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 𝑓(𝑧) is a function of the form

Let Ξ£ denote the class of meromorphic functions 𝑓(𝑧) normalized by

𝑓 (𝑧) =

𝑔 (𝑧) , β„Ž (𝑧)

(1)

where 𝑔(𝑧) and β„Ž(𝑧) are entire functions with β„Ž(𝑧) =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/𝑧) on the punctured disk π‘ˆβˆ— = {𝑧 : 0 < |𝑧| < 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 πΏπ‘‘π‘Ž (𝛼, 𝛽)𝑓(𝑧) will be discussed.

𝑓 (𝑧) =

1 ∞ + βˆ‘ π‘Ž 𝑧𝑛 , 𝑧 𝑛=1 𝑛

(2)

which are analytic in the punctured unit disk π‘ˆβˆ— . For 0 ≀ 𝛽, we denote by π‘†βˆ— (𝛽) and π‘˜(𝛽) the subclasses of Ξ£ consisting of all meromorphic functions which are, respectively, starlike of order 𝛽 and convex of order 𝛽 in π‘ˆβˆ— . For functions 𝑓𝑗 (𝑧) (𝑗 = 1; 2) defined by 𝑓𝑗 (𝑧) =

1 ∞ + βˆ‘ π‘Ž 𝑧𝑛 , 𝑧 𝑛=1 𝑛,𝑗

(3)

we denote the Hadamard product (or convolution) of 𝑓1 (𝑧) and 𝑓2 (𝑧) by (𝑓1 βˆ— 𝑓2 ) =

1 ∞ + βˆ‘ π‘Ž π‘Ž 𝑧𝑛 . 𝑧 𝑛=1 𝑛,1 𝑛,2

(4)

Μƒ 𝛽; 𝑧) by Let us define the function πœ™(𝛼, 1 ∞ (𝛼) πœ™Μƒ (𝛼, 𝛽; 𝑧) = + βˆ‘ 𝑛+1 𝑧𝑛 , 𝑧 𝑛=0 (𝛽)𝑛+1

(5)

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for 𝛽 =ΜΈ 0, βˆ’1, βˆ’2, . . ., and 𝛼 ∈ C/{0}, where (πœ†)𝑛 = πœ†(πœ† + 1)𝑛+1 is the Pochhammer symbol. We note that

where π‘Ž > 0, and 1 1 when 𝑧 ∈ πœ•π‘ˆ). Important special cases of the function Ξ¦(𝑧, 𝑑, π‘Ž) include, for example, the Riemann zeta function 𝜁(𝑑) = Ξ¦(1, 𝑑, 1), the Hurwitz zeta function 𝜁(𝑑, π‘Ž) = Ξ¦(1, 𝑑, π‘Ž), the Lerch zeta function 𝑙𝑑 (𝜁) = Ξ¦(exp2πœ‹π‘–πœ‰ , 𝑑, 1), (πœ‰ ∈ R, R(𝑑) > 1), and the polylogarithm 𝐿𝑖𝑑 (𝑧) = 𝑧Φ(𝑧, 𝑑, π‘Ž). Recent results on Ξ¦(𝑧, 𝑑, π‘Ž) can be found in the expositions [5, 6]. By making use of the following normalized function we define 𝐺𝑑,π‘Ž (𝑧) = (1 + π‘Ž)𝑑 [Ξ¦ (𝑧, 𝑑, π‘Ž) βˆ’ π‘Žπ‘‘ + ∞

1+π‘Ž 𝑑 𝑛 1 = + βˆ‘( )𝑧 , 𝑧 𝑛=1 𝑛 + π‘Ž

1 ] 𝑧(1 + π‘Ž)𝑑

(𝑧 ∈ π‘ˆ ) .

(10)

σΈ€ 

= 𝛼 (πΏπ‘‘π‘Ž (𝛼 + 1, 𝛽) 𝑓 (𝑧)) βˆ’ (𝛼 + 1) πΏπ‘‘π‘Ž (𝛼, 𝛽) 𝑓 (𝑧) .

(11)

In order to prove our main results, we recall the following lemma according to Yang [13]. Lemma 1. Let π‘ž(𝑧) = 1 + π‘žπ‘› 𝑧𝑛 + π‘žπ‘›+1 𝑧𝑛+1 + β‹… β‹… β‹… be analytic functions in π‘ˆ = π‘ˆβˆ— βˆͺ {0} with π‘ž(𝑧) =ΜΈ 0 for 𝑧 ∈ π‘ˆ. If π‘§π‘ž (𝑧) } < 𝑀, π‘ž2 (𝑧)

Theorem 2. Let 𝛼 + 1 > 0, πΏπ‘‘π‘Ž (𝛼, 𝛽)𝑓(𝑧)/πΏπ‘‘π‘Ž (𝛼 + 1, 𝛽)𝑓(𝑧) =ΜΈ 0 for 𝑧 ∈ π‘ˆβˆ— and suppose that R {1 + βˆ’

πΏπ‘‘π‘Ž (𝛼 + 1, 𝛽) 𝑓 (𝑧) π›ΌπΏπ‘‘π‘Ž (𝛼 + 1, 𝛽) 𝑓 (𝑧) (1 + ) πΏπ‘‘π‘Ž (𝛼, 𝛽) 𝑓 (𝑧) (𝛼 + 1) πΏπ‘‘π‘Ž (𝛼, 𝛽) 𝑓 (𝑧)

πΏπ‘‘π‘Ž (𝛼 + 2, 𝛽) 𝑓 (𝑧) } < 𝑀, πΏπ‘‘π‘Ž (𝛼, 𝛽) 𝑓 (𝑧)

(𝑧 ∈ π‘ˆ) ,

(15)

11βˆ’ log 2, 𝑛

(17) βˆ—

(𝑧 ∈ π‘ˆ ) .

The bound in (17) is the best possible. Proof. Define the function π‘ž(𝑧) by

The meromorphic functions with the generalized hypergeometric functions were considered recently by many others; see, for example, [7–12]. It follows from (10) that

R {1 + π‘Ž

We begin with the following theorem.

Then

(𝑧 ∈ π‘ˆβˆ— ) .

𝑧(πΏπ‘‘π‘Ž (𝛼, 𝛽) 𝑓 (𝑧))

(14)

3. Main Results

βˆ—

1 ∞ (𝛼)𝑛+1 1 + π‘Ž 𝑑 𝑛 ( +βˆ‘ )π‘Žπ‘§ . 𝑧 𝑛=1 (𝛽)𝑛+1 𝑛 + π‘Ž 𝑛

(𝑧 ∈ π‘ˆ) .

The bound in (14) is the best possible.

(9)

πΏπ‘‘π‘Ž (𝛼, 𝛽) 𝑓 (𝑧) = πœ™ (𝛼, 𝛽; 𝑧) βˆ— 𝐺𝑑,π‘Ž (𝑧)

σΈ€ 

2 (𝑀 βˆ’ 1) 1 }>1βˆ’ log 2, π‘ž (𝑧) π‘›π‘Ž

where

Corresponding to the functions 𝐺𝑑,π‘Ž (𝑧) and using the Hadamard product for 𝑓(𝑧) ∈ Ξ£, we define a new linear operator 𝐿 𝑑,π‘Ž (𝛼, 𝛽) 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¦ (𝑧, 𝑑, π‘Ž) =

(13)

then

where

∞

π‘›π‘Ž , 2 log 2

(12)

π‘ž (𝑧) =

πΏπ‘‘π‘Ž (𝛼, 𝛽) 𝑓 (𝑧) . πΏπ‘‘π‘Ž (𝛼 + 1, 𝛽) 𝑓 (𝑧)

(18)

Then, clearly π‘ž(𝑧) = 1+π‘žπ‘› 𝑧𝑛 +π‘žπ‘›+1 𝑧𝑛+1 +β‹… β‹… β‹… analytic function in π‘ˆβˆ— with π‘ž(𝑧) =ΜΈ 0 for 𝑧 ∈ π‘ˆβˆ— . It follows from (18) and (11) that σΈ€ 

σΈ€ 

𝑧(πΏπ‘‘π‘Ž (𝛼 + 1, 𝛽) 𝑓 (𝑧)) π‘§π‘žσΈ€  (𝑧) 𝑧(πΏπ‘‘π‘Ž (𝛼, 𝛽) 𝑓 (𝑧)) βˆ’ = π‘ž (𝑧) πΏβˆ—π‘ (π‘Ž + 1, 𝑐) 𝑓 (𝑧) πΏπ‘‘π‘Ž (𝛼, 𝛽) 𝑓 (𝑧)

(19) by making use of the familiar identity (11) in (19), we obtain πΏπ‘‘π‘Ž (𝛼 + 2, 𝛽) 𝑓 (𝑧) π‘§π‘žσΈ€  (𝑧) 1 1 βˆ’ = + πΏπ‘‘π‘Ž (𝛼 + 1, 𝛽) 𝑓 (𝑧) 𝛼 + 1 (𝛼 + 1) π‘ž (𝑧) (𝛼 + 1) π‘ž (𝑧) (20)

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3

or, equivalent,

where

1 π‘§π‘žσΈ€  (𝑧) 1+ (𝛼 + 1) π‘ž2 (𝑧) =1+

1 0, 𝐺𝑑,π‘Ž (𝑧) =ΜΈ 0 for 𝑧 ∈ π‘ˆβˆ— and suppose that σΈ€ 

σΈ€ σΈ€ 

𝑧(𝐺𝑑,π‘Ž (𝑧)) + (1/2) (𝐺𝑑,π‘Ž (𝑧)) R {(𝑧𝐺𝑑,π‘Ž (𝑧)) ( )} < 𝑀, 𝐺𝑑,π‘Ž (𝑧) (33) 𝛿

where 1 0, π‘§πΏπ‘‘π‘Ž (𝛼+1, 𝛽)𝑓(𝑧) =ΜΈ 0 for 𝑧 ∈ π‘ˆβˆ— and suppose that R {(π‘§πΏπ‘‘π‘Ž

(31)

or, equivalent 1+

Then

(30)

by making use of the familiar identity (11) in (30), we get

where

σΈ€ 

(29)

σΈ€ 

σΈ€ σΈ€ 

𝑧(𝐺𝑑,π‘Ž (𝑧)) 4 (𝑀 βˆ’ 1) }>1βˆ’ R{ log 2, 𝐺𝑑,π‘Ž (𝑧) 𝑛

(28)

(𝑧 ∈ π‘ˆ ) .

π‘§π‘žσΈ€  (𝑧) 𝑧(πΏπ‘‘π‘Ž (𝛼 + 1, 𝛽) 𝑓 (𝑧)) βˆ’ 1. = π›Ώπ‘ž (𝑧) πΏπ‘‘π‘Ž (𝛼 + 1, 𝛽) 𝑓 (𝑧)

𝑧(𝐺𝑑,π‘Ž (𝑧)) + (1/2) (𝐺𝑑,π‘Ž (𝑧)) } < 𝑀, βˆ’ 𝐺𝑑,π‘Ž (𝑧)

1 1 βˆ’

4𝛿 (𝑀 βˆ’ 1) log 2, 𝑛

(𝑧 ∈ π‘ˆβˆ— ) .

(35)

The bound in (35) is the best possible. Letting 𝛿 = 1, 𝑀 = 1 + 𝑛/8 log 2, and 𝑑 = 0 in Corollary 6, we have the following.

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Corollary 7. Let 𝑓󸀠 (𝑧) =ΜΈ 0 for 𝑧 ∈ π‘ˆβˆ— and suppose that R {𝑧𝑓(𝑧)σΈ€  (1 +

𝑧𝑓󸀠󸀠 (𝑧) 𝑛 )} < 1 + . 2𝑓󸀠 (𝑧) 8 log 2

Letting 𝛼 = 𝛽 = 1 in Theorem 8, we have the following. (36)

Corollary 9. Let πœ‰ > 0, 𝑧(𝐺𝑑,π‘Ž (𝑧))σΈ€  /𝐺𝑑,π‘Ž (𝑧) =ΜΈ 0 for 𝑧 ∈ π‘ˆβˆ— and suppose that πœ‰

Then R {𝑧𝑓(𝑧)σΈ€  } > 0,

(𝑧 ∈ π‘ˆβˆ— ) .

(37)

{ 𝐺𝑑,π‘Ž (𝑧) R {1 + ( ) σΈ€  𝑧(𝐺𝑑,π‘Ž (𝑧)) {

The result is sharp. σΈ€ 

Theorem 8. Let πœ‰ > 0, 𝑧(πΏπ‘‘π‘Ž (𝛼 + 1, 𝛽)𝑓(𝑧)) /πΏπ‘‘π‘Ž (𝛼, 𝛽)𝑓(𝑧) =ΜΈ 0 for 𝑧 ∈ π‘ˆβˆ— and suppose that { 𝐿𝑑 (𝛼, 𝛽) 𝑓 (𝑧) R {1 + ( 𝑑 π‘Ž ) 𝐿 π‘Ž (𝛼 + 1, 𝛽) 𝑓 (𝑧) { Γ—(

βˆ’

πœ‰

where 11βˆ’ log 2 𝑛

(44)

The bound in (45) is the best possible.

Then πΏπ‘‘π‘Ž (𝛼, 𝛽) 𝑓 (𝑧) ) πΏπ‘‘π‘Ž (𝛼 + 1, 𝛽) 𝑓 (𝑧)

𝑛 . 2πœ‰ log 2

Then R(

where

R(

πœ‰

σΈ€ σΈ€ 

} 𝑧(𝐺𝑑,π‘Ž (𝑧)) 𝐺𝑑,π‘Ž (𝑧) Γ— (1 + ) )} < 𝑀, βˆ’( σΈ€  (𝐺𝑑,π‘Ž (𝑧)) 𝑧(𝐺𝑑,π‘Ž (𝑧)) } (43)

Both authors read and approved the final paper.

(𝛼 + 1) πΏπ‘‘π‘Ž (𝛼 + 2, 𝛽) 𝑓 (𝑧) πΏπ‘‘π‘Ž (𝛼 + 1, 𝛽) 𝑓 (𝑧)

(42)

Acknowledgment The work here is supported by a special grant: DIP-2013-1.

πœ‰

βˆ’π›Ό(

πΏπ‘‘π‘Ž (𝛼, 𝛽) 𝑓 (𝑧) ) βˆ’ 1) . (𝛼 + 1, 𝛽) 𝑓 (𝑧)

πΏπ‘‘π‘Ž

Applying Lemma 1, with π‘Ž = 1/πœ‰, we get the required result.

References [1] S. G. Krantz, β€œMeromorphic functions and singularities at infinity,” in Handbook of Complex Variables, pp. 63–68, BirkhΒ¨aauser, Boston, Mass, USA, 1999.

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, β€œAn integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination,” Integral Transforms and Special Functions, vol. 18, no. 3, pp. 207–216, 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, β€œTwo-sided inequalities for the extended Hurwitz-Lerch Zeta function,” Computers and Mathematics with Applications, vol. 62, no. 1, pp. 516–522, 2011. [6] H. M. Srivastava, R. K. Saxena, T. K. Pogany, and R. Saxena, β€œIntegral transforms and special functions,” Applied Mathematics and Computation, vol. 22, no. 7, pp. 487–506, 2011. [7] J. Dziok and H. M. Srivastava, β€œCertain subclasses of analytic functions associated with the generalized hypergeometric function,” Integral Transforms and Special Functions, vol. 14, no. 1, pp. 7–18, 2003. [8] F. Ghanim and M. Darus, β€œA new class of meromorphically analytic functions with applications to the generalized hypergeometric functions,” Abstract and Applied Analysis, vol. 2011, Article ID 159405, 10 pages, 2011. [9] F. Ghanim and M. Darus, β€œSome properties of certain subclass of meromorphically multivalent functions defined by linear operator,” Journal of Mathematics and Statistics, vol. 6, no. 1, pp. 34–41, 2010. [10] F. Ghanim and M. Darus, β€œSome properties on a certain class of meromorphic functions related to Cho-Kwon-Srivastava operator,” Asian-European Journal of Mathematics, vol. 5, no. 4, Article ID 1250052, pp. 1–9, 2012. [11] J.-L. Liu and H. M. Srivastava, β€œCertain properties of the DziokSrivastava operator,” Applied Mathematics and Computation, vol. 159, no. 2, pp. 485–493, 2004. [12] J.-L. Liu and H. M. Srivastava, β€œClasses of meromorphically multivalent functions associated with the generalized hypergeometric function,” Mathematical and Computer Modelling, vol. 39, no. 1, pp. 21–34, 2004. [13] D. Yang, β€œSome criteria for multivalent starlikeness,” Southeast Asian Bulletin of Mathematics, vol. 24, no. 3, pp. 491–497, 2000.

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New result of analytic functions related to Hurwitz zeta function.

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 f...
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