Harmonic multivalent meromorphic functions. defined by an integral operator

Transcription

1 Journal of Applied Mathematics & Bioinformatics, vol.2, no.3, 2012, ISSN: (print), (online) Scienpress Ltd, 2012 Harmonic multivalent meromorphic functions defined by an integral operator Amit Kumar Yadav 1 Abstract The object of this article is to study a class M H (p) of harmonic multivalent meromorphic functions of the form f(z) h(z)+g(z), 0 < z < 1, where h and g are meromorphic functions. An integral operator is considered and is used to define a subclass M H (p,α,m,c) of M H (p). SomepropertiesofM H (p)arestudiedwiththepropertieslikecoefficient condition, bounds, extreme points, convolution condition and convex combination for functions belongs to M H (p,α,m,c) class. Mathematics Subject Classification: 30C45, 30C50, 30C55 Keywords: Meromorphic Function, Starlike functions, Integral Operator 1 Introduction and Prelimnaries A function f u + iv, which is continuos complex-valued harmonic in a domain D C, if both u and v are real harmonic in D. Cluine and Sheil- 1 Department of Information Technology Ibra College of Technology, PO Box : 327, Postal Code: 400, Alsharqiyah, Ibra, Sultanate of Oman, amitkumar@ict.edu.om Article Info: Received : July 2, Revised : August 2, 2012 Published online : December 30, 2012

2 100 Harmonic multivalent meromorphic functions... small 3] investigated the family of all complex valued harmonic mappings f definedontheopenunitdisku,whichadmitstherepresentationf(z) h(z)+ g(z) where h and g are analytic univalent in U. Hengartner and Schober 4] considered the class of functions which are harmonic, meromorphic, orientation preserving and univalent in Ũ {z : z > 1} so that f( ). Such functions admit the representation f(z) h(z)+g(z)+alog z, (1) where h(z) αz + a n z n, g(z) βz + b n z n are analytic in Ũ {z z > 1}, α,β,a C with 0 β α and w(z) f z /f z is analytic with w(z) < 1 for z Ũ. H denotes a class of functions of the form (1) with α 1,β 0. The class H has been studied in various research papers such as 5], 6] and 7]. Theorem ] If f H, then the diameter D f of C\f(U), satisfies D f 2 1+b 1. This estimate is sharp for f(z) z +b 1 /z +Alog z whenever b 1 < 1 and A ( 1 b 1 2) / 1+b 1, b 1 1 and A 0, or b 1 1 and A 2. A function is said to be meromorphic if poles are its only singularities in the complex plane C. Let M p (p N 0 {1,2,...}) be a class of multivalent meromorphic functions of the form: h(z) a p a p n+p 1 z n+p 1, a p 0,a n+p 1 C,z U U\{0}. (2)

3 Amit Kumar Yadav 101 Definition 1.2. A Bernardi type integral operator I m p,c(m 0,c > p) for meromorphic multivalent function h M p is defined as : I 0 p,ch(z) h(z) I 1 p,ch(z) c p+1 z c+1 I m p,ch(z) c p+1 z c+1 z 0 z 0 t c I 0 p,ch(t)dt t c I m 1 p,c h(t)dt, m 1. The Series expansion of I m p,ch(z) for h(z) of the form (2) is given by Ip,ch(z) m a p θ m (n) a p n+p 1 z n+p 1 (c > p,m 0), (3) where ( ) m c p+1 θ m (n). (4) n+p+c ( Note that 0 < θ m (n) < θ m c p+1 m. (1) 1+p+c) For fixed integer p 1, denote by M H (p), a family of harmonic multivalent meromorphic functions of the form f(z) h(z)+g(z), z U, (5) where h M p with a p > 0 of the form (2) and g M p of the form g(z) b n+p 1 z n+p 1,b n+p 1 C, z U. (6) In the expression (5), h is called a meromorphic part and g co-meromorphic part of f. The class M H (p) with its subclass has been studied in 1] and 8] for a p 1. A quite similar to the above mentioned integral operator was used for harmonic analytic functions in 2]. In terms of the operator defined in Defininition 1.2, an operator f M H (p) is defined as follows: Definition 1.3. Let f h + g be of the form (5), the integral operator I m p,cf(z) is defined as I m p,cf(z) I m p,ch(z)+( 1) m I m p,cg(z),z U (7)

4 102 Harmonic multivalent meromorphic functions... the series expansions for Ip,ch(z) m is given by (3) with a p > 0 and for Ip,cg(z) m with g(z) of the form (6) is given as: Ip,cg(z) m θ m (n)b n+p 1 z n+p 1 (c > p,m 0). (8) Involving operator I m p,c define by (7), a class M H (p,α,m,c) of functions f M H (p) is defined as follows: Definition 1.4. A function f M H (p) is said to be in M H (p,α,m,c) if and only if it satisfies { I m } p,c f(z) Re > α, z U,0 α < 1, c > p, m N. (9) Ip,c m+1 f(z) Let M H (p,α,m,c) be a subclass of M H (p,α,m,c), consists of harmonic multivalent meromorphic functions f m h m + g m, where h m and g m are of the form h m (z) a p z a p n+p 1 z n+p 1, a p > 0 and (10) g m (z) ( 1) m+1 b n+p 1 z n+p 1. 2 Some results for the class M H (p) In this section, some results for M H (p) class are derived. Theorem 2.1. Let f M H (p) be of the form (5), then the diameter D f of clf(u ) satisfies D f 2 a p. This estimate is sharp for f(z) a p z p.

5 Amit Kumar Yadav 103 Proof. Let D f (r) denotes the diameter of clf(u r)(r), where U r(r) {z : 0 < z < r, 0 < r < 1} and Since D f (r) D f(r), it follows that D f (r) ] 2 4 2π 1 2π 0 a p r 2p a p 2 Df(r) : max f(z) f( z). z r f(re iθ ) f( re iθ ) 2 dθ r 2p + 4 a p 2 r 2p, ] ( a2n+p b 2n+p 2 2) r 2(2n+p 2), p is odd ] ( a2n+p b 2n+p 1 2) r 2(2n+p 1), p is even noted that as r 1, D f (r) decreases to D f, which concludes the result. Remark ]Taking p 1,a p 1 and w(z) : f(1/z) in Theorem 2.1, same result as Theorem 1.1 has been obtained for w(z), which is of the form (1) with A 0. Theorem 2.3. If f M H (p) be of the form (5), then (n+p 1) ( a n+p 1 2 b n+p 1 2) p a p 2. Equality occurs if and only if C\f(U ) has zero area. Proof. The area of the ommited set is 1 1 lim fdf lim r 12i z r r 1 2i z rhh dz + 1 ] gg 2i dz z r π p a p 2 + (n+p 1) ( a n+p 1 2 b n+p 1 2)] 0.

6 104 Harmonic multivalent meromorphic functions... Remark ]Taking p 1,a p 1 and w(z) : f(1/z) in Theorem 2.3, then for w H,with A 0, it follows that n ( a n 2 b n 2) 1+2Rb 1. 3 Coefficient Conditions In this section, sufficient coefficient condition for a functiom f M H (p) to beinm H (p,α,m,c)classisestablishedandthenitisshownthatthiscoefficient condition is necessary for its subclass M H (p,α,m,c). Theorem 3.1. Let f(z) h(z)+g(z) be the form (5) and θ m (n) is given by (4) if θ m (n) ( 1 αθ 1 (n) ) a n+p 1 + ( 1+αθ 1 (n) ) b n+p 1 ], (11) holds for 0 α < 1 and m N, then f(z) is harmonic in U and f M H (p,α,m,c). Proof. Let the function f(z) h(z)+g(z) be the form (5) satisfying (11). In order to show f M H (p,α,m,c), it suffices to show that or, Re { I m } p,c f(z) Re > α (12) Ip,c m+1 f(z) { I m p,c h(z)+( 1) m I m p,cg(z) I m+1 p,c h(z)+( 1) m+1 I m+1 p,c g(z) where z re iθ, 0 < r 1, 0 θ 2π and 0 α < 1. Let } > α A(z) : I m p,ch(z)+( 1) m I m p,cg(z) (13) and B(z) : I m+1 p,c h(z)+( 1) m+1 I m+1 p,c g(z). (14)

7 Amit Kumar Yadav 105 It is observed that (12) holds if A(z)+(1 α)b(z) A(z) (1+α)B(z) 0. (15) From (13) and (14), it follows that A(z)+(1 α)b(z) and Ip,ch(z)+( 1) m m Ip,cg(z)+(1 α) m ( I m+1 p,c h(z)+( 1) m+1 I m+1 p,c g(z)) (2 α)a p θ m (n)+(1 α)θ m+1 (n) ] a p n+p 1 z n+p 1 (2 α)a p z p +( 1) m θ m (n) (1 α)θ m+1 (n) ] b n+p 1 z n+p 1 θ m (n) 1+(1 α)θ 1 (n) ] a n+p 1 z n+p 1 θ m (n) 1 (1 α)θ 1 (n) ] b n+p 1 z n+p 1 A(z) (1+α)B(z) Ip,ch(z)+( 1) m m Ip,cg(z) (1+α) m ( I m+1 p,c h(z)+( 1) m+1 I m+1 p,c g(z)) ( α)a p θ m (n) (1+α)θ m+1 (n) ] a p n+p 1 z n+p 1 +( 1) m θ m (n)+(1+α)θ m+1 (n) ] b n+p 1 z n+p 1 αa p z θ m (n) (1+α)θ m+1 (n) ] a p n+p 1 z n+p 1 α a p z p + ( 1) m θ m (n)+(1+α)θ m+1 (n) ] b n+p 1 z n+p 1 θ m (n) 1 (1+α)θ 1 (n) ] a n+p 1 z n+p 1 + θ m (n) 1+(1+α)θ 1 (n) ] b n+p 1 z n+p 1.

8 106 Harmonic multivalent meromorphic functions... Thus A(z)+(1 α)b(z) A(z) (1+α)B(z) 2 z p 2 θ m (n) 1 αθ 1 (n) ] a n+p 1 z n+p 1 2 θ m (n) 1+αθ 1 (n) ] b n+p 1 z n+p 1 { 2 z p a p (1 α) 2 0, { θ m (n) 1 αθ 1 (n) ] a n+p 1 z n 1 θ m (n) 1+αθ 1 (n) ] } b n+p 1 z n 1 θ m (n) ( 1 αθ 1 (n) ) a n+p 1 + ( 1+αθ 1 (n) ) b n+p 1 ]} if (11) holds. This proves the Theorem. Theorem 3.2. Let f m h m +g m where h m and g m are of the form (10) then f m M H (p,α,m,c) under the same parametric conditions taken in Theorem 3.1, if and only if the inequality (11) holds. Proof. Since M H (p,α,m,c) M H (p,α,m,c), if part is proved in Theorem 3.1. It only needs to prove the only if part of the Theorem. For this, it suffices to show that f m / M H (p,α,m,c) if the condition (11) does not hold. If f m M H (p,α,m,c), } then writing corresponding series expansions in (9), it follows that Re 0 for all values of z in U where and { ξ(z) η(z) ξ(z) (1 α) a p θ m (n) 1 αθ 1 (n) ] a p n+p 1 z n+p 1 θ m (n) 1+αθ 1 (n) ] b n+p 1 z n+p 1 η(z) a p θ m+1 (n) a p n+p 1 z n+p 1 θ m+1 (n) b n+p 1 z n+p 1.

9 Amit Kumar Yadav 107 Since hence the condition Re { } ξ(z) ξ(z) η(z) Re 0, η(z) } 0 holds if { ξ(z) η(z) θ m (n)(1 αθ 1 (n)) a n+p 1 +(1+αθ 1 (n)) b n+p 1 ]r n+2p 1 a p + θ m+1 (n) a n+p 1 r n+2p θ m+1 (n) b n+p 1 r n+2p 1 (16) Now if the condition (11) does not holds then the numerator of above equation is non-positive for r sufficiently close to 1, which contradicts that f m M H (p,α,m,c) and this proves the required result. 4 Bounds and Extreme Points In this section, bounds for functions belonging to the class M H (p,α,m,c) are obtained and also provide extreme points for the same class. Theorem 4.1. If f m h m +g m M H (p,α,m,c) for 0 α < 1, 0 < z r < 1, and θ m (n) is given by (4), under the same parametric conditions taken in Theorem 3.1 then a p r p rp 1 αθ 1 (1)] I m p,cf k (z) a p r p +rp 1 αθ 1 (1)]. Proof. Let f m h m +g m M H (p,α,m,c). Taking the absolute value of I m p,cf m from (7), it follows that I m p,cf m (z) a p θ m (n) a p n+p 1 z n+p 1 +( 1) 2m+1 θ m (n) b n+p 1 z n+p 1 a p z + θ m (n)( a p n+p 1 b n+p 1 )z n+p 1

10 108 Harmonic multivalent meromorphic functions... I m p,cf m (z) a p r p +rp and I m p,cf m (z) a p r p +rp a p r p + a p r p + θ m (n) αθ m+1 (n)] θ m (n) αθ m+1 (n)] θm (n)( a n+p 1 + b n+p 1 ) θ m (n) αθ m+1 (n)] θ m (n)1 αθ 1 (n)] θm (n)( a n+p 1 + b n+p 1 ) r p θ m (n) αθ m+1 (n) ] ( a 1 αθ 1 n+p 1 + b n+p 1 ) (1)] r p θ m (n) ( 1 αθ 1 (n) ) a 1 αθ 1 n+p 1 ] (1)] a p r p +rp 1 αθ 1 (1)] I m p,cf m (z) a p r p rp + θ m (n) ( 1+αθ 1 (n) ) b n+p 1 ]] a p θ m (n) a p n+p 1 z n+p 1 +( 1) 2m+1 θ m (n) b n+p 1 z n+p 1 a p z + θ m (n)( a p n+p 1 b n+p 1 )z n+p 1 a p r p rp a p r p θ m (n) αθ m+1 (n)] θ m (n) αθ m+1 (n)] θm (n)( a n+p 1 + b n+p 1 ) θ m (n) αθ m+1 (n)] θ m (n)1 αθ 1 (n)] θm (n)( a n+p 1 + b n+p 1 ) r p θ m (n) αθ m+1 (n) ] ( a 1 αθ 1 n+p 1 + b n+p 1 ) (1)] a p r r p θ m (n) ( 1 αθ 1 (n) ) a p 1 αθ 1 n+p 1 ] (1)] + θ m (n) ( 1+αθ 1 (n) ) b n+p 1 ]] a p r p rp 1 αθ 1 (1)] This proves the required result.

11 Amit Kumar Yadav 109 Corollary 4.2. Let f m h m +g m M H (p,α,m,c), for z U and θ m (n) is given by (4), under the same parametric conditions taken in Theorem 3.1 then { w : w < a p (1 α)a } p f(u ). 1 αθ 1 (1)] Theorem 4.3. Let f m h m +g m, where h m and g m are of the form (10) then f m M H (p,α,m,c) and θ m (n) is given by (4), under the same parametric conditions taken in Theorem 3.1, if and only if f m can be expressed as: f m (z) where z U and ( xn+p 1 h mn+p 1(z)+y n+p 1 g ) mn+p 1(z), (17) n0 h mp 1 (z) a p z, h p m n+p 1 (z) a p z + p θ m (n) αθ m+1 (n) zn+p 1, (18) g mp 1 (z) a p z, g p m n+p 1 (z) a p z p +( 1)m+1 θ m (n)+αθ m+1 (n) zn+p 1 (19) for n 1,2,3,..., and Proof. Let f m (z) (x n+p 1 +y n+p 1 ) 1, x n+p 1, y n+p 1 0. (20) ( xn+p 1 h mn+p 1(z)+y n+p 1 g ) mn+p 1(z) n0 ( ) a p x p 1 h mp 1 +y p 1 g mp 1 + x n+p 1 z + p θ m (n) αθ m+1 (n) zn+p 1 ( ) a p + y n+p 1 z p +( 1)m+1 θ m (n)+αθ m+1 (n) zn+p 1 f m (z) (x n+p 1 +y n+p 1 ) a p {( ) p θ m (n) αθ m+1 (n) x n+p 1 n0 } +( 1) m+1 θ m (n)+αθ m+1 (n) y n+p 1 z n+p 1.

12 110 Harmonic multivalent meromorphic functions... Thus by Theorem 3.2, f m M H (p,α,m,c), since, { ( ) θ m (n) αθ m+1 (n) (1 α)a p θ m (n) αθ m+1 (n) x n+p 1 ( )} θm (n)+αθ m+1 (n) θ m (n)+αθ m+1 (n) y n+p 1 (x n+p 1 +y n+p 1 ) (1 x p 1 y p 1 ) 1. Conversly, suppose that f m M H (p,α,m,c), then (11) holds. Setting which satisfy (20), thus x n+p 1 θm (n) αθ m+1 (n) a n+p 1 y n+p 1 θm (n)+αθ m+1 (n) b n+p 1 f m (z) a p a p n+p 1 z n+p 1 +( 1) m+1 b n+p 1 z n+p 1 a p p θ m (n)+αθ m+1 (n) x n+p 1z n+p 1 +( 1) m+1 θ m (n)+αθ m+1 (n) y n+p 1z n+p 1 a p h p mn+p 1 a ] p x z p n+p 1 + g mn+p 1 a ] p y z p n+p 1 a p 1 x z p n+p 1 y n+p 1 ]+ h mn+p 1 x n+p 1 +g mn+p 1 y n+p 1 x p 1 h mp 1 +y p 1 g mp 1 + h mn+p 1 x n+p 1 + g mn+p 1 y n+p 1 ( xn+p 1 h mn+p 1(z)+y n+p 1 g ) mn+p 1(z) This proves the Theorem.

13 Amit Kumar Yadav 111 Remark 4.4. The extreme points for the class M H (p,α,m,c) are given by (18) and (19). 5 Convolution and Integral Convolution In this section, convolution and integral convolution properties of the class M H (p,α,m,c) are established. Let f m,f m M H (p) be defined as follows: f m (z) a p a p n+p 1 z n+p 1 +( 1) m+1 b n+p 1 z n+p 1, z U, F m (z) a p A p n+p 1 z n+p 1 +( 1) m+1 B n+p 1 z n+p 1, z U. The convolution of f m and F m for m N, z U is defined as: (f m F m )(z) f m (z) F m (z) a p a p n+p 1 A n+p 1 z n+p 1 + (21) +( 1) m+1 b n+p 1 B n+p 1 z n+p 1. The integral convolution of f m and F m for m N, z U is defined as: (f m F m )(z) f m (z) F m (z) a p p a n+p 1 A n+p 1 z n+p 1 (22) p n+p 1 +( 1) m+1 p b n+p 1 B n+p 1 z n+p 1. n+p 1 Theorem 5.1. For 0 α < 1, m N. If f m,f m M H (p,α,m,c) and θ m (n) is given by (4) then (f m F m ) M H (p,α,m,c).

14 112 Harmonic multivalent meromorphic functions... Proof. Since F m M H (p,α,m,c), then by Theorem 3.2, A n+p 1 1 and B n+p 1 1, hence, { 1 αθ θ m 1 (n) (n) (1 α)a p { 1+αθ 1 (n) + { } 1 αθ θ m 1 (n) (n) a n+p 1 (1 α)a p { 1+αθ 1 (n) + 1 } A n+p 1 a n+p 1 } ] B n+p 1 b n+p 1 } ] b n+p 1 as f m M H (p,α,m,c). Thus by the Theorem 3.2, (f m F m ) M H (p,α,m,c). Theorem 5.2. For 0 α < 1, m N. If f m,f m M H (p,α,m,c) and θ m (n) is given by (4) then (f m F m ) M H (p,α,m,c). Proof. Since F m M H (p,α,m,c), then by Theorem 3.2, A n+p 1 1, p B n+p 1 1 and 1 hence, n+p 1 { } 1 αθ θ m 1 (n) p an+p 1 A n+p 1 (n) (1 α)a p n+p 1 { } ] 1+αθ 1 (n) p bn+p 1 B n+p 1 + n+p 1 { } 1 αθ θ m 1 (n) (n) a n+p 1 (1 α)a p { } ] 1+αθ 1 (n) + b n+p 1 1 as f m M H (p,α,m,c). Thus by the Theorem 3.2, (f m F m ) M H (p,α,m,c).

15 Amit Kumar Yadav Convex Combination In this section, it is proved that the class M H (p,α,m,c) is closed under convex linear combination of its members. Theorem 6.1. Let the functions f mj (z) defined as: f mj (z) a p a p n+p 1,j z n+p 1 +( 1) m+1 b n+p 1,j z n+p 1, z U. be in the class M H (p,α,m,c) for every j 1,2,3..., then the function ψ(z) c j f mj (z) j1 is also in the class M H (p,α,m,c), where c j 1 for c j 0 (j 1,2,3...). Proof. From the definition ψ(z) a ( p ) ( ) c p j a n+p 1,j z n+p 1 +( 1) m+1 c j b n+p 1,j z n+p 1. j1 j1 j1 (23) Since f mj (z) M H (p,α,m,c) for every j 1,2,3..., then by Theorem 3.2, it follows that {1 αθ } ( θ m 1 ) (n) (n) c j a n+p 1,j j1 { } ( 1+αθ 1 )] (n) + c j b n+p 1,j ( { 1 αθ 1 (n) c j (1 α)a p { 1+αθ 1 (n) + c j.1 1. j1 j1 j1 } a n+p 1,j } ) b n+p 1,j Hence ψ(z) M H (p,α,m,c), which is the desired result.

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