On a New Subclass of Meromorphically Multivalent Functions Defined by Linear Operator

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1 International Mathematical Forum, Vol. 8, 213, no. 15, HIKARI Ltd, On a New Subclass of Meromorphically Multivalent Functions Defined by Linear Operator Ali Hussein Battor, Waggas Galib Atshan and Ahmed Abbas Abdalreda Department of Mathematics, College of Education for Girls University of Kufa, Najaf-Iraq Department of Mathematics College of Computer Science and Mathematics University of Al-Qadisiya, Diwaniya -Iraq battor ali@yahoo.com waggashnd@gmail.com, waggas hnd@yahoo.com ahmedaljnahe@yahoo.com Copyright c 213 Ali Hussein Battor et al. 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. Abstract Making use a linear operator, which is defined here by the convolution (or Hadamard product) involving the generalized hypergeometric function, we discuss the new subclass Σ p (q, s, α 1,σ) of meromorphically multivalent functions with negative coefficients in the punctured unit disk U. In the present paper, we attempt to give the various important properties of this new subclass Σ p(q, s, α 1,σ), like, coefficient inequality, convex set, growth and distortion bounds. We also derive some interesting results of our class, like, integral representation, neighborhoods of the class Σ p(q, s, α 1,σ), radii of starlikeness and convexity and integral operator. Mathematics Subject Classification: Primary 3C45; Secondary 3C5 Keywords: Meromorphic Multivalent Function, Linear Operator, Distortion bounds, Neighborhoods, Radii of starlikeness, Generalized hypergeometric function, Integral Operator

2 714 Ali Hussein Battor et al. 1 Introduction Let Σ p denote the class of functions of the form: f(z) = 1 z a p n z n ; (,p N = {1, 2, }), (1.1) which are analytic and meromorphic multivalent in the punctured unit disk U = {z C :< z < 1} = U {}. Recall [3,9] that a function f Σ p is said to be meromorphically multivalent starlike of order ρ if it is satisfying the following condition :- { } zf (z) Re >ρ; ( ρ<p; z U ). (1.2) f(z) Similarly, recall[14] a function f Σ p is said to be meromorphically multivalent convex of order ρ if it is satisfying the following condition :- { } Re 1+ zf (z) >ρ; ( ρ<p; z U ). (1.3) f (z) For functions f Σ p given by (1.1) and g Σ p given by g(z) = 1 z b p n z n, (b n,p N), we define the Hadamard product (or convolution) of f and g by (f g) = 1 z a p n b n z n =(g f)(z). (1.4) For positive real values of α 1,,α q and β 1,,β s (β j, 1, ; j = 1, 2,,s), we now define the generalized hypergeometric function qf s (α 1,,α q ; β 1,,β s ; z) by qf s (α 1,,α q ; β 1,,β s ; z) = n= (α 1 ) n (α q ) n z n (β 1 ) n (β s ) n n!, (1.5) (q s +1;q, s N = N {}; z U), where (θ) v is the Pochhammer symbol defined 1 v = (θ) v = (1.6) θ(θ + 1)(θ +2)(θ + v 1), v N.

3 Meromorphically multivalent functions 715 Corresponding to the function h p (α 1,,α q ; β 1,,β s ; z), defined by h p (α 1,,α q ; β 1,,β s ; z) =z p qf s (α 1,,α q ; β 1,,β s ; z); (1.7) we consider a linear operator H p (α 1,,α q ; β 1,,β s ): Σ p Σ p, which is defined by means of the following Hadamard product (or convolution): H p (α 1,,α q ; β 1,,β s )f(z) =h p (α 1,,α q ; β 1,,β s ; z) f(z). (1.8) We observe that, for a function f(z) of the form (1.1); we have H p (α 1,,α q ; β 1,,β s )f(z) = 1 z p If, for convenience, we write (α 1 ) n (α q ) n (β 1 ) n (β s ) n n! zn. (1.9) H p,q,s (α 1 )=H p (α 1,,α q ; β 1,,β s ), (1.1) then one can easily verify from the definition (1.8) that z(h p,q,s (α 1 )f(z)) = α 1 H p,q,s (α 1 +1)f(z) (α 1 + p)h p,q,s (α 1 )f(z). (1.11) The linear operator H p,q,s (α 1 ) was investigated recently by Lin and Srivastava [11]. Some interesting subclasses of analytic functions associated with the generalized hypergeometric function, were considered recently by (for example) Dziok and Srivastava {[5] and [6]}, Gangadharan et. al. [7] and Liu [1]. Definition 1.1 : Let Σ p (q, s, α 1,σ) denote the new class of functions f Σ p, which satisfy the condition, z(h p,q,s(α 1 )f(z)) (H p,q,s(α 1 +(1+p) )f(z)) z(h p,q,s(α 1 )f(z)) (H p,q,s(α 1 (1 + p)+2σ )f(z)) < 1 (1.12) where σ < p, p N and z U. Such type of study was carried out by several different authors for another classes, like, Aouf [1,2], Mogra et. al. [13], Miller [12] and Cho et.al. [4]. 2 Main Results First, we derive the coefficient inequality for the class Σ p (q, s, α 1,σ) contained in:-

4 716 Ali Hussein Battor et al. Theorem 2.1 : Let f Σ p. Then f is in the class Σ p (q, s, α 1,σ) if and only if [n + σ 1] (α 1) n (α q ) n p((p +1) σ), (2.1) (β 1 ) n (β s ) (n 1)! where σ < p, p N. The result is sharp for the function f(z) = 1 p((p +1) σ)(n 1)! + zp [n + σ 1] (α 1) n(α q) n z n, (n p). (2.2) Proof : Suppose that the inequality (2.1) holds true and z = 1. Then, we have z(h p,q,s (α 1 )f(z)) +(1+p)(H p,q,s (α 1 )f(z)) z(h p,q,s (α 1 )f(z)) +(2σ (1 + p))(h p,q,s (α 1 )f(z)) = n(n + p) (α 1) n (α q ) n (β 1 ) n (β s ) n n! zn 1 2p(p +1) σ)z (p+1) n(n +2(σ 1) p) (α 1) n (α q ) n (β 1 ) n (β s ) n n(n + p) (α 1) n (α q ) n (β 1 ) n (β s ) n n! z n 1 2p((p +1) σ) = hence + n[n +2(σ 1) p] (α 1) n (α q ) n (β 1 ) n (β s ) n n! z n 1 2n(n + σ 1) (α 1) n (α q ) n 2p((p +1) σ), (β 1 ) n (β s ) n n! (n + σ 1) (α 1) n (α q ) n (β 1 ) n (β s ) n (n 1)! p(p +1) σ), n! zn 1 by hypothesis. Thus by maximum modulus principle, f Σ p(q, s, α 1,σ). To show the converse, suppose that f Σ p (q, s, α 1,σ). Then from (1.12), we have z(h p,q,s(α 1 )f(z)) (H p,q,s(α 1 +(1+p) )f(z)) z(h p,q,s(α 1 )f(z)) (H p,q,s(α 1 (1 + p)+2σ )f(z)) n[n + p] (α 1) n(α q) n n! zn 1 = 2p((p +1) σ)z (p+1) n[n +2(σ 1) p] (α 1) n(α q) n < 1. n! zn 1

5 Meromorphically multivalent functions 717 Since Re(z) z for all z (z U), we have Re 2p((p +1) σ)z (p+1) n(n + p) (α 1) n(α q) n n! zn 1 n[n +2(σ 1) p] (α 1) n(α q) n n! zn 1 < 1. (2.3) We choose the value of z on the real axis so that z(hp,q,s(α 1)f(z)) (H p,q,s(α 1 is real. Upon )f(z)) clearing the denominator of (2.3) and letting z 1, through real values so we can write (2.3) as [n + σ 1] (α 1) n (α q ) n p((p +1) σ). (β 1 ) n (β s ) n (n 1)! Sharpness of the result follows by setting f(z) = 1 p((p +1) σ)(n 1)! + zp [n + σ 1] (α 1) n(α q) n z n, (n 1). Corollary 2.2 : Let f Σ p (q, s, α 1,σ). Then p((p +1) σ)(n 1)! [n + σ 1] (α 1) n(α q) n, (n 1). Theorem 2.3 : The class Σ p(q, s, α 1,σ) is a convex set. Proof : Let f 1 and f 2 be the arbitrary elements of Σ p (q, s, α 1,σ). Then for every t ( t 1), we show that (1 t)f 1 + tf 2 Σ p (q, s, α 1,σ). Thus, we have (1 t)f 1 + tf 2 = 1 z + [(1 t)a p n + tb n ]z n. Hence, (n + σ 1) (α 1) n (α q ) n [(1 t) + tb n ] (β 1 ) n (β s ) n (n 1)! =(1 t) +t (n + σ 1) (α 1) n (α q ) n (β 1 ) n (β s ) n (n 1)! (n + σ 1) (α 1) n (α q ) n b n (β 1 ) n (β s ) n (n 1)! (1 t)p((p +1) σ)+tp((p +1) σ) =p((p +1) σ). This completes the proof.

6 718 Ali Hussein Battor et al. 3 Growth and Distortion Bounds Next, we obtain the growth and distortion bounds for the linear operator H p,q,s (α 1 ). Theorem 3.1 :Iff Σ p (q, s, α 1,σ), then 1 +1) σ) p((p r p H rp p,q,s (α 1 )f(z) 1 +1) σ) +p((p r p, ( z = r<1). [p + σ 1] rp [p + σ 1] (3.1) The result is sharp for the function f(z) = 1 p((p +1) σ)(p 1)! + zp [p + σ 1] (α 1) p(α q) p z p, (3.2) (β 1 ) p(β s) p Proof : Let f Σ p(q, s, α 1,σ). Then by Theorem 2.1, we get [p + σ 1] (α 1) p (α q ) p 1 (β 1 ) p (β s ) p (p 1)! [n + σ 1] (α 1) n (α q ) n p(p +1) σ), (β 1 ) n (β s ) n (n 1)! or p((p +1) σ)(p 1)! [p + σ 1] (α 1) p(α q) p (3.3) (β 1 ) p(β s) p H p,q,s (α 1 )f(z) 1 z p + 1 z p + (α 1 ) n (α q ) n (β 1 ) n (β s ) n (α 1 ) p (α q ) p (β 1 ) p (β s ) p = 1 r + (α 1) p (α q ) p r p p (β 1 ) p (β s ) p (p 1)! (n 1)! z n z p (p 1)! 1 p((p +1) σ) + r p. (3.4) rp [p + σ 1]

7 Meromorphically multivalent functions 719 Similarly, H p,q,s (α 1 )f(z) 1 z p (α 1 ) n (α q ) n (β 1 ) n (β s ) n 1 z (α 1) p (α q ) p z p p (β 1 ) p (β s ) p (p 1)! = 1 r (α 1) p (α q ) p r p p (β 1 ) p (β s ) p (p 1)! (n 1)! z n 1 p((p +1) σ) r p. (3.5) rp [p + σ 1] From (3.4) and (3.5),we get (3.1) and the proof is complete. Theorem 3.2 :Iff Σ p (q, s, α 1,σ), then 1 r p2 ((p +1) σ) r p 1 (H p+1 p,q,s (α 1 )f(z)) [p + σ 1] 1 r + p2 ((p +1) σ) r p 1, ( z = r<1).(3.6) p+1 [p + σ 1] The result is sharp for the function f is given by (3.2). The proof is similar to that of Theorem (3.1). 4 Integral Representation In the following theorem, we obtain integral representation for H p,q,s (α 1 )f(z). Theorem 4.1 : Let f Σ p(q, s, α 1,σ). Then H p,q,s (α 1 )f(z) = z [ z exp ] (2σ (1 + p))ψ(t) (1 + p) dt dt, t(1 ψ(t)) where ψ(t) < 1,z U. Proof : By letting z(hp,q,s(α 1)f(z)) (H p,q,s(α 1 = η(z) in (1.12), we have )f(z)) or equivalently η(z)+(1+p) η(z) (1+p)+2σ < 1, η(z) + (1 + p) η(z) (1 + p)+2σ = ψ(z), ( ψ(z) < 1,z U). So (H p,q,s (α 1 )f(z)) (H p,q,s (α 1 )f(z)) = (2σ (1 + p))ψ(z) (1 + p), z(1 ψ(z))

8 72 Ali Hussein Battor et al. after integration, we have log((h p,q,s (α 1 )f(z)) )= z (2σ (1 + p))ψ(t) (1 + p) dt. t(1 ψ(t)) Therefore, [ z (H p,q,s (α 1 )f(z)) = exp ] (2σ (1 + p))ψ(t) (1 + p) dt. t(1 ψ(t)) After integration, we have H p,q,s (α 1 )f(z) = z [ z exp and this gives the required result. ] (2σ (1 + p))ψ(t) (1 + p) dt dt, t(1 ψ(t)) 5 Neighborhoods for the class Σ p (q, s, α 1,σ) Following the earlier works on neighborhoods of analytic functions by Goodman [8] and Ruscheweyh [15], we begin by introducing here the δ-neighborhood of a function f Σ p of the form (1.1) by means of the definition below:- { } N δ (f) = g Σ p : g(z) =z p b nz n and n b n δ, δ<1. (5.1) Particulary for the identity function e(z) =z p, we have { } N δ (e) = g Σ p : g(z) =z p b nz n and n b n δ. (5.2) Definition 5.1 : A function f Σ p is said to be in the class Σ p,y (q, s, α 1,σ) if there exists function g Σ p(q, s, α 1,σ) such that f(z) g(z) 1 < 1 y, (z U, y<1). Theorem 5.1 :Ifg Σ p(q, s, α 1,σ) and then N δ (g) Σ p,y(q, s, α 1,σ). p((p +1) σ)(p 1)! y =1 [p + σ 1] (α 1) n(α q) n, (5.3)

9 Meromorphically multivalent functions 721 Proof : Let f N δ (g). Then, we find from (5.1) that n b n δ, which implies the coefficient inequality b n δ, (n p). Since g Σ p (q, s, α 1,σ), then by using Theorem 2.1, we get so that f(z) g(z) 1 < p((p +1) σ)(p 1)! b n [p + σ 1] (α 1) n(α q) n, b n 1 b n δ[p + σ 1] (α 1) n(α q) n =1 y. [p + σ 1] (α 1) n(α q) n p((p +1) σ)(p 1)! Hence, by Definition 5.1, f Σ p,y (q, s, α 1,σ) for y given by (5.3). This completes the proof. Theorem 5.2 :If p((p +1) σ) δ = [p + σ 1] (α 1) p(α q) p, (β 1 ) p(β s) p then Σ p (q, s, α 1,σ) N δ (e). Proof : Let f Σ p,y (q, s, α 1,σ). Then by using Theorem 2.1, we have [p + σ 1] (α 1) p (α q ) p 1 (β 1 ) p (β s ) p (p 1)! p((p +1) σ). (5.4) On the other hand, from (2.1) and (5.4) that (α 1 ) p (α q ) p (β 1 ) p (β s ) p +1) σ)(p 1)! p((p [p + σ 1] (α 1) p(α q) p (β 1 ) p(β s) p p((p +1) σ) =. [p + σ 1] n p((p +1) σ) [p + σ 2] (α 1) p (α q ) p (β 1 ) p (β s ) p 1 (p 1)!

10 722 Ali Hussein Battor et al. That is, n p((p +1) σ) [p + σ 1] (α 1) p(α q) p = δ. (β 1 ) p(β s) p Thus, by the definition given by (5.2), f N δ (e). This completes the proof. 6 Radii of Starlikeness and Convexity In the following theorems, we discuss the radii of starlikeness and convexity. Theorem 6.1 : If f Σ p(q, s, α 1,σ), then f is multivalent meromorphic starlike of order ϕ ( ϕ<p) in the disk z <r 1, where r 1 = inf n { (p ϕ)[n + σ 1] (α 1 ) n(α q) n } 1 n+p, n p. (n ϕ +2p)p((p +1) σ)(n 1)! The result is sharp for the function f given by (2.2). Proof : It is sufficient to show that zf (z) f(z) + p p ϕ for z <r 1. (6.1) But zf (z)+pf(z) f(z) Thus, (6.1) will be satisfied if (n + p) z n+p (n + p) z n+p 1. z n+p 1 p ϕ, z n+p or if Since f Σ p (q, s, α 1,σ), we have (n ϕ +2p) p ϕ z n+p 1. (6.2) [n + σ 1] (α 1) n(α q) n p((p +1) σ)(n 1)! 1.

11 Meromorphically multivalent functions 723 Hence, (6.2) will be true if or equivalently (n ϕ +2p) p ϕ z n+p [n + σ 1] (α 1) n(α q) n p((p +1) σ)(n 1)!, z { (p ϕ)[n + σ 1] (α 1 ) n(α q) n } 1 n+p, n p (n ϕ +2p)p((p +1) σ)(n 1)! which follows the result. Theorem 6.2 : If f Σ p (q, s, α 1,σ), then f is multivalent meromorphic convex of order ϕ ( ϕ<p) in the disk z <r 2, where r 2 = inf n { (p ϕ)[n + σ 1] (α 1 ) p(α q) p } 1 n+p (β 1 ) p(β s) p. n(n ϕ +2p)((p +1 σ)(n 1)! The result is sharp for the function f given by (2.2). Proof : It is sufficient to show that zf (z) f (z) +1+p p ϕ for z <r 2. (6.3) But n(n + p)a zf (z) f (z) +1+p = zf (z)+(p +1)f (z) n z n+p f (z) p. n z n+p Thus, (6.3) will be satisfied if or if n(n + p) z n+p n p p ϕ, z n+p Since f Σ p (q, s, α 1,σ), we have n(n ϕ +2p) z n+p 1. (6.4) p(p ϕ) [n + σ 1] (α 1) n(α q) n p((p +1) σ)(n 1)! 1.

12 724 Ali Hussein Battor et al. Hence, (6.4) will be true if or equivalently z n(n ϕ +2p) p(p ϕ) which follows the result. z n+p [n + σ 1] (α 1) n(α q) n p((p +1) σ)(n 1)!, { (p ϕ)[n + σ 1] (α 1 ) n(α q) n } 1 n+p, n p, n(n ϕ +2p)((p +1) σ)(n 1)! 7 Integral Operator Theorem 7.1 : Let the function f be given by (1.1) in the class Σ p(q, s, α 1,σ). Then, the integral operator w(z) =μ 1 is in the class Σ p (q, s, α 1,τ), where u μ f(uz)du, ( <u 1, <μ< ), (7.1) τ = (p + 1)(μ + p + 1)[p + σ 1] μ[p 1]((p +1) σ). μ((p +1) σ)+(μ + p + 1)[p + σ 1] The result is sharp for the function f given by (3.2). Proof : Let f(z) = 1 + a z p n z n, in the class Σ p(q, s, α 1,σ). Then w(z) = μ It is enough to show that = μ 1 1 = 1 z p u μ f(uz)du ( u μ 1 z n ) u n+μ z n du μ μ + n +1 z n. μ[n + τ 1] (α 1) n(α q) n (μ + n +1)p((p +1) τ)(n 1)! 1. (7.2)

13 Meromorphically multivalent functions 725 Since f Σ p (q, s, α 1,σ), then by Theorem 2.1, we get Note that (7.2) is satisfied if or equivalently [n + σ 1] (α 1) n(α q) n p((p +1) σ)(n 1)! 1. μ[n + τ 1] (α 1) n(α q) n (μ + n +1)p((p +1) τ)(n 1)! (α [n + σ 1] 1 ) n(α q) n (β 1 ) n(β s) n p((p +1) σ)(n 1)! τ (p + 1)(μ + n + 1)[n + σ 1] μ[n 1]((p +1) σ) μ((p +1) σ)+(μ + n + 1)[n + σ 1] = w(n). A simple computation will show that w(n) is increasing function of n. This means that w(n) w(p). Using this, we obtain the result. References [1] M. K. Aouf, A certain subclass of meromorphically starlike functions with positive coefficients, Rendiconti di Mathematica, serie VII, 9, Roma (1989), [2] M. K. Aouf, On a certain class of meromorphic univalent functions with positive coefficients, Rendiconti di Mathematica, Serie VII, 11, Roma (1991), [3] M. K. Aouf and H. M. Hossen, New criteria for meromorphic p-valent Starlike functions, Tsukuba Journal of Mathematics, 17(2) (1993), [4] N. E. Cho, S. H. Lee and S. Owa, A class of meromorphic univalent functions with positive coefficients, Koebe J. Math., 4 (1987), [5] J. Dziok and H. M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput., 13 (1999), [6] J. Dziok and H. M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transform. Spec. Funct., 14 (23),7-18. [7] A. Gangahharan, T. N. Shanamugam and H. M. Srivastava, Generalized hypergeometric functions associated with k-uniformly convex functions, Comput. Math. Appl., 44(2) (22), [8] A. W. Goodman, Univalent functions and non-analytic curves, Proc. Amer. Math. Soc., 8 (1975),

14 726 Ali Hussein Battor et al. [9] S. S. Kumar, V. Ravichandran and G. Murugusundaramoorthy, Classes of meromorphic p-valent parabolic starlike functions with positive coefficients, The Australian Journal of Mathematical Analysis and Applications, 2(2) (25), 1-9. [1] J.-L. Liu, Strongly starlike functions associated with the Dziok-Srivastava Operator, Tamkang J. Math., 35(1) (24), [11] J.-L. Liu and H. M. Srivastava, Classes of meromorphically multivalent functions associated with the generalized hypergeometric function, Math. Comput. Modelling, 39 (24), [12] J. E. Miller, Convex meromorphic mapping and related functions, Proc. Amer. Math. Soc., 25 (197), [13] M. L. Mogra, T. R. Reddy and O. P. Juneja, Meromorphic univalent functions with positive coefficients, Bull. Austral. Math. Soc., 32 (1985), [14] M. Nunokawa and O. P. Ahuja, On meromorphic starlike and convex functions, Indian Journal of Pure and Applied Mathematics, 32(7) (21), [15] S. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc., 81 (1981), Received: January 5, 213