Some basic properties of certain subclasses of meromorphically starlike functions

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1 Wang et al. Journal of Inequalities Applications 2014, 2014:29 R E S E A R C H Open Access Some basic properties of certain subclasses of meromorphically starlike functions Zhi-Gang Wang 1*, HM Srivastava 2 Shao-Mou Yuan 3 * Correspondence: zhigangwang@foxmail.com 1 School of Mathematics Statistics, Anyang Normal University, Anyang, Henan , People s Republic of China Full list of author information is available at the end of the article Abstract In this paper, we introduce investigate certain subclasses of meromorphically starlike functions. Such results as coefficient inequalities, neighborhoods, partial sums, inclusion relationships are derived. Relevant connections of the results derived here with those in earlier works are also pointed out. MSC: Primary 30C45; secondary 30C80 Keywords: meromorphic function; starlike function; Hadamard product or convolution; neighborhood; partial sum 1 Introduction Let denote the class of functions f of the form f z= 1 z + a k z k, 1.1 which are analytic in the punctured open unit disk U := { z : z C 0 < z <1 } =: U \{0}. Afunctionf is said to be in the class MS αofmeromorphically starlike functions of order α if it satisfies the inequality zf z R < α z U;0 α <1. f z Let P denote the class of functions p given by pz=1+ p k z k z U, 1.2 which are analytic in U satisfy the condition R pz >0 z U. For some recent investigations on analytic starlike functions, see for example the earlier works [1 14] the references cited in each of these earlier investigations Wang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited.

2 Wang et al. Journal of Inequalities Applications 2014, 2014:29 Page 2 of 13 Given two functions f, g,wheref is given by 1.1g is given by gz= 1 z + b k z k, the Hadamard product or convolution f g is defined by f gz:= 1 z + a k b k z k =: g f z. Afunctionf is said to be in the class Hβ, λ if it satisfies the condition zf z R + β z2 f z < βλ λ β λ z U, 1.3 f z f z 2 2 where throughout this paper unless otherwise mentioned the parameters β λ are constrained as follows: β 0 1 λ < Clearly, we have H0, λ=ms λ. In a recent paper, Wang et al. [15] had proved that if f Hβ, λ, then f MS λ, which implies that the class Hβ, λ is a subclass of the class MS λ of meromorphically starlike functions of order λ. Let H + β, λ denote the subset of Hβ, λ such that all functions f Hβ, λ havingthe following form: f z= 1 z a k z k a k In the present paper, we aim at proving some coefficient inequalities, neighborhoods, partial sums inclusion relationships for the function classes Hβ, λh + β, λ. 2 Preliminary results In order to prove our main results, we need the following lemmas. Lemma 2.1 See [16] If the function p P is given by 1.2, then p k 2 k N.

3 Wang et al. Journal of Inequalities Applications 2014, 2014:29 Page 3 of 13 Lemma 2.2 Let β >0 >0.Suppose also that the sequence {A k } is defined by A 1 = A k+1 = 21 γ 2β 1 2β +βk +1k A l k N. 2.1 Then A k = k 1 j=1 3 6β 2γ + jβj + 1 2β +βj +1j +1 k N \{1}. 2.2 Proof By virtue of 2.1, we easily get [ 1 2β +βk +1k +1 ] Ak+1 =2 1+ A l, 2.3 [ ] k 1 1 2β +βk +k Ak =2 1+ A l. 2.4 Combining , we obtain A k+1 A k = 3 6β 2γ + kβk β +βk +1k +1 Thus, for k 2, we deduce from 2.5that A k = A k A3 A2 k 1 A 1 = A k 1 A 2 A 1 j=1 3 6β 2γ + jβj β +βj +1j +1 The proof of Lemma 2.2 is evidently completed. The following two lemmas can be derived from [17, Theorem 1] see also [18], we here choose to omit the details of proof. Lemma 2.3 Let 1+βλ λ + 1 λ 3 β > Suppose also that f is given by 1.1. If [ ] k + βkk 1+γ ak, 2.7

4 Wang et al. Journal of Inequalities Applications 2014, 2014:29 Page 4 of 13 where throughout this paper unless otherwise mentioned the parameter γ is constrained as follows: γ := λ βλ λ + 1 β 2 2, 2.8 then f Hβ, λ. Lemma 2.4 Let f be given by 1.5. Suppose also that γ is defined by 2.8 the condition 2.6 holds true. Then f H + β, λ if only if [ ] k + βkk 1+γ ak Main results We begin by proving the following coefficient estimates for functions belonging to the class Hβ, λ. Theorem 3.1 Let γ be defined by 2.8. If f Hβ, λ with 0<β < 2/5, then a 1, a k k 1 j=1 3 6β 2γ + jβj + 1 2β +βj +1j +1 k N \{1}. Proof Suppose that qz:= zf z f z β z2 f z f z + βλ λ β λ Then, by the definition of the function class Hβ, λ, we know that q is analytic in U R qz >0 z U with q0 = 1 γ 2β >0. It follows from that qzf z= zf z βz 2 f z γ f z. 3.2 By noting that hz= qz P,

5 Wang et al. Journal of Inequalities Applications 2014, 2014:29 Page 5 of 13 if we put qz=c 0 + c k z k c 0 =, by Lemma 2.1,weknowthat c k 21 γ 2β k N. It follows from 3.2that c 0 + c k z k 1 z + a k z k 1 = z ka k z k 2β z + β kk 1a k z k 1 γ z + a k z. k 3.3 In view of 3.3, we get 1 γ 2βa 1 + c 2 = a 1 γ a c k+2 +a k+1 + a l c k+1 l = k +1a k+1 βkk +1a k+1 γ a k+1 k N. 3.5 From 3.4, we obtain a Moreover, we deduce from 3.5that a k+1 21 γ 2β β +βk +1k +1 a l k N. 3.7 Next,wedefinethesequence{A k } as follows: A 1 = Inordertoprovethat A k+1 = 21 γ 2β β +βk +1k +1 A l k N. 3.8 a k A k k N,

6 Wang et al. Journal of Inequalities Applications 2014, 2014:29 Page 6 of 13 we make use of the principle of mathematical induction. By noting that a 1 A 1 =. Therefore, assuming that a l A l l =1,2,3,...,k; k N. Combining , we get a k+1 21 γ 2β β +βk +1k γ 2β β +βk +1k +1 a l A l = A k+1 k N. Hence, by the principle of mathematical induction, we have a k A k k N 3.9 as desired. By means of Lemma , we know that 2.2 holds true. Combining , we readily get the coefficient estimates asserted by Theorem 3.1. Following the earlier works based upon the familiar concept of neighborhood of analytic functions by Goodman [19] Ruscheweyh[20], more recently by Altintaş et al. [21 24], Cǎtaş [25], Cho et al. [26], Liu Srivastava [27 29], Frasin [30], Keerthi et al. [31], Srivastava et al. [32] Wang et al. [33]. Assuming that γ is given by 2.8 the condition 2.6 of Lemma 2.3 holds true, we here introduce the δ-neighborhood of a function f of the form 1.1 by means of the following definition: N δ f := { g : gz= 1 z + b k z k k + βkk 1+γ a k b k δ δ 0 } By making use of the definition 3.10, we now derive the followingresult. Theorem 3.2 Let the condition 2.6 hold true. If f satisfies the condition f z+εz 1 Hβ, λ 1+ε ε C; ε < δ; δ >0, 3.11 then N δ f Hβ, λ. 3.12

7 Wang et al. Journal of Inequalities Applications 2014, 2014:29 Page 7 of 13 Proof By noting that the condition 1.3canbewrittenas zf z f z + β z2 f z +1 f z + β z2 f z <1 z U, γ 1 f z zf z f z we easily find from 3.13 that a function g Hβ, λifonlyif zg z+βz 2 g z+gz zg z+βz 2 g z+2γ 1gz σ z U; σ C; σ =1, which is equivalent to where g hz z 1 0 z U, 3.14 hz= 1 z + It follows from 3.15that c k z k k + βkk 1+1 [k + βkk 1+2γ 1]σ c k := [β +1 γ βσ ] c k = k + βkk 1+1 [k + βkk 1+2γ 1]σ 2[β +1 γ βσ ] = k + βkk 1+1+[k + βkk 1+2γ 1] σ 21 γ 2β σ k + βkk 1+γ σ =1. If f given by 1.1 satisfies the condition 3.11, we deduce from 3.14that f hz z 1 ε ε < δ; δ >0, or equivalently, f hz z 1 δ z U; δ > We now suppose that qz= 1 z + d k z k N δ f. It follows from 3.10that q f hz z 1 = d k a k c k z k+1 z k + βkk 1+γ d k a k < δ. 3.17

8 Wang et al. Journal of Inequalities Applications 2014, 2014:29 Page 8 of 13 Combining , we easilyfind that q hz z 1 = [f +q f ] hz z 1 f hz z 1 q f hz z 1 >0, which implies that q hz z 1 0 z U. Therefore, we have qz N δ f Hβ, λ. The proof of Theorem 3.2 is thus completed. Next, we derive the partial sums of the class Hβ, λ. For some recent investigations involving the partial sums in analytic function theory, one can find in [28, 29, 34, 35] the references cited therein. Theorem 3.3 Let f be given by 1.1 define the partial sums f n z of f by If f n z= 1 n z + a k z k n N k + βkk 1+γ a k 1, 3.19 where γ is given by 2.8 the condition 2.6 holds true, then 1. f Hβ, λ; 2. f z n + βnn +1+2β +2γ R n N; z U, 3.20 f n z n + βnn +1+1+γ fn z R f z n + βnn +1+1+γ n + βnn β n N; z U The boundsin are sharp. Proof First of all, we suppose that f 1 z= 1 z. We know that f 1 z+εz 1 = 1 Hβ, λ. 1+ε z

9 Wang et al. Journal of Inequalities Applications 2014, 2014:29 Page 9 of 13 From 3.19, we easily find that k + βkk 1+γ a k 0 1, which implies that f N 1 z 1. By virtue of Theorem 3.2,wededucethat f N 1 z 1 Hβ, λ. Next, it is easy to see that n +1+βnn +1+γ Therefore, we have > n + βnn 1+γ >1 n N. n a k + n + βnn +1+1+γ a k k=n+1 k + βkk 1+γ a k We now suppose that n + βnn +1+1+γ f z h 1 z= f n z k=n+1 a kz k+1 =1+ n+βnn+1+1+γ It follows from that n + βnn +1+2β +2γ n + βnn +1+1+γ 1+ n a kz k h 1 z 1 h 1 z+1 n+βnn+1+1+γ 2 2 n a k n+βnn+1+1+γ k=n+1 a k k=n+1 a k 1 z U, which shows that R h 1 z 0 z U Combining , we deduce that the assertion 3.20 holdstrue. Furthermore, if we put then f z= 1 z n + βnn +1+1+γ zn+1, 3.25 f z f n z =1 n + βnn +1+2β +2γ n + βnn +1+1+γ zn+2 n + βnn +1+1+γ z 1, which implies that the bound in 3.20isthebestpossibleforeachn N.

10 Wang et al. Journal of Inequalities Applications 2014, 2014:29 Page 10 of 13 Similarly, we suppose that n + βnn β fn z h 2 z= f z k=n+1 a kz k+1 =1 n+βnn+1+2 2β n + βnn +1+1+γ n + βnn β 1+ a kz k In view of , we conclude that h 2 z 1 h 2 z+1 n+βnn+1+2 2β 2 2 n a k n+βnn+1+2β+2γ k=n+1 a k k=n+1 a k 1 z U, which implies that R h 2 z 0 z U Combining , we readily get the assertion 3.21 oftheorem3.3. Thebound in 3.21 is sharp with the extremal function f given by We thus complete the proof of Theorem 3.3. In what follows, we turn to quotients involving derivatives. The proof of Theorem 3.4 below is similar to that of Theorem 3.3, we here choose to omit the analogous details. Theorem 3.4 Let f be given by 1.1 define the partial sums f n z of f by If the conditions hold, where γ isgivenby 2.8,then f z n +2γ +n +1n +2β R f n z n + βnn +1+1+γ n N; z U, 3.28 f R n z f z n + βnn +1+1+γ n 2n +1β +2n +1 nγ n N; z U The bounds in are sharp with the extremal function given by Finally, we prove the following inclusion relationship for the function class Hβ, λ. Theorem 3.5 Let Then β 1 β λ 1 λ 2 <1. Hβ 1, λ 1 Hβ 2, λ Proof Suppose that f Hβ 1, λ 1. Then zf z z 2 f z R + β 1 < λ 1 [β 1 λ ] 1 + β 1 f z f z 2 2 z U. 3.31

11 Wang et al. Journal of Inequalities Applications 2014, 2014:29 Page 11 of 13 Since β 1 β 2 1 1/2 λ 1 λ 2 <1,wefindthat [ λ 1 β 1 λ ] 1 + β λ 2 [β 1 λ ] 1 + β It follows from that zf z z 2 f z R + β 1 < λ 2 [β 1 λ ] 1 + β 1 f z f z 2 2 z U, 3.33 which shows that f Hβ 1, λ 2, subsequently, we see that f MS λ 2, that is, zf z R < λ 2 z U f z Now, by setting so that μ = β 2 β 1, 0<μ 1, we easily find from that zf z R f z zf z = μr f z <0 z U, z 2 f z + β 2 f z + β 1 z 2 f z f z [ λ 2 β 2 λ ] 1 β ] [ λ 2 β 1 λ β 1 zf z +1 μr + λ 2 2 f z that is, f Hβ 2, λ 2. Therefore, the assertion 3.30of Theorem 3.5holdstrue. From Theorem 3.5 the definition of the function class H + β, λ, we easily get the following inclusion relationship. Corollary 3.6 Let Then β 1 β λ 1 λ 2 <1. H + β 1, λ 1 H + β 2, λ 2 MS λ 2.

12 Wang et al. Journal of Inequalities Applications 2014, 2014:29 Page 12 of 13 By virtue of Lemma 2.4, we obtain the following result. Corollary 3.7 Let f H + β, λ. Suppose also that γ is defined by 2.8 the condition 2.6 holds true. Then a k k + βkk 1+γ. Each of these inequalities is sharp, with the extremal function given by f k z= 1 z k + βkk 1+γ zk. Competing interests The authors declare that they have no competing interests. Authors contributions The authors completed the paper together. They also read approved the final manuscript. Author details 1 School of Mathematics Statistics, Anyang Normal University, Anyang, Henan , People s Republic of China. 2 Department of Mathematics Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada. 3 School of Mathematics Computing Science, Changsha University of Science Technology, Changsha, Hunan , People s Republic of China. Acknowledgements The present investigation was supported by the National Natural Science Foundation under Grant nos , , , the Foundation for Excellent Youth Teachers of Colleges Universities of Henan Province under Grant no. 2013GGJS-146, the NaturalScienceFoundationofEducationalCommitteeofHenanProvince under Grant no. 14B of the People s Republic of China. The authors would like to thank the referees for their valuable comments suggestions, which essentially improved the quality of this paper. Received: 9 October 2013 Accepted: 5 January 2014 Published: 24 Jan 2014 References 1. Ali, RM, Ravichran, V: Classes of meromorphic α-convex functions. Taiwan. J. Math. 14, Ali, RM, Ravichran, V, Seenivasagan, N: Subordination superordination of the Liu-Srivastava linear operator on meromorphic functions. Bull. Malays. Math. Soc. 31, Ali, RM, Ravichran, V, Seenivasagan, N: On subordination superordination of the multiplier transformation for meromorphic functions. Bull. Malays. Math. Soc. 33, Liu, J-L, Srivastava, HM: Some convolution conditions for starlikeness convexity of meromorphically multivalent functions. Appl. Math. Lett. 16, Liu, M-S, Zhu, Y-C, Srivastava, HM: Properties characteristics of certain subclasses of starlike functions of order β. Math. Comput. Model. 48, Mohd, MH, Ali, RM, Keong, LS, Ravichran, V: Subclasses of meromorphic functions associated with convolution. J. Inequal. Appl. 2009, Article ID Nunokawa, M, Ahuja, OP: On meromorphic starlike convex functions. Indian J. Pure Appl. Math. 32, Silverman, H, Suchithra, K, Stephen, BA, Gangadharan, A: Coefficient bounds for certain classes of meromorphic functions. J. Inequal. Appl. 2008, Article ID Srivastava, HM, Khairnar, SM, More, M: Inclusion properties of a subclass of analytic functions defined by an integral operator involving the Gauss hypergeometric function. Appl. Math. Comput. 218, Srivastava, HM, Yang, D-G, Xu, N: Some subclasses of meromorphically multivalent functions associated with a linear operator. Appl. Math. Comput. 195, Sun, Y, Kuang, W-P, Liu, Z-H: Subordination superordination results for the family of Jung-Kim-Srivastava integral operators. Filomat 24, Tang, H, Deng, G-T, Li, S-H: On a certain new subclass of meromorphic close-to-convex functions. J. Inequal. Appl. 2013, Wang, Z-G, Sun, Y, Xu, N: Some properties of certain meromorphic close-to-convex functions. Appl. Math. Lett. 25, Yuan, S-M, Liu, Z-M, Srivastava, HM: Some inclusion relationships integral-preserving properties of certain subclasses of meromorphic functions associated with a family of integral operators. J. Math. Anal. Appl. 337, Wang,Z-G,Liu,Z-H,Xiang,R-G:Somecriteriaformeromorphicmultivalentstarlikefunctions.Appl.Math.Comput. 218,

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