satisfied. □
The next theorem aims to provide coefficient inequalities for functions \(f(z)\) belonging to the class \(\mathcal{S}_{(\lambda_{p}),(\mu _{q}),b}^{s,a,\lambda,*}(\alpha)\).
Theorem 2
Let \(\alpha\in[0,1)\). If \(f(z) \in\mathcal{S}_{(\lambda_{p}),(\mu _{q}),b}^{s,a,\lambda,*}(\alpha)\), then
The result is sharp.
Proof
Let
Then \(p(z)\) is analytic and
We note easily that
With the help of (1.17), we find
for \(k \in\mathbb{N}\setminus\{1\}\).
By making use of the Carathéodory lemma [29, p.41], we have
We have to prove that inequality (2.4) holds true for \(k \in \mathbb{N}\setminus\{1\}\). We will proceed by the principle of mathematical induction. If \(k=2\) in (2.6), we obtain
Now suppose that (2.4) is satisfied for \(k\leqq n\). Then, from (2.4) and (2.6), we have that
whence
The result is sharp for the function \(f(z)\) given by
□
3 Distortion inequalities for the function class \(\mathcal{S}_{(\lambda_{p}),(\mu_{q}),b}^{s,a,\lambda,*}(\alpha)\)
In this section, we establish distortion inequalities for functions belonging to the class \(\mathcal{S}_{(\lambda_{p}),(\mu _{q}),b}^{s,a,\lambda,*}(\alpha)\). These inequalities are given in the following theorem.
Theorem 3
Let \(f(z) \in\mathcal{S}_{(\lambda_{p}),(\mu_{q}),b}^{s,a,\lambda ,*}(\alpha)\) and \(0\leqq\alpha<1\). Then
and
Proof
Let \(f(z) \in\mathcal{A}\) be given by (1.1). Then, making use of Theorem 2, we find
and
From (1.1), we also have that
and
We thus obtain the results (3.1) and (3.2) asserted by Theorem 3. □
4 Extreme points
This section is devoted to presenting the extreme points of the function class \(\mathcal{S}_{(\lambda_{p}),(\mu_{q}),b}^{s,a,\lambda ,*}(\alpha)\). Let \(\mathcal{\widetilde{S}}_{(\lambda_{p}),(\mu _{q}),b}^{s,a,\lambda,*}(\alpha)\) be the subclass of \(\mathcal {S}_{(\lambda_{p}),(\mu_{q}),b}^{s,a,\lambda,*}(\alpha)\) that consists in all functions \(f(z) \in\mathcal{A}\), which satisfy inequality (2.1). Then the extreme points of \(\mathcal{\widetilde{S}}_{(\lambda _{p}),(\mu_{q}),b}^{s,a,\lambda,*}(\alpha)\) are given by the following theorem.
Theorem 4
Let
and
Then
if and only if it can be expressed in the following form:
Proof
Suppose that
Then
Thus, by the definition of the function class \(\mathcal{\widetilde {S}}_{(\lambda_{p}),(\mu_{q}),b}^{s,a,\lambda,*}(\alpha)\), we have
Conversely, if
then, by using (2.1), we may set
which implies that
The proof of Theorem 4 is thus completed. □
5 The Fekete-Szegö problem
In this section, we shall obtain the Fekete-Szegö inequality for functions in the class \(\mathcal{S}_{(\lambda_{p}),(\mu _{q}),b}^{s,a,\lambda,*}(\alpha)\) when
We need to recall an important lemma due to Ma and Minda [30] in order to prove our result involving Fekete-Szegö inequality.
Lemma 1
If
is an analytic function in \(\mathbb{U}\) such that
then
When \(\nu<0\) or \(\nu>1\), the equality holds true if and only if
or one of its rotations. If \(0<\nu<1\), then the equality holds true if and only if
or one of its rotations. If \(\nu=0\), then the equality holds true if and only if
or one of its rotations. If \(\nu=1\), 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 \(\nu=0\).
Theorem 5
Let
and
If \(f \in\mathcal{S}_{(\lambda_{p}),(\mu_{q}),b}^{s,a,\lambda,*}(\alpha )\), then
where
and
The result is sharp.
Proof
For \(f \in\mathcal{S}_{(\lambda_{p}),(\mu_{q}),b}^{s,a,\lambda ,*}(\alpha)\), let
Then, with the help of (2.5), we have
and
Therefore, we find
We thus write
where
The result asserted by Theorem 5 follows by applying Lemma 1.
Moreover, if \(\tau<\sigma_{1}\) or \(\tau>\sigma_{2}\), then the equality holds true if and only if
For \(\sigma_{1}<\tau<\sigma_{2}\), the equality holds true if and only if
If \(\tau=\sigma_{1}\), then the equality holds true if and only if
Finally, when \(\tau=\sigma_{2}\), the equality holds true if and only if \(J_{(\lambda_{p}),(\tau_{q}),b}^{s,a,\lambda}(f)(z)\) satisfies the following condition:
where
□
We conclude this paper by mentioning that, by suitably specializing the parameters involved, our main results (Theorem 1 to Theorem 5) 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.
References
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)
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)
Rǎducanu, 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)
Srivastava, HM, Răducanu, D, Sălăgean, 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)
Srivastava, HM, Choi, J: Series Associated with Zeta and Related Functions. Kluwer Academic, Dordrecht (2001)
Srivastava, HM: Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Camb. Philos. Soc. 129, 77-84 (2000)
Srivastava, HM, Choi, J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam (2012)
Srivastava, HM: A new family of the λ-generalized Hurwitz-Lerch zeta functions with applications. Appl. Math. Inf. Sci. 8, 1485-1500 (2014)
Srivastava, HM: Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 5, 390-444 (2011)
Srivastava, HM: Generating relations and other results associated with some families of the extended Hurwitz-Lerch zeta functions. SpringerPlus 2, Article ID 67 (2013)
Srivastava, HM, Gaboury, S: New expansion formulas for a family of the λ-generalized Hurwitz-Lerch zeta functions. Int. J. Math. Math. Sci. 2014, Article ID 131067 (2014)
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)
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)
Srivastava, HM, Manocha, HL: A Treatise on Generating Functions. Halsted, New York (1984)
Mathai, AM, Saxena, RK, Haubold, HJ: The H-Function: Theory and Applications. Springer, New York (2010)
Srivastava, HM, Gupta, KC, Goyal, SP: The H-Functions of One and Two Variables with Applications. South Asian Publishers, New Delhi (1982)
Prajapat, JK, Bulboacă, T: Double subordination preserving properties for a new generalized Srivastava-Attiya operator. Chin. Ann. Math. 33, 569-582 (2012)
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)
Choi, JH, Saigo, M, Srivastava, HM: Some inclusion properties of a certain family of integral operators. J. Math. Anal. Appl. 276, 432-445 (2002)
Cho, NE, Srivastava, HM: Argument estimation of certain analytic functions defined by a class of multiplier transformation. Math. Comput. Model. 37, 39-49 (2003)
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)
Bernardi, SD: Convex and starlike univalent functions. Trans. Am. Math. Soc. 135, 429-446 (1969)
Carlson, BC, Shaffer, DB: Starlike and prestarlike hypergeometric functions. SIAM J. Math. Anal. 15, 737-745 (1984)
Owa, S, Srivastava, HM: Univalent and starlike generalized hypergeometric functions. Can. J. Math. 39, 1057-1077 (1987)
Dziok, J, Srivastava, HM: Classes of analytic functions associated with the generalized hypergeometric function. Appl. Math. Comput. 103, 1-13 (1999)
Dziok, J, Srivastava, HM: Certain subclasses of analytic functions associated with the generalized hypergeometric function. Integral Transforms Spec. Funct. 14, 7-18 (2003)
Hohlov, YE: Operators and operations in the class of univalent functions. Izv. Vysš. Učebn. Zaved., Mat. 10, 83-89 (1978)
Ruscheweyh, S: New criteria for univalent functions. Proc. Am. Math. Soc. 49, 109-115 (1975)
Duren, PL: Univalent Functions. Grundlehren der Mathematischen Wissenschaften, vol. 259. Springer, Berlin (1983)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
Open Access 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.
About this article
Cite this article
Srivastava, H.M., Gaboury, S. A new class of analytic functions defined by means of a generalization of the Srivastava-Attiya operator. J Inequal Appl 2015, 39 (2015). https://doi.org/10.1186/s13660-015-0573-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-015-0573-z