Some subclasses of meromorphic functions involving the Hurwitz-Lerch Zeta function

Transcription

1 Hacettepe Journal of Mathematic and Statitic Volume , Some ubclae of meromorphic function involving the Hurwitz-Lerch Zeta function Zhi-Gang Wang and Lei Shi Abtract The main purpoe of thi paper i to invetigate ome ubclae of meromorphic function involving the meromorphic modied verion of the familiar Srivatava-Attiya operator. Such reult a incluion relationhip, convolution propertie, coecient inequalitie, integralpreerving propertie, ubordination and uperordination propertie are proved. Keyword: Analytic function; Meromorphic function; Hurwitz-Lerch Zeta function; Srivatava-Attiya opertor; Dierential ubordination. 2 AMS Claication: Primary 3C45; Secondary 3C8. Received :.3.25 Accepted : Doi :.5672/HJMS Introduction Let Σ denote the cla of function of the form. fz = a k z k, k= which are analytic in the punctured open unit dik U := {z : z C and < z < } =: U\{}. Let f, g Σ, where f i given by. and g i dened by gz = b k z k. k= Correponding Author. School of Mathematic and Computing Science, Hunan Firt Normal Univerity, Changha 425, Hunan, People' Republic of China. wangmath@63.com School of Mathematic and Statitic, Anyang Normal Univerity, Anyang 455, Henan, People' Republic of China. himath@63.com

2 45 Then the Hadamard product or convolution f g of the function f and g i dened by f gz := a k b k z k =: g fz. k= Let P denote the cla of function of the form p z = + p k z k, k= which are analytic and convex in U, and atify the condition Rp z > z U. For two function f and g, analytic in U, the function f i aid to be ubordinate to g in U, or the function g i aid to be uperordinate to f in U, and write fz gz z U, if there exit a Schwarz function ω, which i analytic in U with uch that ω = and ωz < z U fz = g ωz Indeed, it i known that z U. fz gz z U = f = g and fu gu. Furthermore, if the function g i univalent in U, then we have the following equivalence: fz gz z U f = g and fu gu. The following we recall a general Hurwitz-Lerch Zeta function Φz,, a dened by cf., e.g., [2, p. 2 et ep.] z k.2 Φz,, a := k + a k= a C \ Z ; C when z < ; R > when z =, where, a uual, Z := Z \ N Z := {, ±, ±2,...}; N := {, 2, 3,...}. Several intereting propertie and characteritic of the Hurwitz-Lerch Zeta function Φz,, a can be found in the recent invetigation by for example Choi and Srivatava [], Ferreira and López [4], Garg et al. [5], Lin et al. [7], Luo and Srivatava [], Srivatava et al. [2], Ghanim [6] and other. By making ue of the Hurwitz-Lerch Zeta function Φz,, a, Srivatava and Attiya [9] ee alo [8, 9, 4, 7, 22, 23, 24, 27, 28, 29, 3] recently introduced and invetigated the integral operator + b J, b fz = z + c k z k b C \ Z ; C; z U. k + b k=2 Motivated eentially by the above-mentioned Srivatava-Attiya operator J, b, we now introduce the linear operator W, b : Σ Σ dened, in term of the Hadamard product or convolution, by.3 W, b fz := Θ, b z fz b C \ {Z {}}; C; f Σ; z U,

3 45 where, for convenience,.4 Θ, b z := b [Φz,, b b + It can eaily be een from. to.4 that.5 W, b fz = b a k z k. k= ] z b z U. Indeed, the operator W, b can be dened for b C \ {Z {}}, where W, fz := lim b {W, b fz}. We oberve that.6 W, b fz = fz, and.7 W, γfz = γ z γ z t γ ftdt Rγ >. Furthermore, from the denition.5, we nd that.8 W +, b fz = b z b z t b W, b ftdt Rb >. Dierentiating both ide of.8 with repect to z, we get the following ueful relationhip:.9 z W +, b f z = b W, b fz b W +, b fz. By uing the integral operator.5, we now introduce the following ubclae of the cla Σ of meromorphic function... Denition. A function f Σ i aid to be in the cla MS, b η; φ if it atie the ubordination. z W, bf z η φz η W, b fz C; Rb > ; η [, ; φ P; z U..2. Denition. A function f Σ i aid to be in the cla MC, b ; φ if it atie the condition. z W +, b fz + z W, b fz φz, C; Rb > ; φ P; z U. For ome recent invetigation on meromorphic function, ee for example the earlier work [2, 3, 5, 6, 25, 26, 3] and the reference cited therein. In thi paper, we aim at deriving the incluion relationhip, convolution propertie, coecient inequalitie, integral-preerving propertie, ubordination and uperordination propertie for the function clae MS, b η; φ and MC, b ; φ.

4 Preliminary reult The following lemma will be required in the proof of our main reult. 2.. Lemma. [] Let ϑ, γ C. Suppoe that ψ i convex and univalent in U with ψ = and Rϑψz + γ > z U. If p i analytic in U with p =, then the following ubordination implie that pz + pz ψz zp z ψz z U ϑpz + γ z U Lemma. Let α <, C and Rb >. Suppoe alo that the equence {A k } k= i dened by 2. A = α b + b, A k+ = Then 2.2 A k = α b + k b j= 2 α k + 2 j 2α + 3 j b Proof. From 2., we nd that 2.3 k + 2 b + A k+ = 2 α + and 2.4 k + b A k = 2 α + Combining 2.3 and 2.4, we get A k+ 2.5 = k 2α + 3 A k k b + j + b + j. k m= m= k b b + m A m k N. m= k b b + m A m, b b + m A m. Thu, for k 2, we deduce from 2.5 that A k = A k A3 A2 A = α A k A 2 A b + k b The proof of Lemma 2.2 i completed. j= j 2α + 3 j + 2 b + j + b + j Lemma. [2] Let the function Ω be analytic and convex univalent in U with Ω =. Suppoe alo that the function Θ given by Θz = + d nz n + d n+z n+ + i analytic in U. If 2.6 Θz + zθ z ζ then Ωz Θz ϖz = ζ n z ζ n z Rζ > ; ζ ; z U, t ζ n Ωtdt Ωz z U,

5 453 and ϖ i the bet dominant of Lemma. [8] Let q be a convex univalent function in U and let σ, η C with { } R + zq z σ > max, R. q z η If p i analytic in U and σp z + ηzp z σqz + ηzq z, then p q and q i the bet dominant. Denote by Q the et of all function f that are analytic and injective on U Ef, where { } Ef = ε U : lim fz =, z ε and uch that f ε for ε U Ef. Let HU denote the cla of analytic function in U and let H[a, p] denote the ubcla of the function f HU of the form: fz = a + a pz p + a p+z p+ + a C; p N Lemma. [3] Let q be convex univalent in U and κ C. Further aume that Rκ >. If p H[q, ] Q, and p + κzp i univalent in U, then qz + κzq z p z + κzp z implie q p and q i the bet ubordinant. 3. Main reult Firtly, we derive the following incluion relationhip for the function cla MS, b η; φ. 3.. Theorem. Let η < and φ P with 3. R ηφz + η b < z U. Then 3.2 MS, b η; φ MS +, b η; φ. Proof. Let f MS, b η; φ and uppoe that 3.3 ϕz := z W +, bf z η η W +, b fz z U. Then ϕ i analytic in U with ϕ =. By virtue of.9 and 3.3, we get 3.4 b W, bfz = ηϕz η + b. W +, b fz Dierentiating both ide of 3.4 with repect to z logarithmically and uing 3.3, we have z W, bf z zϕ 3.5 z η = ϕz + η W, b fz ηϕz η + b φz. By mean of 3., an application of Lemma 2. to 3.5 yield ϕz = z W +, bf z η φz, η W +, b fz that i f MS +, b η; φ, which implie that the aertion 3.2 of Theorem 3. hold.

6 454 Next, we derive ome convolution propertie of the cla MS, b η; φ Theorem. Let f MS, b η; φ. Then 3.6 fz = [ z z exp η where ω i analytic in U with ω = and ωz < z U. ] φ ωξ dξ ξ Proof. Suppoe that f MS, b η; φ. We nd from. that 3.7 z W, b f z W, b fz = η φ ωz η, z + k= z k, b where ω i analytic in U with ω = and ωz < z U. From 3.7, we get 3.8 W, b f z W, b fz + z which, upon integration, yield 3.9 log z W, b fz = η It follow from 3.9 that 3. W, b fz = z exp φ ωz = η, z z z η φ ωξ dξ. ξ φ ωξ dξ. ξ The aertion 3.6 of Theorem 3.2 can directly be derived from.5 and Theorem. Let f Σ and φ P. Then f MS, b η; φ if and only if 3. { { f z k= k b [ z k η φ e iθ ] η }} b z k k= z U ; θ < 2π. Proof. Suppoe that f MS, b η; φ. We know that.6 i equivalent to 3.2 z W, bf z η φ e iθ z U; θ < 2π. η W, b fz It i eay to ee that the condition 3.2 can be written a follow: { [ 3.3 z W, b f z η φ e iθ ] } η W, b fz z U ; θ < 2π. z On the other hand, we nd from.5 that 3.4 z W, b f z = b k a k z k. k= Combining.5, 3.3 and 3.4, we get the aertion 3. of Theorem 3.3.

7 Theorem. If f MS, b ; [ + 2αz]/ z, then a α b + b, and a k α b + k b j= j 2α + 3 j + 2 b + j + b + j k N\{}. Proof. Suppoe that 3.5 hz := zw, bf z W, b fz α α = + c z + c 2z 2 +. It follow from f MS, b ; [ + 2αz]/ z that h P, and ubequently one ha c k 2 for k N. By virtue of 3.5, we know that 3.6 z W, b f z = [α hz α]w, b fz. It now follow from.5, 3.5 and 3.6 that 3.7 k k= b a k z k = [ + α c z + c 2z 2 + ] [ k= By evaluating the coecient of z k in both ide of 3.7, we get 3.8 k [ b b a k = a k + α c k+ + k By oberving the fact that c k 2 for k N, we nd from 3.8 that 2 α 3.9 a k k + k + b b b + m a m. 3.2 m= Now, we dene the equence {A k } k= a follow: A = α b + b, A k+ = In order to prove that a k A k k N, 2 α k + 2 l= + + b ] b a k z k. b c l a k l]. l k b b + m A m k N. m= we make ue of the principle of mathematical induction. By noting that a A = α b + b. Therefore, auming that a m A m m =, 2, 3,, k; k N.

8 456 Combining 3.9 and 3.2, we get 2 α a k+ k b 2 α k b = A k+. k b b + m a m k b b + m A m m= m= Hence, by the principle of mathematical induction, we have 3.2 a k A k k N a deired. By virtue of Lemma 2.2 and 3.2, we know that 2.2 hold. Combining 3.2 and 2.2, we readily get the coecient etimate aerted by Theorem 3.4. In what follow, we derive ome integral-preerving propertie for the cla MS, b η; φ Theorem. Let f MS, b η; φ with R ηφz + η µ < z U; Rµ >. Then the integral operator F dened by 3.22 F z := µ z t µ ftdt z U ; Rµ > z µ belong to the cla MS, b η; φ. Proof. Let f MS, b η; φ. We then nd from 3.22 that 3.23 z W, b F z + µw, b F z = µ W, b fz. By etting 3.24 qz := η z W, bf z η, W, b F z we oberve that q i analytic in U with q =. It follow from 3.23 and 3.24 that 3.25 ηqz η + µ = µ W, bfz W, b F z. Dierentiating both ide of 3.25 with repect to z logarithmically and uing 3.24, we get zq z 3.26 qz + ηqz η + µ = z W, bf z η φz. η W, b fz Since R ηφz η + µ > z U, by virtue of Lemma 2. and 3.26, we obtain z W, bf z η φz, η W, b F z which implie that the aertion of Theorem 3.5 hold.

9 Theorem. Let f MS, b η; φ with R ηδ φz + η δ µ < z U; δ ; µ C. Then the function K Σ dened by µ δ z /δ 3.27 W, b Kz := t µ W, b ft dt δ z U ; δ z µ belong to the cla MS, b η; φ. Proof. Let f MS, b η; φ and uppoe that 3.28 ϱz := z W, bk z η η W, b Kz In view of 3.27 and 3.28, we have 3.29 µ η δ ηδ ϱz = µ δ z U. δ W, b fz. W, b Kz Now, by mean of 3.27, 3.28 and 3.29, we obtain 3.3 ϱz + Since zϱ z µ η δ ηδ ϱz = η Rµ η δ ηδ φz > z U, z W, bf z η W, b fz φz. it follow from 3.3 and Lemma 2. that ϱz φz, that i K MS, b η; φ. We thu complete the proof of Theorem 3.6. Now, we derive the following ubordination property for the cla MC, b ; φ Theorem. Let f MC, b ; φ with R/b >. Then 3.3 z W +, b fz b b 2 z 2 z Proof. Let f MC, b ; φ and uppoe that 3.32 hz := z W +, b fz z U. t b 2 φtdt φz. Then h i analytic in U. By virtue of.5,. and 3.32, we nd that 3.33 hz + b zh z = z W +, b fz + z W, b fz φz. Thu, an application of Lemma 2.3 to 3.33 yield the deired aertion 3.3 of Theorem Theorem. Let 2 >. Then MC, b 2; φ MC, b ; φ. Proof. Suppoe that f MC, b 2; φ. It follow that z W +, b fz + 2z W, b fz φz z U. Since 2 < and the function φ i convex and univalent in U, we deduce from 3.3 and 3.34 that z W +, b fz + z W, b fz = 2 [ 2z W +, b fz + 2z W, b fz] + z W +, b fz φz, 2

10 458 which implie that f MC, b ; φ. The proof of Theorem 3.8 i thu completed Theorem. Let f MC, b ; φ. If the function F Σ i dened by 3.22, then 3.35 z W +, b F z φz z U. Proof. Let f MC, b ; φ and uppoe that 3.36 χz := z W +, b F z z U. From 3.22, we nd that 3.37 z W +, b F z + µ W +, b F z = µ W +, b fz. By virtue of 3.3, 3.36 and 3.37, we have 3.38 χz + µ z χ z = z W +, b fz φz. Thu, an application of Lemma 2.3 to 3.38, we get the aertion of Theorem Theorem. Let q be univalent in U. Suppoe alo that q atie the condition { } 3.39 R + zq z b q z > max, R. If f Σ atie the following ubordination 3.4 z W +, b fz + z W, b fz q z + b zq z, then z W +, b fz q z, and q i the bet dominant. Proof. Let the function h be dened by We know that 3.33 hold. Combining 3.33 and 3.4, we nd that 3.4 hz + b zh z q z + b zq z. By Lemma 2.4 and 3.4, we obtain the aertion of Theorem 3.. We now derive the following uperordination reult for the cla MC, b ; φ. 3.. Theorem. Let q 2 be convex univalent in U, C with R >. Alo let z W +, b fz H[q 2, ] Q and z W +, b fz + z W, b fz be univalent in U. If then q 2z + b zq 2z z W +, b fz + z W, b fz, q 2z z W +, b fz, and q 2 i the bet ubordinant. Proof. Let the function h be dened by Then q 2z + b zq 2z z W +, b fz + z W, b fz = hz + b zh z. Thu, an application of Lemma 2.5, yield the aertion of Theorem 3.. Finally, combining the above-mentioned ubordination and uperordination reult, we obtain the following andwich type reult.

11 Corollary. Let q 3 be convex univalent and let q 4 be univalent in U, C with R >. Suppoe alo that q 4 atie the condition { } R + zq 4 z b q 4 z > max, R. If z W +, b fz H[q 3, ] Q and z W +, b fz+z W, b fz i univalent in U, alo then q 3z + b zq 3z z W +, b fz + z W, b fz q 4z + b zq 4z, q 3z z W +, b fz q 4z, and q 3 and q 4 are, repectively, the bet ubordinant and the bet dominant. Acknowledgment The preent invetigation wa upported by the National Natural Science Foundation under Grant no. 38, the Natural Science Foundation of Hunan Province under Grant no. 26JJ236, the Foundation for Excellent Youth Teacher of College and Univeritie of Henan Province under Grant no. 23GGJS-46, the Foundation of Educational Committee of Henan Province under Grant no. 7A4. The author would like to thank the referee for their valuable comment and uggetion, which eentially improved the quality of thi paper. Reference [] Choi, J., and Srivatava, H. M. Certain familie of erie aociated with the Hurwitz-Lerch Zeta function, Appl. Math. Comput. 7 25, [2] Dziok, J. Clae of meromorphic function aociated with conic region, Acta Math. Sci. Ser. B Engl. Ed , [3] Dziok, J. Clae of multivalent analytic and meromorphic function with two xed point, Fixed Point Theory Appl. 23, 23: 86, pp. 8. [4] Ferreira, C., and López, J. L. Aymptotic expanion of the Hurwitz-Lerch Zeta function, J. Math. Anal. Appl , [5] Garg, M., Jain, K., and Srivatava, H. M. Some relationhip between the generalized Apotol-Bernoulli polynomial and Hurwitz-Lerch Zeta function, Integral Tranform Spec. Funct. 7 26, [6] Ghanim, F. A tudy of a certain ubcla of Hurwitz-Lerch-Zeta function related to a linear operator, Abtr. Appl. Anal. vol. 23, Article ID , 7 page, 23. [7] Lin, S.-D., and Srivatava, H. M. Some familie of the Hurwitz-Lerch Zeta function and aociated fractional derivative and other integral repreentation, Appl. Math. Comput , [8] Liu, J.-L. Sucient condtion for trongly tarlike function involving the generalized Srivatava-Attiya operator, Integral Tranform Spec. Funct. 22 2, 799. [9] Liu, Z.-H., Wang, Z.-G., Wen, F.-H., and Sun, Y. Some ubclae of analytic function involving the generalized Srivatava-Attiya operator, Hacet. J. Math. Stat. 4 22, [] Luo, Q.-M., and Srivatava, H. M. Some generalization of the Apotol-Bernoulli and Apotol-Euler polynomial, J. Math. Anal. Appl , [] Miller S. S., and Mocanu, P. T. On ome clae of rt order dierential ubordination, Michigan Math. J , [2] Miller, S. S., and Mocanu, P. T. Dierential Subordination: Theory and Application, in: Serie in Pure and Applied Mathematic, Vol. 225, Marcel Dekker, New York, 2.

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