Convolution Properties for Certain Meromorphically Multivalent Functions

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1 Filomat 31:1 (2017), DOI /FIL L Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: htt:// Convolution Proerties for Certain Meromorhically Multivalent Functions Jin-Lin Liu a, Reha Srivastava b a Deartment of Mathematics, Yangzhou University, Yangzhou , P. R. China b Deartment of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada Abstract. Let Σ denote the class of functions of the form f (z) z + a n z n ( N). Two new subclasses H, (λ, A, B) and Q, (λ, A, B) of meromorhically multivalent functions starlie resect to -symmetric oints in Σ are investigated. Certain convolution roerties for these subclasses are obtained. 1. Introduction and Preliminaries Throughout this aer we assume that N {1, 2, 3, }, N \ {1}, 1 B < 0, B < A B and 1 λ < +. (1.1) For functions f and analytic in the oen unit dis U {z : z < 1}, the function f is said to be subordinate to, written f (z) (z) (z U), if there exists an analytic function w in U w(0) 0 and w(z) < 1 such that f (z) (w(z)) in U. Let Σ denote the class of functions of the form f (z) z + a n z n ( N), (1.2) which are analytic in the unctured oen unit dis U 0 U \ {0}. Let f j (z) z + a n,j z n Σ (j 1, 2). The Hadamard roduct (or convolution) of f 1 and f 2 is defined by ( f 1 f 2 )(z) z + a n,1 a n,2 z n Mathematics Subject Classification. Primary 30C45, 30C80. Keywords. Meromorhically multivalent function; Hadamard roduct (or convolution); subordination; -symmetric oint. Received: 09 January 2017; Acceted: 13 January 2017 Communicated by Hari M. Srivastava This wor is suorted by National Natural Science Foundation of China(Grant No ) and Natural Science Foundation of Jiangsu Province(Grant No. BK ). addresses: jlliu@yzu.edu.cn (Jin-Lin Liu), rehas@math.uvic.ca (Reha Srivastava)

2 J.-L. Liu, R. Srivastava / Filomat 31:1 (2017), In [3] (see also [4]) Dzio obtained the following result. Lemma. Let f Σ defined by (1.2) satisfy [(1 A)δ n,, + ()(nλ + (λ 1))] a n (A B), (1.3) where Then δ n,, 0 1 ( n+ N ), ( n+ N ). (1.4) (1 λ) f (z) λz f (z) f, (z) 1 + Az 1 + Bz (z U), (1.5) where f, (z) 1 1 j0 ε j f (εj z) and ε ex ( ) 2πi. (1.6) Recently, Liu and Srivastava [9] introduced and investigated two new subclasses H, (λ, A, B) and Q, (λ, A, B) of Σ as follows. A function f Σ is said to be in the class H, (λ, A, B) if and only if it satisfies the coefficient inequality (1.3). Also, a function f Σ is said to be in the class Q, (λ, A, B) if and only if it satisfies the coefficient inequality n[(1 A)δ n,, + ()(nλ + (λ 1))] a n 2 (A B). (1.7) For f Σ, one can see that f Q, (λ, A, B) if and only if 2z + z f (z) H, (λ, A, B). (1.8) In [9] the authors ointed out that each function in the classes H, (λ, A, B) and Q, (λ, A, B) is meromorhically starlie resect to -symmetric oints. The subject of meromorhically univalent and multivalent functions continue to receive a great deal of attention. Very recently, Srivastava et al. (see, e.g., [14, 15, 16, 17, 18, 19, 20, 22]) investigated various subclasses of meromorhic functions. Certain roerties such as distortion bounds, inclusion relations and integral transforms for these subclasses are studied. Motivated essentially by these wors and some other wors (see also, e.g., [1], [2], [5-13], [21] and [23-26]), we aim at investigating here convolution roerties of the subclasses H, (λ, A, B) and Q, (λ, A, B).

3 J.-L. Liu, R. Srivastava / Filomat 31:1 (2017), Convolution Proerties for the Classes H, (λ, A, B) and Q, (λ, A, B) In this section we assume that 1 B j < 0 and B j < A j B j (j 1, 2). (2.1) Furthermore, we denote by λ 1 the root in (1, + ) of the equation: h(λ) aλ 2 + bλ + c 0, where We also denote and a ( 1 )( 2 ), b [( 1 )( 2 ) ( 1 )(A 2 B 2 ) ( 2 )(A 1 )], c (A 1 )(A 2 B 2 ). A(B) B + Ã(B) B + Theorem 1. Let 2λ 2 () (2 + 1)λ 2 (2.2) (2.3) j. (2.4) f j H, (λ, A j, B j ) (j 1, 2) 2 N and 1 B max{b 1, B 2 }. Then we have the following: (i) If (1 A 1 )(1 A 2 ) ( 1 )( 2 ) and λ 1, then f 1 f 2 H, (λ, A(B), B). (ii) If (1 A 1 )(1 A 2 ) > ( 1 )( 2 ) and λ λ 1, then f 1 f 2 H, (λ, A(B), B). (iii) If (1 A 1 )(1 A 2 ) > ( 1 )( 2 ) and 1 λ < λ 1, then f 1 f 2 H, (λ, Ã(B), B). In all cases (i)-(iii) the numbers A(B) and Ã(B) are otimal in the sense that they cannot be decreased for each B. Proof. Suose that 1 B max{b 1, B 2 } B j (j 1 or 2). It follows from (2.1) and (2.3) that A(B) B 2λ j 2λ j j j (2λ 1) j λ 1 λ j 2B > 0, 1 A j + 1

4 J.-L. Liu, R. Srivastava / Filomat 31:1 (2017), which imlies that B < A(B) B. Also, (2.1) and (2.4) give that Ã(B) B (2 + 1)λ j j 2B > 0, which imlies that B < Ã(B) B. Let 2 N and f j (z) z + a n,j z n H, (λ, A j, B j ) (z U 0 ; j 1, 2). Then (1 A j )δ n,, + ( j )(nλ + (λ 1)) ( ) a n,1a n,2 (1 A j )δ n,, + ( j )(nλ + (λ 1)) a n,j 1. (2.5) ( ) Also, f 1 f 2 H, (λ, A, B) if and only if (1 A)δ n,, + ()(nλ + (λ 1)) a n,1 a n,2 1. (2.6) (A B) In order to rove Theorem 1, it follows from (2.5) and (2.6) that we only need to find the smallest A such that (1 A)δ n,, + ()(nλ + (λ 1)) (A B) (1 A j )δ n,, + ( j )(nλ + (λ 1)) ( ) (n ). (2.7) For n and n+ N, (2.7) is equivalent to λ(n+) λ(n+) ϕ 1 (n). (2.8) It can be verified that ϕ 1 (n) (n, λ 1) is decreasing in n and so, in view of 2 N, ϕ 1 (n) ϕ 1 () B + For n and n+ 2λ 2 N, (2.7) becomes 2. (2.9) nλ+(λ1) 2 ψ 1B 1 (n) (2.10) j and we have ψ 1 (n) ψ 1 ( + 1) B + (2+1)λ 2. (2.11)

5 J.-L. Liu, R. Srivastava / Filomat 31:1 (2017), Now 2λ j h(λ) λ(a 1 )(A 2 B 2 ), j + 1 (2 + 1)λ λ j (2.12) where h(λ) λ( λ)( 1 )( 2 ) λ[( 1 )(A 2 B 2 ) + ( 2 )(A 1 )] + (A 1 )(A 2 B 2 ) aλ 2 + bλ + c (2.13) a ( 1 )( 2 ), b [( 1 )( 2 ) ( 1 )(A 2 B 2 ) ( 2 )(A 1 )], c (A 1 )(A 2 B 2 ). Note that a < 0, h(0) c > 0 and h(1) ( 1)( 1 )( 2 ) [( 1 )(A 2 B 2 ) + ( 2 )(A 1 )] + (A 1 )(A 2 B 2 ) (1 A 1 )(1 A 2 ) ( 1 )( 2 ). (2.14) Therefore, if (i) or (ii) is satisfied, then it follows from (2.7) to (2.14) that h(λ) 0, ψ 1 ( + 1) ϕ 1 () A(B), and f 1 f 2 H, (λ, A(B), B). Furthermore, for B < A 0 < A(B), we have 1 A + ()(2λ 1) A 0 B > 1 A(B) + ()(2λ 1) A(B) B Hence for functions f j (z) z + we have f 1 f 2 H, (λ, A 0, B). 1 A j + ( j )(2λ 1) 1 A j + ( j )(2λ 1) 1. 1 A j + ( j )(2λ 1) z H, (λ, A j, B j ) (j 1, 2) (iii) If (1 A 1 )(1 A 2 ) > ( 1 )( 2 ) and 1 λ < λ 1, then we have h(λ) > 0, ϕ 1 () < ψ 1 ( + 1) Ã(B), and f 1 f 2 H, (λ, Ã(B), B). Furthermore, the number Ã(B) cannot be decreased as can be seen from the functions f j defined by f j (z) z + ( ) ((2 + 1)λ )( j ) z+1 H, (λ, A j, B j ) (j 1, 2). Theorem 2. Let f 1 H, (λ, A 1, B 1 ) and f 2 Q, (λ, A 2, B 2 )

6 J.-L. Liu, R. Srivastava / Filomat 31:1 (2017), N and 1 B max{b 1, B 2 }. Also let A(B), Ã(B) and λ 1 be given as in Theorem 1. Then we have the following: (i) If (1 A 1 )(1 A 2 ) ( 1 )( 2 ) and λ 1, then f 1 f 2 Q, (λ, A(B), B). (ii) If (1 A 1 )(1 A 2 ) > ( 1 )( 2 ) and λ λ 1, then f 1 f 2 Q, (λ, A(B), B). (iii) If (1 A 1 )(1 A 2 ) > ( 1 )( 2 ) and 1 λ < λ 1, then f 1 f 2 Q, (λ, Ã(B), B). In all cases (i)-(iii) the numbers A(B) and Ã(B) are otimal in the sense that they cannot be decreased for each B. Proof. Since f 1 H, (λ, A 1, B 1 ), 2z + z f 2 (z) H, (λ, A 2, B 2 ) (see (1.8)), and ( f 1 (z) 2z + z f 2 (z) ) 2z + z( f 1 f 2 ) (z) the assertion of the theorem follows from Theorem 1. Next, we denote by λ 2 the root in (1, + ) of the equation: h 1 (λ) a 1 λ 2 + b 1 λ + c 1 0, (z U 0 ), where a 1 (3 + 1)( 1 )( 2 ), b 1 ( + 1)( 1 )( 2 ) 2 [( 1 )(A 2 B 2 ) + ( 2 )(A 1 )], c 1 2 (A 1 )(A 2 B 2 ). (2.15) We also denote à 1 (B) B + Theorem 3. Let 2 () ( + 1)((2 + 1)λ ) j. (2.16) f 1 H, (λ, A 1, B 1 ) and f 2 Q, (λ, A 2, B 2 ) 2 N and 1 B max{b 1, B 2 }. Then we have the following: (i) If 2 (1 A 1 )(1 A 2 ) (2 + 1)( 1 )( 2 ) and λ 1, then f 1 f 2 H, (λ, A(B), B). (ii) If 2 (1 A 1 )(1 A 2 ) > (2 + 1)( 1 )( 2 ) and λ λ 2, then f 1 f 2 H, (λ, A(B), B).

7 J.-L. Liu, R. Srivastava / Filomat 31:1 (2017), (iii) If 2 (1 A 1 )(1 A 2 ) > (2 + 1)( 1 )( 2 ) and 1 λ < λ 2, then f 1 f 2 H, (λ, Ã1(B), B). In all cases (i)-(iii) the numbers A(B) and Ã1(B) are otimal in the sense that they cannot be decreased for each B. Proof. It can be verified that à 1 (B) B ( + 1)((2 + 1)λ ) 2 j > and so B < Ã1(B) < B. In order to rove Theorem 3, we only need to find the smallest A such that j 2B > 0 (1 A)δ n,, + ()(nλ + (λ 1)) n (A B) (1 A j )δ n,, + ( j )(nλ + (λ 1)) ( ) (2.17) for all n. For n and n+ N, (2.17) is equivalent to λn(n+) 2 2 n 2 ϕ 2 (n). (2.18) Define Then (λ, x) x λx(x + ) (λ, x) λ(2x + ) 3λ 2 3λ 1 3λ 1 2 j 1 j 1 j j x j j j 1 A j > 0 (x ; λ 1), 1 A j (x ; λ 1). which imlies that ϕ 2 (n) defined by (2.18) is decreasing in n (n ). Hence, in view of 2 ϕ 2 (n) ϕ 2 () B + For n and n+ 2λ 2 2 N, (2.17) reduces to A(B). n(nλ+(λ1)) 2 2 ψ 2 (n) N, we have

8 and, in view of 2 N, we have J.-L. Liu, R. Srivastava / Filomat 31:1 (2017), ψ 2 (n) ψ 2 ( + 1) B + (+1)((2+1)λ). 2 2 Now 2λ j h 1 (λ) λ 2 (A 1 )(A 2 B 2 ), j + 1 ( + 1)((2 + 1)λ ) λ 2 j (2.19) where h 1 (λ) a 1 λ 2 + b 1 λ + c 1 and a 1, b 1, c 1 are given by (2.15). Note that a 1 < 0, h 1 (0) c 1 > 0 and h 1 (1) 2 2 ( 1 )( 2 ) 2 [( 1 )(A 2 B 2 ) + ( 2 )(A 1 )] + 2 (A 1 )(A 2 B 2 ) ( + 1) 2 ( 1 )( 2 ) 2 (1 A 1 )(1 A 2 ) (2 + 1)( 1 )( 2 ). The remaining art of the roof of Theorem 3 is similar to that as in Theorem 1 and hence we omit it. The roof of the Theorem is comleted. From Theorem 3 we have the following theorem at once. Theorem 4. Let f j Q, (λ, A j, B j ) (j 1, 2) 2 N and 1 B max{b 1, B 2 }. Also let A(B), Ã1(B) and λ 2 be given as in Theorem 3. Then we have the following: (i) If 2 (1 A 1 )(1 A 2 ) (2 + 1)( 1 )( 2 ) and λ 1, then f 1 f 2 Q, (λ, A(B), B). (ii) If 2 (1 A 1 )(1 A 2 ) > (2 + 1)( 1 )( 2 ) and λ λ 2, then f 1 f 2 Q, (λ, A(B), B). (iii) If 2 (1 A 1 )(1 A 2 ) > (2 + 1)( 1 )( 2 ) and 1 λ < λ 2, then f 1 f 2 Q, (λ, Ã1(B), B). In all cases (i)-(iii) the numbers A(B) and Ã1(B) are otimal in the sense that they cannot be decreased for each B. Finally, we denote by λ 3 the root in (1, + ) of the equation: h 2 (λ) a 2 λ 2 + b 2 λ + c 2 0, where a 2 [( + 1) 2 (2 + 1) 2 3 ]( 1 )( 2 ), b 2 ( + 1) 2 ( 1 )( 2 ) 3 [( 1 )(A 2 B 2 ) + ( 2 )(A 1 )], c 2 3 (A 1 )(A 2 B 2 ). (2.20)

9 J.-L. Liu, R. Srivastava / Filomat 31:1 (2017), We also denote 3 () Ã 2 (B) B + ( + 1) 2 ((2 + 1)λ ) j. (2.21) Theorem 5. Let f j Q, (λ, A j, B j ) (j 1, 2) 2 N and 1 B max{b 1, B 2 }. Then we have the following: (i) If 3 (1 A 1 )(1 A 2 ) ( )( 1 )( 2 ) and λ 1, then f 1 f 2 H, (λ, A(B), B). (ii) If 3 (1 A 1 )(1 A 2 ) > ( )( 1 )( 2 ) and λ λ 3, then f 1 f 2 H, (λ, A(B), B). (iii) If 3 (1 A 1 )(1 A 2 ) > ( )( 1 )( 2 ) and 1 λ < λ 3, then f 1 f 2 H, (λ, Ã2(B), B). In all cases (i)-(iii) the numbers A(B) and Ã2(B) are otimal in the sense that they cannot be decreased for each B. Proof. It can be seen that B < Ã2(B) < B. In order to rove Theorem 5, we only need to find the smallest A such that ( ) (1 A)δ n,, + ()(nλ + (λ 1)) 2 n (1 A j )δ n,, + ( j )(nλ + (λ 1)) (2.22) (A B) ( ) for n. For n and n+ N, (2.22) can be written as λn 2 (n+) 3 2 n n2 + 2 λ(n+) A simle calculation shows that ϕ 3 (n) (n, λ 1) is decreasing in n. Therefore ϕ 3 (n) ϕ 3 () B + A(B). 2λ 2 2 ϕ 3 (n). (2.23) For n and n+ N, (2.22) becomes n 2 (nλ+(λ1)) 2 3 ψ 3 (n) and we have ψ 3 (n) ψ 3 ( + 1) B + (+1) 2 ((2+1)λ). 2 3

10 J.-L. Liu, R. Srivastava / Filomat 31:1 (2017), Now 2λ j h 2 (λ) λ 3 (A 1 )(A 2 B 2 ), j ( + 1)2 ((2 + 1)λ ) 3 j where h 2 (λ) a 2 λ 2 + b 2 λ + c 2 and a 2, b 2, c 2 are given by (2.20). Note that a 2 < 0, h 2 (0) c 2 > 0 and h 2 (1) 3 (1 A 1 )(1 A 2 ) ( )( 1 )( 2 ). The remaining art of the roof is similar to that of Theorem 1 and thus we omit it. ACKNOWLEDGEMENT The authors would lie to exress sincere thans to the referee for careful reading and suggestions which heled us to imrove the aer. References [1] M. K. Aouf, J. Dzio and J. Sool, On a subclass of strongly starlie functions, Al. Math. Lett. 24 (2011) [2] N. E. Cho, O. S. Kwon and S. Owa, Certain subclasses of Saaguchi functions, Southeast Asian Bull. Math. 17 (1993) [3] J. Dzio, Classes of meromorhic functions associated conic regions, Acta Math. Sci. Ser. B Engl. Ed. 32 (2012) [4] J. Dzio, Classes of multivalent analytic and meromorhic functions two fixed oints, Fixed Point Theory Al. (2013) 2013:86. [5] J. Dzio and J. Sool, Some inclusion roerties of certain class of analytic functions, Taiwanese J. Math. 13 (2009) [6] J. Dzio, R. K. Raina and J. Sool, On alha-convex functions related to shell-lie functions connected Fibonacci numbers, Al. Math. Comut. 218 (2011) [7] S. A. Halim, Functions starlie resect to other oints, Int. J. Math. Math. Sci. 14 (1991) [8] O. S. Kwon, Y. J. Sim, N. E. Cho and H. M. Srivastava, Some radius roblems related to a certain subclass of analytic functions, Acta Math. Sinica 30 (2014) [9] J.-L. Liu and R. Srivastava, Inclusion and distortion roerties for certain meromorhically multivalent functions, Quaestiones Math., to aear. [10] A. K. Mishra, T. Panigrahi and R. K. Mishra, Subordination and inclusion theorems for subclasses of meromorhic functions alications to electromagnetic cloaing, Math. Comut. Modelling 57 (2013) [11] R. Pavatham and S. Radha, On α-starlie and α-close-to-convex functions resect to n-symmetric oints, Indina J. Pure Al. Math. 16 (1986) [12] Z.-G Peng and Q.-Q Han, On the coefficients of several classes of bi-univalent functions, Acta Math. Scientia 34 (2014) [13] J. Sool, A certain class of starlie functions, Comut. Math. Al. 62 (2011) [14] H. M. Srivastava, K. R. Alhindi and M. Darus, An investigation into the olylogarithm function and its associated class of meromorhic functions, Maejo Internat. J. Sci. Tech. 10 (2016) [15] H. M. Srivastava, R. M. El-Ashwah and N. Breaz, A certain subclass of multivalent functions involving higher-order derivatives, Filomat 30 (2016) [16] H. M. Srivastava, S. Gaboury and F. Ghanim, Partial sums of certain classes of meromorhic functions related to the Hurwitz-Lerch zeta function, Moroccan J. Pure Al. Anal. 1 (2015) [17] H. M. Srivastava, S. Gaboury and F. Ghanim, Some further roerties of a linear oerator associated the λ-generalized Hurwitz-Lerch zeta function related to the class of meromorhically univalent functions, Al. Math. Comut. 259 (2015) [18] H. M. Srivastava, S. Gaboury and F. Ghanim, Certain subclasses of meromorhically univalent functions defined by a linear oerator associated the λ-generalized Hurwitz-Lerch function, Integral Transforms Sec. Funct. 26 (2015) [19] H. M. Srivastava, S. B. Joshi, S. S. Joshi and H. Pawar, Coefficient estimates for certain subclasses of meromorhically bi-univalent functions, Palest J. Math. 5(Secial Issue: 1) (2016) [20] H. M. Srivastava, D.-G. Yang and N-E. Xu, Some subclasses of meromorhically multivalent functions associated a linear oerator, Al. Math. Comut. 195 (2008) [21] J. Stantiewics, Some remars on functions starlie resect to symmetric oints, Ann. Univ. Mariae Curie-Slodowsa 19 (1965) [22] H. Tang, H. M. Srivastava, S.-H. Li and L.-N. Ma, Third-order differential subordination and suerordination results for meromorhically multivalent functions associated the Liu-Srivastava oerator, Abstr. Al. Anal (2014) Article ID

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