Hindawi Publishing Corporation The Scientific World Journal Volume 2013, Article ID 475643, 5 pages http://dx.doi.org/10.1155/2013/475643
Research Article New Result of Analytic Functions Related to Hurwitz Zeta Function F. Ghanim1 and M. Darus2 1 2
Department of Mathematics, College of Sciences, University of Sharjah, Sharjah, United Arab Emirates School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia
Correspondence should be addressed to F. Ghanim;
[email protected] Received 25 September 2013; Accepted 11 November 2013 Academic Editors: L. Gosse, T. Li, and P. Wang Copyright Β© 2013 F. Ghanim and M. Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using a linear operator, we obtain some new results for a normalized analytic function f defined by means of the Hadamard product of Hurwitz zeta function. A class related to this function will be introduced and the properties will be discussed.
1. Introduction
2. Preliminaries
A meromorphic function is a single-valued function, that is, analytic in all but possibly a discrete subset of its domain, and at those singularities it must go to infinity like a polynomial (i.e., these exceptional points must be poles and not essential singularities). A simpler definition states that a meromorphic function π(π§) is a function of the form
Let Ξ£ denote the class of meromorphic functions π(π§) normalized by
π (π§) =
π (π§) , β (π§)
(1)
where π(π§) and β(π§) are entire functions with β(π§) =ΜΈ 0 (see [1, page 64]). A meromorphic function therefore may only have finite-order, isolated poles and zeros and no essential singularities in its domain. A meromorphic function with an infinite number of poles is exemplified by csc(1/π§) on the punctured disk πβ = {π§ : 0 < |π§| < 1}. An equivalent definition of a meromorphic function is a complex analytic map to the Riemann sphere. For example, the Gamma function is meromorphic in the whole complex plane; see [1, 2]. In the present paper, we will derive some properties of univalent functions defined by means of the Hadamard product of Hurwitz Zeta function; a class related to this function will be introduced and the properties of the liner operator πΏπ‘π (πΌ, π½)π(π§) will be discussed.
π (π§) =
1 β + β π π§π , π§ π=1 π
(2)
which are analytic in the punctured unit disk πβ . For 0 β€ π½, we denote by πβ (π½) and π(π½) the subclasses of Ξ£ consisting of all meromorphic functions which are, respectively, starlike of order π½ and convex of order π½ in πβ . For functions ππ (π§) (π = 1; 2) defined by ππ (π§) =
1 β + β π π§π , π§ π=1 π,π
(3)
we denote the Hadamard product (or convolution) of π1 (π§) and π2 (π§) by (π1 β π2 ) =
1 β + β π π π§π . π§ π=1 π,1 π,2
(4)
Μ π½; π§) by Let us define the function π(πΌ, 1 β (πΌ) πΜ (πΌ, π½; π§) = + β π+1 π§π , π§ π=0 (π½)π+1
(5)
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for π½ =ΜΈ 0, β1, β2, . . ., and πΌ β C/{0}, where (π)π = π(π + 1)π+1 is the Pochhammer symbol. We note that
where π > 0, and 1 1 when π§ β ππ). Important special cases of the function Ξ¦(π§, π‘, π) include, for example, the Riemann zeta function π(π‘) = Ξ¦(1, π‘, 1), the Hurwitz zeta function π(π‘, π) = Ξ¦(1, π‘, π), the Lerch zeta function ππ‘ (π) = Ξ¦(exp2πππ , π‘, 1), (π β R, R(π‘) > 1), and the polylogarithm πΏππ‘ (π§) = π§Ξ¦(π§, π‘, π). Recent results on Ξ¦(π§, π‘, π) can be found in the expositions [5, 6]. By making use of the following normalized function we define πΊπ‘,π (π§) = (1 + π)π‘ [Ξ¦ (π§, π‘, π) β ππ‘ + β
1+π π‘ π 1 = + β( )π§ , π§ π=1 π + π
1 ] π§(1 + π)π‘
(π§ β π ) .
(10)
σΈ
= πΌ (πΏπ‘π (πΌ + 1, π½) π (π§)) β (πΌ + 1) πΏπ‘π (πΌ, π½) π (π§) .
(11)
In order to prove our main results, we recall the following lemma according to Yang [13]. Lemma 1. Let π(π§) = 1 + ππ π§π + ππ+1 π§π+1 + β
β
β
be analytic functions in π = πβ βͺ {0} with π(π§) =ΜΈ 0 for π§ β π. If π§π (π§) } < π, π2 (π§)
Theorem 2. Let πΌ + 1 > 0, πΏπ‘π (πΌ, π½)π(π§)/πΏπ‘π (πΌ + 1, π½)π(π§) =ΜΈ 0 for π§ β πβ and suppose that R {1 + β
πΏπ‘π (πΌ + 1, π½) π (π§) πΌπΏπ‘π (πΌ + 1, π½) π (π§) (1 + ) πΏπ‘π (πΌ, π½) π (π§) (πΌ + 1) πΏπ‘π (πΌ, π½) π (π§)
πΏπ‘π (πΌ + 2, π½) π (π§) } < π, πΏπ‘π (πΌ, π½) π (π§)
(π§ β π) ,
(15)
11β log 2, π
(17) β
(π§ β π ) .
The bound in (17) is the best possible. Proof. Define the function π(π§) by
The meromorphic functions with the generalized hypergeometric functions were considered recently by many others; see, for example, [7β12]. It follows from (10) that
R {1 + π
We begin with the following theorem.
Then
(π§ β πβ ) .
π§(πΏπ‘π (πΌ, π½) π (π§))
(14)
3. Main Results
β
1 β (πΌ)π+1 1 + π π‘ π ( +β )ππ§ . π§ π=1 (π½)π+1 π + π π
(π§ β π) .
The bound in (14) is the best possible.
(9)
πΏπ‘π (πΌ, π½) π (π§) = π (πΌ, π½; π§) β πΊπ‘,π (π§)
σΈ
2 (π β 1) 1 }>1β log 2, π (π§) ππ
where
Corresponding to the functions πΊπ‘,π (π§) and using the Hadamard product for π(π§) β Ξ£, we define a new linear operator πΏ π‘,π (πΌ, π½) on Ξ£ by the following series:
=
R{
(7)
is the well-known Gaussian hypergeometric function. We recall here a general Hurwitz-Lerch-Zeta function, which is defined in [3, 4] by the following series: Ξ¦ (π§, π‘, π) =
(13)
then
where
β
ππ , 2 log 2
(12)
π (π§) =
πΏπ‘π (πΌ, π½) π (π§) . πΏπ‘π (πΌ + 1, π½) π (π§)
(18)
Then, clearly π(π§) = 1+ππ π§π +ππ+1 π§π+1 +β
β
β
analytic function in πβ with π(π§) =ΜΈ 0 for π§ β πβ . It follows from (18) and (11) that σΈ
σΈ
π§(πΏπ‘π (πΌ + 1, π½) π (π§)) π§πσΈ (π§) π§(πΏπ‘π (πΌ, π½) π (π§)) β = π (π§) πΏβπ (π + 1, π) π (π§) πΏπ‘π (πΌ, π½) π (π§)
(19) by making use of the familiar identity (11) in (19), we obtain πΏπ‘π (πΌ + 2, π½) π (π§) π§πσΈ (π§) 1 1 β = + πΏπ‘π (πΌ + 1, π½) π (π§) πΌ + 1 (πΌ + 1) π (π§) (πΌ + 1) π (π§) (20)
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or, equivalent,
where
1 π§πσΈ (π§) 1+ (πΌ + 1) π2 (π§) =1+
1 0, πΊπ‘,π (π§) =ΜΈ 0 for π§ β πβ and suppose that σΈ
σΈ σΈ
π§(πΊπ‘,π (π§)) + (1/2) (πΊπ‘,π (π§)) R {(π§πΊπ‘,π (π§)) ( )} < π, πΊπ‘,π (π§) (33) πΏ
where 1 0, π§πΏπ‘π (πΌ+1, π½)π(π§) =ΜΈ 0 for π§ β πβ and suppose that R {(π§πΏπ‘π
(31)
or, equivalent 1+
Then
(30)
by making use of the familiar identity (11) in (30), we get
where
σΈ
(29)
σΈ
σΈ σΈ
π§(πΊπ‘,π (π§)) 4 (π β 1) }>1β R{ log 2, πΊπ‘,π (π§) π
(28)
(π§ β π ) .
π§πσΈ (π§) π§(πΏπ‘π (πΌ + 1, π½) π (π§)) β 1. = πΏπ (π§) πΏπ‘π (πΌ + 1, π½) π (π§)
π§(πΊπ‘,π (π§)) + (1/2) (πΊπ‘,π (π§)) } < π, β πΊπ‘,π (π§)
1 1 β
4πΏ (π β 1) log 2, π
(π§ β πβ ) .
(35)
The bound in (35) is the best possible. Letting πΏ = 1, π = 1 + π/8 log 2, and π‘ = 0 in Corollary 6, we have the following.
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Corollary 7. Let πσΈ (π§) =ΜΈ 0 for π§ β πβ and suppose that R {π§π(π§)σΈ (1 +
π§πσΈ σΈ (π§) π )} < 1 + . 2πσΈ (π§) 8 log 2
Letting πΌ = π½ = 1 in Theorem 8, we have the following. (36)
Corollary 9. Let π > 0, π§(πΊπ‘,π (π§))σΈ /πΊπ‘,π (π§) =ΜΈ 0 for π§ β πβ and suppose that π
Then R {π§π(π§)σΈ } > 0,
(π§ β πβ ) .
(37)
{ πΊπ‘,π (π§) R {1 + ( ) σΈ π§(πΊπ‘,π (π§)) {
The result is sharp. σΈ
Theorem 8. Let π > 0, π§(πΏπ‘π (πΌ + 1, π½)π(π§)) /πΏπ‘π (πΌ, π½)π(π§) =ΜΈ 0 for π§ β πβ and suppose that { πΏπ‘ (πΌ, π½) π (π§) R {1 + ( π‘ π ) πΏ π (πΌ + 1, π½) π (π§) { Γ(
β
π
where 11β log 2 π
(44)
The bound in (45) is the best possible.
Then πΏπ‘π (πΌ, π½) π (π§) ) πΏπ‘π (πΌ + 1, π½) π (π§)
π . 2π log 2
Then R(
where
R(
π
σΈ σΈ
} π§(πΊπ‘,π (π§)) πΊπ‘,π (π§) Γ (1 + ) )} < π, β( σΈ (πΊπ‘,π (π§)) π§(πΊπ‘,π (π§)) } (43)
Both authors read and approved the final paper.
(πΌ + 1) πΏπ‘π (πΌ + 2, π½) π (π§) πΏπ‘π (πΌ + 1, π½) π (π§)
(42)
Acknowledgment The work here is supported by a special grant: DIP-2013-1.
π
βπΌ(
πΏπ‘π (πΌ, π½) π (π§) ) β 1) . (πΌ + 1, π½) π (π§)
πΏπ‘π
Applying Lemma 1, with π = 1/π, we get the required result.
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The Scientific World Journal [2] R. K. Pandey, Applied Complex Analysis, Discovery Publishing House, Grand Rapids, Mich, USA, 2008. [3] H. M. Srivastava and A. A. Attiya, βAn integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination,β Integral Transforms and Special Functions, vol. 18, no. 3, pp. 207β216, 2007. [4] H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic, Boston, Mass, USA, 2001. [5] H. M. Srivastava, D. Jankov, T. K. PogΒ΄any, and R. K. Saxena, βTwo-sided inequalities for the extended Hurwitz-Lerch Zeta function,β Computers and Mathematics with Applications, vol. 62, no. 1, pp. 516β522, 2011. [6] H. M. Srivastava, R. K. Saxena, T. K. Pogany, and R. Saxena, βIntegral transforms and special functions,β Applied Mathematics and Computation, vol. 22, no. 7, pp. 487β506, 2011. [7] J. Dziok and H. M. Srivastava, βCertain subclasses of analytic functions associated with the generalized hypergeometric function,β Integral Transforms and Special Functions, vol. 14, no. 1, pp. 7β18, 2003. [8] F. Ghanim and M. Darus, βA new class of meromorphically analytic functions with applications to the generalized hypergeometric functions,β Abstract and Applied Analysis, vol. 2011, Article ID 159405, 10 pages, 2011. [9] F. Ghanim and M. Darus, βSome properties of certain subclass of meromorphically multivalent functions defined by linear operator,β Journal of Mathematics and Statistics, vol. 6, no. 1, pp. 34β41, 2010. [10] F. Ghanim and M. Darus, βSome properties on a certain class of meromorphic functions related to Cho-Kwon-Srivastava operator,β Asian-European Journal of Mathematics, vol. 5, no. 4, Article ID 1250052, pp. 1β9, 2012. [11] J.-L. Liu and H. M. Srivastava, βCertain properties of the DziokSrivastava operator,β Applied Mathematics and Computation, vol. 159, no. 2, pp. 485β493, 2004. [12] J.-L. Liu and H. M. Srivastava, βClasses of meromorphically multivalent functions associated with the generalized hypergeometric function,β Mathematical and Computer Modelling, vol. 39, no. 1, pp. 21β34, 2004. [13] D. Yang, βSome criteria for multivalent starlikeness,β Southeast Asian Bulletin of Mathematics, vol. 24, no. 3, pp. 491β497, 2000.
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