A NEW SUBCLASS OF MEROMORPHIC FUNCTION WITH POSITIVE COEFFICIENTS

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1 Bulletin of Mathematical Analysis and Applications ISSN: , URL: Volume 3 Issue 32010), Pages A NEW SUBCLASS OF MEROMORPHIC FUNCTION WITH POSITIVE COEFFICIENTS S. KAVITHA, S. SIVASUBRAMANIAN, K. MUTHUNAGAI Abstract. In the present investigation, the authors define a new class of meromorphic functions defined in the punctured unit disk Δ := {z C : 0 < z < 1}. Coefficient inequalities, growth and distortion inequalities, as well as closure results are obtained. We also prove a Property using an integral operator and its inverse defined on the new class. 1. Introduction Let Σ denote the class of normalized meromorphic functions f of the form defined on the punctured unit disk fz) = 1 z + a n z n, 1.1) Δ := {z C : 0 < z < 1}. The Hadamard product or convolution of two functions fz) given by 1.1) and is defined by gz) = 1 z + g n z n 1.2) f g)z) := 1 z + a n g n z n. A function f Σ is meromorphic starlike of order α 0 α < 1) if zf ) z) R > α z Δ := Δ {0}). fz) The class of all such functions is denoted by Σ α). Similarly the class of convex functions of order α is defined. Let Σ P be the class of functions f Σ with a n 0. The subclass of Σ P consisting of starlike functions of order α is denoted by Σ P α). Now, we define a new class of functions in Definition Mathematics Subject Classification. 30C50. Key words and phrases. Meromorphic functions, starlike function, convolution, positive coefficients, coefficient inequalities, integral operator. c 2010 Universiteti i Prishtinës, Prishtinë, Kosovë. Submitted September 14, Published September 3,

2 110 KAVITHA ET AL. Definition 1. Let 0 α < 1. Further, let fz) Σ p be given by 1.1), 0 λ < 1 The class M P α, λ) is defined by { zf ) } z) M P α, λ) = f Σ P : R λ 1)fz) + λzf > α. z) Clearly, M P α, 0) reduces to the class Σ P α). The class Σ P α) and various other subclasses of Σ have been studied rather extensively by Clunie [4], Nehari and Netanyahu [8], Pommerenke [9], [10]), Royster [11], and others cf., e.g., Bajpai [2], Mogra et al. [7], Uralegaddi and Ganigi [16], Cho et al. [3], Aouf [3], and Uralegaddi and Somanatha [15]; see also Duren [[5], pages 29 and 137], and Srivastava and Owa [[13], pages 86 and 429]) see also [1]). In this paper, we obtain the coefficient inequalities, growth and distortion inequalities, as well as closure results for the class M P α, λ). Properties of a certain integral operator and its inverse defined on the new class M P α, λ) are also discussed. 2. Coefficients Inequalities Our first theorem gives a necessary and sufficient condition for a function f to be in the class M P α, λ). Theorem 2.1. Let fz) Σ P be given by 1.1). Then f M P α, λ) if and only if a n. 2.1) Proof. If f M p α, λ), then zf ) z) R λ 1)fz) + λzf z) { = R By letting z 1, we have { 1 + na } n 1 + λ 1 + λn)a n 1 + na nz n λ 1 + λn)a nz n+1 > α. This shows that 2.1) holds. Conversely assume that 2.1) holds. It is sufficient to show that zf z) {λ 1)fz) + λzf z)} zf z) + 1 2α) {λ 1)fz) + λzf z)} < 1 z Δ). } > α. Using 2.1), we see that zf z) {λ 1)fz) + λzf z)} zf z) + 1 2α) {λ 1)fz) + λzf z)} = 1 λ)n + 1)a nz n+1 2) + [{ α)λ}n + 1 2α)λ 1)] a nz n+1 1 λ)n + 1)a n 2) [{ α)λ}n + 1 2α)λ 1)] a 1. n Thus we have f M p α, λ). For the choice of λ = 0, we get the following.

3 A NEW SUBCLASS OF MEROMORPHIC FUNCTION Remark 2.2. Let fz) Σ P be given by 1.1). Then f Σ P α) if and only if n + α)a n. Our next result gives the coefficient estimates for functions in M P α, λ). Theorem 2.3. If f M P α, λ), then a n, n = 1, 2, 3,.... The result is sharp for the functions F n z) given by F n z) = 1 z + zn, n = 1, 2, 3,.... Proof. If f M P α, λ), then we have, for each n, a n a n. Therefore we have a n. Since F n z) = 1 z + zn satisfies the conditions of Theorem 2.1, F n z) M P α, λ) and the equality is attained for this function. For λ = 0, we have the following corollary. Remark 2.4. If f Σ P α), then Theorem 2.5. If f M P α, λ), then a n, n = 1, 2, 3,.... n + α 1 r 1 + α 2αλ r fz) 1 r + r z = r). 1 + α 2αλ The result is sharp for fz) = 1 z + z. 2.2) {1 + α 2αλ} Proof. Since fz) = 1 z + a nz n, we have Since, Using this, we have fz) 1 r + a n r n 1 r + r a n. a n 1 + α 2αλ. fz) 1 r α 2αλ r.

4 112 KAVITHA ET AL. Similarly fz) 1 r The result is sharp for fz) = 1 z + 1+α 2αλ z. 1 + α 2αλ r. Similarly we have the following: Theorem 2.6. If f M P α, λ), then 1 r α 2αλ f z) 1 r α 2αλ The result is sharp for the function given by 2.2). z = r). 3. Neighborhoods for the class M p γ) α, λ) In this section, we determine the neighborhood for the class M p γ) α, λ), which we define as follows: Definition 2. A function f Σ p is said to be in the class M p γ) α, λ) if there exists a function g M p α, λ) such that fz) gz) 1 < 1 γ, z Δ, 0 γ < 1). 3.1) Following the earlier works on neighborhoods of analytic functions by Goodman [6] and Ruscheweyh [14], we define the δ-neighborhood of a function f Σ p by { } N δ f) := g Σ p : gz) = 1 z + b n z n and n a n b n δ. 3.2) Theorem 3.1. If g M p α, λ) and then γ = 1 δ1 + α 2αλ), 3.3) 2α 2αλ N δ g) M p γ) α, λ). Proof. Let f N δ g). Then we find from 3.2) that n a n b n δ, 3.4) which implies the coefficient inequality a n b n δ, n N). 3.5) Since g M p α, λ), we have [cf. equation 2.1)] b n 1 + α 2αλ, 3.6)

5 A NEW SUBCLASS OF MEROMORPHIC FUNCTION so that fz) gz) 1 < = a n b n 1 b n δ1 + α 2αλ) 2α 2αλ = 1 γ, provided γ is given by 3.3). Hence, by definition, f M p γ) α, λ) for γ given by 3.3), which completes the proof. Let the functions F k z) be given by 4. Closure Theorems F k z) = 1 z + f n,k z n, k = 1, 2,..., m. 4.1) We shall prove the following closure theorems for the class M P α, λ). Theorem 4.1. Let the function F k z) defined by 4.1) be in the class M P α, λ) for every k = 1, 2,..., m. Then the function fz) defined by fz) = 1 z + a n z n a n 0) belongs to the class M P α, λ), where a n = 1 m m k=1 f n,k n = 1, 2,..) Proof. Since F n z) M P α, λ), it follows from Theorem 2.1 that f n,k 4.2) for every k = 1, 2,.., m. Hence 1 m a n = f n,k m k=1 = 1 m ) f n,k m k=1. By Theorem 2.1, it follows that fz) M P α, λ). Theorem 4.2. The class M P α, λ) is closed under convex linear combination. Proof. Let the function F k z) given by 4.1) be in the class M P α, λ). Then it is enough to show that the function Hz) = λf 1 z) + 1 λ)f 2 z) 0 λ 1) is also in the class M P α, λ). Since for 0 λ 1, Hz) = 1 z + [λf n,1 + 1 λ)f n,2 ]z n,

6 114 KAVITHA ET AL. we observe that [λf n,1 + 1 λ)f n,2 ] = λ f n,1 + 1 λ) f n,2. By Theorem 2.1, we have Hz) M P α, λ). Theorem 4.3. Let F 0 z) = 1 z and F nz) = 1 z + {n+α αλ1+n)} zn for n = 1, 2,.... Then fz) M P α, λ) if and only if fz) can be expressed in the form fz) = n=0 λ nf n z) where λ n 0 and n=0 λ n = 1. Proof. Let Then λ n fz) = = 1 z + λ n F n z) n=0 λ n ) zn. ) = λ n = 1 λ 0 1. By Theorem 2.1, we have fz) M P α, λ). Conversely, let fz) M P α, λ). From Theorem 2.3, we have a n for n = 1, 2,.. we may take and Then λ n = a n for n = 1, 2,... λ 0 = 1 fz) = λ n. λ n F n z). n=0

7 A NEW SUBCLASS OF MEROMORPHIC FUNCTION Partial Sums Silverman [12] determined sharp lower bounds on the real part of the quotients between the normalized starlike or convex functions and their sequences of partial sums. As a natural extension, one is interested to search results analogous to those of Silverman for meromorphic univalent functions. In this section, motivated essentially by the work of silverman [12] and Cho and Owa [3] we will investigate the ratio of a function of the form fz) = 1 z + a n z n, 5.1) to its sequence of partial sums f 1 z) = 1 z and f kz) = 1 z + k a n z n 5.2) when the coefficients are sufficiently small to satisfy the condition analogous to a n. For the sake of brevity we rewrite it as d n a n, 5.3) where d n := n + α αλ1 + n) 5.4) More precisely we will determine sharp lower bounds for R{fz)/f { k z)} and } R{f k z)/fz)}. In this connection we make use of the well known results thatr 1+wz) 1 wz) > 0 z Δ) if and only if ωz) = c n z n satisfies the inequality ωz) z. Unless otherwise stated, we will assume that f is of the form 1.1) and its sequence of partial sums is denoted by f k z) = 1 z + k a nz n. Theorem 5.1. Let fz) M P α, λ) be given by 5.1)satisfies condition, 2.1) { } fz) Re d k+1λ, α) 1 + α z U) 5.5) f k z) d k+1 λ, α) where d n λ, α) {, if n = 1, 2, 3,..., k d k+1 λ, α), if n = k + 1, k + 2,.... The result 5.5) is sharp with the function given by fz) = 1 z + Proof. Define the function wz) by 1 + wz) 1 wz) = d k+1λ, α) 5.6) d k+1 λ, α) zk ) [ fz) f k z) d ] k+1λ, α) 1 + α d k+1 λ, α)

8 116 KAVITHA ET AL. = 1 + k a n z n+1 + dk+1 λ,α) a n z n+1 5.8) a n z n k It suffices to show that wz) 1. Now, from 5.8) we can write Hence we obtain Now wz) 1 if dk+1 λ,α) wz) = k a n z n+1 + wz) 2 2 k a n dk+1 λ,α) dk+1 λ,α) k=n+1 dk+1 λ,α) a n z n+1 ) k=n+1 a n ) a n z n+1. a n or, equivalently, dk+1 λ, α) 2 k a n 2 2 a n k a n + d k+1λ, α) a n 1. From the condition 2.1), it is sufficient to show that k which is equivalent to a n + d k+1λ, α) a n d n λ, α) a n k dn λ, α) 1 + α a n dn λ, α) d k+1 λ, α) + a n 0 5.9) To see that the function given by 5.7) gives the sharp result, we observe that for z = re iπ/k fz) f k z) = 1 + d k+1 λ, α) zn 1 d k+1 λ, α) = d k+1λ, α) 1 + α when r 1. d k+1 λ, α)

9 A NEW SUBCLASS OF MEROMORPHIC FUNCTION which shows the bound 5.5) is the best possible for each k N. The proof of the next theorem is much akin to that of the earlier theorem and hence we state the theorem without proof. Theorem 5.2. Let fz) M P α, λ) be given by 5.1)satisfies condition, 2.1) { } fk z) d k+1 λ, α) Re z U) 5.10) fz) d k+1 λ, α) + where d k+1 λ, α) {, if n = 1, 2, 3,..., k d n λ, α) 5.11) d k+1 λ, α), if n = k + 1, k + 2,.... The result 5.10) is sharp with the function given by fz) = 1 z + d k+1 λ, α) zk ) 6. Radius of meromorphic starlikeness and meromorphic convexity The radii of starlikeness and convexity for the class are given by the following theorems for the class M P α, λ). Theorem 6.1. Let the function f be in the class M P α, λ). Then f is meromorphically starlike of order ρ0 ρ < 1), in z < r 1 α, λ, ρ), where [ ] 1 n+1 1 ρ)) r 1 α, λ, ρ) = inf, 6.1) n 1 n + 2 ρ) Proof. Since, we get fz) = 1 z + a n z n, f z) = 1 z 2 + na n z n 1. It is sufficient to show that zf z) fz) 1 1 ρ 6.2) or equivalently or n + 1)a n z n zf z) fz) + 1 = 1 z 1 ρ a n z n n + 2 ρ a n z n+1 1, 1 ρ for 0 ρ < 1, and z < r 1 α, λ, ρ). By Theorem 2.1, 6.2) will be true if n + 2 ρ z n+1 1 ρ

10 118 KAVITHA ET AL. or, if [ z 1 ρ)) n + 2 ρ) This completes the proof of Theorem 6.1. ] 1 n+1, n ) Theorem 6.2. Let the function f in the class M P α, λ). Then f is meromorphically convex of order ρ, 0 ρ < 1), in z < r 2 α, λ, ρ), where [ ] 1 n+1 1 ρ)) r 2 α, λ, ρ) = inf, n 1, 6.4) n 1 n n + 2 ρ) Proof. Since, we get fz) = 1 z a n z n, f z) = 1 z 2 na n z n 1. It is sufficient to show that 1 zf z) f z) 1 1 ρ or equivalently 6.5) nn + 1)a n z n 1 zf z) f z) + 2 = 1 z 2 1 ρ or na n z n 1 nn + 2 ρ) a n z n+1 1, 1 ρ for 0 ρ < 1, and z < r 2 α, λ, ρ). By Theorem 2.1, 6.5) will be true if nn + 2 ρ) z n+1 ) 1 ρ or, if [ z 1 ρ)) nn + 2 ρ) This completes the proof of Theorem Integral Operators ] 1 n+1, n ) In this section, we consider integral transforms of functions in the class M p α, λ). Theorem 7.1. Let the function fz) given by 1) be in M p α, λ). Then the integral operator F z) = c is in M p δ, λ), where δ = 1 0 u c fuz)du 0 < u 1, 0 < c < ) c + 2) {1 + α 2αλ} c) c) {1 2λ} α) {1 2λ} c + 2).

11 A NEW SUBCLASS OF MEROMORPHIC FUNCTION The result is sharp for the function fz) = 1 z + Proof. Let fz) M p α, λ). Then F z) = c = c It is sufficient to show that = 1 z + Since f M p α, λ), we have Note that 7.1) is satisfied if u c fuz)du u c 1 c {n + δ δλ1 + n)} c + n + 1)1 δ) Rewriting the inequality, we have z + {1+α 2αλ} z. ) f n u n+c z n du c c + n + 1 f nz n. c {n + δ δλ1 + n)} a n ) c + n + 1)1 δ) a n 1. ). ) c {n + δ δλ1 + n)} ) c + n + 1)1 δ). Solving for δ, we have δ c + n + 1) cn) = F n). c) {1 λ1 + n)} + {n + α αλ1 + n)} c + n + 1) A simple computation will show that F n) is increasing and F n) F 1). Using this, the results follows. For the choice of λ = 0, we have the following result of Uralegaddi and Ganigi [15]. Remark 7.2. Let the function fz) defined by 1) be in Σ pα). Then the integral operator F z) = c 1 0 u c fuz)du 0 < u 1, 0 < c < ) is in Σ pδ), where δ = 1+α+cα 1+α+c. The result is sharp for the function Also we have the following: fz) = 1 z α z.

12 120 KAVITHA ET AL. Theorem 7.3. Let fz), given by 1), be in M p α, λ), F z) = 1 c [c + 1)fz) + zf z)] = 1 z + Then F z) is in M p α, λ) for z rα, λ, β) where c + n + 1 f n z n, c > ) c ) 1/n+1) c1 β) rα, λ, β) = inf, n = 1, 2, 3,.... n )c + n + 1) {n + β βλ1 + n)} The result is sharp for the function f n z) = 1 z + {n+α αλ1+n)} zn, n = 1, 2, 3,.... Proof. Let w = zf z) λ 1)fz)+λzf z). Then it is sufficient to show that w 1 w + 1 2β < 1. A computation shows that this is satisfied if {n + β βλ1 + n)} c + n + 1) a n z n ) 1 β)c Since f M p α, λ), by Theorem 2.1, we have a n. The equation 7.3) is satisfied if {n + β βλ1 + n)} c + n + 1) 1 β)c a n z n+1 a n. Solving for z, we get the result. For the choice of λ = 0, we have the following result of Uralegaddi and Ganigi [15]. Remark 7.4. Let the function fz) defined by 1) be in Σ pα) and F z) given by 7.2). Then F z) is in Σ pα) for z rα, β) where ) 1/n+1) c1 β)n + α) rα, β) = inf, n = 1, 2, 3,.... n )c + n + 1)n + β) The result is sharp for the function f n z) = 1 z + n+α zn, n = 1, 2, 3,.... Acknowledgments. The authors thank Prof.G. Murugusundaramoorthy for his suggestions and comments on the original manuscript in particular on the partial sums chapter). The authors also would like to thank the anonymous referee for his/her comments.

13 A NEW SUBCLASS OF MEROMORPHIC FUNCTION References [1] M. K. Aouf, On a certain class of meromorphic univalent functions with positive coefficients, Rend. Mat. Appl. 7 11), 1991), no. 2, [2] S. K. Bajpai, A note on a class of meromorphic univalent functions, Rev. Roumaine Math. Pures Appl ), no. 3, [3] N. E. Cho, S. H. Lee, and S. Owa, A class of meromorphic univalent functions with positive coefficients, Kobe J. Math ), no. 1, [4] J. Clunie, On meromorphic schlicht functions, J. London Math. Soc ), [5] P. L. Duren, Univalent Functions, Grundlehren der mathematischen Wissenschaften, vol. 259, Springer-Verlag, New York, [6] A. W. Goodman, Univalent functions and nonanalytic curve, Proc. Amer. Math. Soc. 8, , 1957). [7] M. L. Mogra, T. R. Reddy, O. P. Juneja, Meromorphic univalent functions with positive coefficients, Bull. Austral. Math. Soc ), [8] Z. Nehari and E. Netanyahu, On the coefficients of meromorphic schlicht functions, Proc. Amer. Math. Soc ), [9] Ch. Pommerenke, ber einige Klassen meromorpher schlichter Funktionen, Math. Z ), German). [10] Ch. Pommerenke, On meromorphic starlike functions, Pacific J. Math ), [11] W. C. Royster, Meromorphic starlike multivalent functions, Trans. Amer. Math. Soc ), [12] H. Silverman, Partial sums of starlike and convex functions, J.Math. Anal. Appl., ), [13] H. M. Srivastava and S. Owa eds.), Current Topics in Analytic Function Theory, World Scientific Publishing, New Jersey, [14] S. Ruscheweyh, Neighborhoods of univalent funtions, Proc. Amer. Math. Soc. 81, , 1981). [15] B A Uralegaddi, M D Ganigi, A certain class of meromorphically starlike functions with positive coefficients, Pure Appl. Math. Sci ), [16] B. A. Uralegaddi and C. Somanatha, Certain differential operators for meromorphic functions, Houston J. Math ), no. 2, S.Kavitha Department of Mathematics, Madras Christian College East Tambaram, Chennai , India address: kavithass19@rediffmail.com S. Sivasubramanian Department of Mathematics, University College of Engineering, Tindivanam Anna University Chennai, Saram , India address: sivasaisastha@rediffmail.com K. Muthunagai Department of Mathematics, Sri Sairam Engineering College Tambaram, Chennai-44, India address: muthunagaik@yahoo.com